不動点理論と最適化
明治⼤学理⼯学部情報科学科
飯塚秀明
連続最適化および関連分野に関する
夏季学校
2021年8⽉23⽇, 24⽇© Meiji University,All rights reserved.

謝辞

連続最適化および関連分野に関する
夏季学校
m統計数理研究所 ⽥中未来 先⽣m慶應義塾⼤学 成島康史 先⽣
m関⻄⼤学 檀寛成 先⽣
m京都⼤学 佐藤寛之 先⽣
この場を借りて感謝申し上げます。
© Meiji University,All rights reserved.

本講義の⽬的q不動点とは何か︖
q不動点を⾒つけるとなぜ嬉しいのか︖という素朴な問に対する回答例の提⽰
n 講義 1: ヒルベルト空間、便利な不等式
n 講義 2: 不動点, ⾮拡⼤写像, 最適化
n 講義 3: 不動点近似法© Meiji University,All rights reserved.

参考⽂献

q⾼橋渉著 ⾮線形・凸解析学⼊⾨

高橋渉先生 東京工業大学名誉教授
1944.01.22〜2020.11.19
ご専門：非線形解析
論文数：５００以上 引用数：１００００以上
博士学生：４３名© Meiji University,All rights reserved.

不動点理論を学んで良かったこと
q ⾼橋先⽣の教え: 数学の問題を⾼い⽴場で捉える
q 最適化や⼯学 (機械学習) の研究者と異なる視点

均衡問題
不動点問題
最適化
© Meiji University,All rights reserved.

深層学習に
現れる
非凸最適化
や均衡問題

講義１

q有限次元ユークリッド空間とヒルベルト空間qよく使う不等式・等式© Meiji University,All rights reserved.

３次元ユークリッド空間

H : 集合3<latexit sha1_base64="eQeP6+Orkd7qFOUndHRPlHB0eqc=">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</latexit><latexit sha1_base64="hK6NMpA9gfe7ze8ZX43cNjlKw0w=">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</latexit>

スカラー倍：↵x 2 R<latexit sha1_base64="ZiBI8HfOf1eznCSCmS2XntwQlR4=">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</latexit>3を定義すると、性質①から⑧を満たす。

R

3 : ベクトル空間<latexit sha1_base64="KEXLzJxzgakGJypUgWVs+iKqowc=">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</latexit>

2<latexit sha1_base64="A4EZRzZgRrxrc56SGLvxNyPp4sA=">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</latexit>x : ベクトル<latexit sha1_base64="L9Gy2YbShKizzyo8amkjBe1omKM=">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</latexit>(I) x, y 2 H, ↵ 2 R,
和：x+y 2H<latexit sha1_base64="1PRgE3bMnHhfmdNvSl/3m6MW9Dg=">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</latexit><latexit sha1_base64="vgJQ8Ie6tYloMzZXlGbQ1ULJHQg=">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</latexit>

スカラー倍：↵x 2 H<latexit sha1_base64="k09rovrkmyHjIEw4WwUAOZWK7vA=">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</latexit>を定義して、性質①から⑧を満たすとする。

H : 線形空間<latexit sha1_base64="eQeP6+Orkd7qFOUndHRPlHB0eqc=">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</latexit>

23(I) x, y 2 R , ↵ 2 R,
和：x + y 2 R3<latexit sha1_base64="3w2WFe8m978PhebcyMeYQlzAcMs=">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</latexit><latexit sha1_base64="KEXLzJxzgakGJypUgWVs+iKqowc=">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</latexit>

R<latexit sha1_base64="A4EZRzZgRrxrc56SGLvxNyPp4sA=">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</latexit>

ヒルベルト空間x : ベクトル<latexit sha1_base64="L9Gy2YbShKizzyo8amkjBe1omKM=">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</latexit><latexit sha1_base64="zl2VnunAW3rsyyX0VZkkyUUibpU=">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</latexit>

x = (a1 , a2 , a3 ), y = (b1 , b2 , b3 )x + y := (a1 + b1 , a2 + b2 , a3 + b3 ),↵x := (↵a1 , ↵a2 , ↵a3 )3x, y, z 2 R , ↵,<latexit sha1_base64="PdmJ6CnKlMxE5ypu/kSYjDInDcw=">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</latexit><latexit sha1_base64="vFdFPTYGupV8e0LjzENf2qwWIgw=">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</latexit>

2R<latexit sha1_base64="k+iNdlPH7qQFEY7gWYgH2nOTEXY=">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</latexit>

x = (a1 , a2 , a3 )

① (x + y) + z = x + (y + z)
② x+y =y+x
③ x + 0 = x 0 = (0, 0, 0)
x = ( a1 ,④ x + ( x) = 0⑤ (↵ )x = ↵( x)⑥ 1x = x⑦ ↵(x + y) = ↵x + ↵y
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a2 , a3 )３次元ユークリッド空間

R<latexit sha1_base64="A4EZRzZgRrxrc56SGLvxNyPp4sA=">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</latexit><latexit sha1_base64="MUFKaMlcnO9DvwUE4TDryxocwHc=">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</latexit>

ヒルベルト空間

3

H : 集合<latexit sha1_base64="eQeP6+Orkd7qFOUndHRPlHB0eqc=">AAAChXichVG7SgNBFD2uj8T4SNRGsAkGxcYwCUHFxqCNpSYmEVRkd53EIZvdZXcS0OAPiK1aWClYiB/gB9j4Axb5BLGMYGPh3c2CqBjvMDNnztxzZw5Xsw3hSsZaPUpvX/9AKDwYGRoeGY3GxsaLrlV3dF7QLcNytjXV5YYweUEKafBt2+FqTTN4SauuefelBndcYZlb8sjmezW1Yoqy0FVJVG49sh9LsCTzI/4bpAKQQBAbVuwBuziABR111MBhQhI2oMKlsYMUGGzi9tAkziEk/HuOE0RIW6csThkqsVVaK3TaCViTzl5N11fr9IpB0yFlHDPsmd2xNnti9+yFffxZq+nX8P5yRLvW0XJ7P3o6mX//V1WjXeLwS9VFoVF2d08SZSz5XgR5s33Gc6l36jeOL9v55dxMc5bdsFfyd81a7JEcmo03/XaT567gNSj1sx2/QTGdTC0k05uZRHY1aFUYU5jGHPVjEVmsYwMFereMM5zjQgkp80pGWeikKj2BZgLfQln5BJsMj9o=</latexit><latexit sha1_base64="MUFKaMlcnO9DvwUE4TDryxocwHc=">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</latexit>(I) 和・スカラー倍を定義すると、(I) 和・スカラー倍を定義して、性質①から⑧を満たすとする。

性質①から⑧を満たす。

H : 線形空間<latexit sha1_base64="eQeP6+Orkd7qFOUndHRPlHB0eqc=">AAAChXichVG7SgNBFD2uj8T4SNRGsAkGxcYwCUHFxqCNpSYmEVRkd53EIZvdZXcS0OAPiK1aWClYiB/gB9j4Axb5BLGMYGPh3c2CqBjvMDNnztxzZw5Xsw3hSsZaPUpvX/9AKDwYGRoeGY3GxsaLrlV3dF7QLcNytjXV5YYweUEKafBt2+FqTTN4SauuefelBndcYZlb8sjmezW1Yoqy0FVJVG49sh9LsCTzI/4bpAKQQBAbVuwBuziABR111MBhQhI2oMKlsYMUGGzi9tAkziEk/HuOE0RIW6csThkqsVVaK3TaCViTzl5N11fr9IpB0yFlHDPsmd2xNnti9+yFffxZq+nX8P5yRLvW0XJ7P3o6mX//V1WjXeLwS9VFoVF2d08SZSz5XgR5s33Gc6l36jeOL9v55dxMc5bdsFfyd81a7JEcmo03/XaT567gNSj1sx2/QTGdTC0k05uZRHY1aFUYU5jGHPVjEVmsYwMFereMM5zjQgkp80pGWeikKj2BZgLfQln5BJsMj9o=</latexit><latexit sha1_base64="JJ0z7lDMaBlmerSZAzeeyPifyoo=">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</latexit><latexit sha1_base64="JJ0z7lDMaBlmerSZAzeeyPifyoo=">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</latexit>(II) 内積(II) 内積hx, yi := a1 b1 + a2 b2 + a3 b3 hx, yi 2 R<latexit sha1_base64="1vPLj3q0th1mBHqZzAjcVoFYHJ0=">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</latexit><latexit sha1_base64="oV6TIhQiQWv0Xn4JUDX8blw5AWw=">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</latexit>を定義すると、性質①から④を満たす。

を定義して、性質①から④を満たすとする。
: 内積空間H<latexit sha1_base64="eQeP6+Orkd7qFOUndHRPlHB0eqc=">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</latexit>

0; hx, xi = 0 () x = 0① hx, xi
② hx + y, zi = hx, zi + hy, zi③ h↵x, yi = ↵hx, yi④ hx, yi = hy, xi<latexit sha1_base64="tLqZ9qhWVZTBhgEaoJgT/A2LkH8=">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</latexit><latexit sha1_base64="Xqu2/m5hYNLZ8AVY13A70ZlLKzo=">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</latexit><latexit sha1_base64="HogNhvjJDcshYIUbqEJHiJnLsco=">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</latexit><latexit sha1_base64="w61b1oSwqj3D1gUsAHcb2cSNy0o=">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</latexit><latexit sha1_base64="OiMzl5pncR8ugMt8nYy5CMwo5Ls=">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</latexit>(III) ノルムkxk :=<latexit sha1_base64="QGhALYIxX19v1BGuaCw87GktLeU=">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</latexit>phx, xiを定義すると、性質①から③を満たす。
0; kxk = 0 () x = 0① kxk② k↵xk = |↵|kxk③ kx + yk  kxk + kyk<latexit sha1_base64="wpJ2SEzFXdT+p7LwUpiJBqDQd4w=">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</latexit><latexit sha1_base64="TZXK6pIYd6OYvki7wcKrXWyR5Rg=">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</latexit><latexit sha1_base64="bJQRPRHdW1nQ0VEQLLcLryaJWk4=">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</latexit>

Cauchy-Schwarz の不等式
|hx, yi|  kxkkyk<latexit sha1_base64="ZZk/WKRTaNpqoPv4J00HSlx2FTI=">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</latexit><latexit sha1_base64="OiMzl5pncR8ugMt8nYy5CMwo5Ls=">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</latexit>(III)ノルム kxk :=<latexit sha1_base64="QGhALYIxX19v1BGuaCw87GktLeU=">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</latexit>phx, xiを定義すると性質①から③を満たす。

H : ノルム空間<latexit sha1_base64="eQeP6+Orkd7qFOUndHRPlHB0eqc=">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</latexit>

Cauchy-Schwarz の不等式が成り⽴つことの証明
内積空間 H 上の任意の x, y に対して

|hx, yi|  kxkkyk<latexit sha1_base64="ZZk/WKRTaNpqoPv4J00HSlx2FTI=">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</latexit>

kxk2 := hx, xi<latexit sha1_base64="zn6jb3t13sGzlzrrpNjELhXvUC8=">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</latexit>

よく使う等式: ２乗ノルム展開<latexit sha1_base64="kXxJbDNg8gmICWZoXjlZ8wxI0XY=">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</latexit>22

kx + yk = kxk + 2hx, yi + kyk* kx + yk2 = hx + y, x + yi

2<latexit sha1_base64="f2qg9yrUyDJpP/d6AH+B78c35ME=">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</latexit><latexit sha1_base64="d/ZJ4okZtU8NtT+binmfs9X9cOk=">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</latexit><latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit>②= hx, x + yi + hy, x + yi
<latexit sha1_base64="yizya08n7gzf7F5s+lThqvP5Wyo=">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</latexit>②
hx + y, zi = hx, zi + hy, zi<latexit sha1_base64="Xqu2/m5hYNLZ8AVY13A70ZlLKzo=">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</latexit>

④

④, ②

hx, yi = hy, xi<latexit sha1_base64="w61b1oSwqj3D1gUsAHcb2cSNy0o=">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</latexit><latexit sha1_base64="LbwQ2+wqa9tkoxjHAD4PnSAK2Hg=">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</latexit>© Meiji University,All rights reserved.

= hx, xi + hy, xi + hx, yi + hy, yi<latexit sha1_base64="FXbCloq59K4L3+1qKyJu0DoifJM=">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</latexit>

④

2

2

= kxk + 2hx, yi + kyk

Cauchy-Schwarz の不等式が成り⽴つことの証明
内積空間 H 上の任意の x, y に対して

kxk2 := hx, xi|hx, yi|  kxkkyk<latexit sha1_base64="ZZk/WKRTaNpqoPv4J00HSlx2FTI=">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</latexit><latexit sha1_base64="zn6jb3t13sGzlzrrpNjELhXvUC8=">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</latexit>

* t2R
2
2
2
0  ktx + yk = ktxk + 2htx, yi + kyk<latexit sha1_base64="Gt2qSNJgT31E3LVjWJ5CZXj7jl4=">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</latexit><latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit><latexit sha1_base64="M8xyhMqUTyDDEymDe+Qt6yWlSkE=">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</latexit><latexit sha1_base64="OhvT5isDJ1vyHVc0EOESX/+s/Ws=">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</latexit>

③, ④

①

<latexit sha1_base64="RKSFrTgdXPI4R5rsM2zv6WfWxts=">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</latexit>

2

2= t kxk + 2thx, yi + kyk③:h↵x, yi = ↵hx, yi<latexit sha1_base64="HogNhvjJDcshYIUbqEJHiJnLsco=">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</latexit><latexit sha1_base64="aD1FET0JH4rqepTr4SFGFLnEHFU=">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</latexit>

22

ktxk = htx, txi = thx, txi④: hx, yi = hy, xi ③<latexit sha1_base64="w61b1oSwqj3D1gUsAHcb2cSNy0o=">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</latexit>= thtx, xi = t2 hx, xi = t2 kxk2

© Meiji University,All rights reserved.

Cauchy-Schwarz の不等式が成り⽴つことの証明
内積空間 H 上の任意の x, y に対して

kxk2 := hx, xi|hx, yi|  kxkkyk<latexit sha1_base64="ZZk/WKRTaNpqoPv4J00HSlx2FTI=">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</latexit><latexit sha1_base64="zn6jb3t13sGzlzrrpNjELhXvUC8=">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</latexit>* t2R<latexit sha1_base64="Gt2qSNJgT31E3LVjWJ5CZXj7jl4=">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</latexit><latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit>

0  t2 kxk2 + 2thx, yi + kyk2
<latexit sha1_base64="nxwVnqcOyjeF2xeiymLCC7UtGlk=">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</latexit><latexit sha1_base64="0TdiFQX3POW9cKdTUPNGWs/tZ2Q=">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</latexit>

2

2

2

D = hx, yi
kxk kyk  0222) hx, yi  kxk kyk<latexit sha1_base64="hLiAYxAKsioShtNmEv3t6yIOyxc=">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</latexit>) |hx, yi|  kxkkyk<latexit sha1_base64="fwAO4CZBJhr3a97obFOv4hOO3Fc=">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</latexit>

演習問題：Cauchy-Schwarz ⇒ 三角不等式© Meiji University,All rights reserved.

３次元ユークリッド空間

R<latexit sha1_base64="A4EZRzZgRrxrc56SGLvxNyPp4sA=">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</latexit><latexit sha1_base64="MUFKaMlcnO9DvwUE4TDryxocwHc=">AAACkHichVHLSsNAFD3Gd31V3QhuikXRTZmIaHGhFTfqSqu1hbaUJE5raJrEZFqsxR/wB1y4UcGF+AF+gBt/wEU/QVwquHHhTRoQFesdZubMmXvuzOGqtqG7grFmh9TZ1d3T29cfGhgcGh4Jj47tu1bV0XhKswzLyaiKyw3d5CmhC4NnbIcrFdXgabW87t2na9xxdcvcE3Wb5ytKydSLuqYIovI5wY9FY3ZzLnIaCRXCURZjfkR+AzkAUQSxbYXvkcMBLGioogIOE4KwAQUujSxkMNjE5dEgziGk+/ccpwiRtkpZnDIUYsu0luiUDViTzl5N11dr9IpB0yFlBNPsid2yV/bI7tgz+/izVsOv4f2lTrva0nK7MHI2sfv+r6pCu8Dhl6qNQqXs9p4Eioj7XnTyZvuM51Jr1a+dnL/uLienGzPsmr2QvyvWZA/k0Ky9aTc7PHkBr0Hyz3b8BvvzMXkxNr+zEE2sBK3qwySmMEv9WEICG9hGit49wjkucSWNSXFpVVprpUodgWYc30La+gQfeJPv</latexit>

ヒルベルト空間3(I) 和・スカラー倍を定義すると、
性質①から⑧を満たす。

H : 集合<latexit sha1_base64="eQeP6+Orkd7qFOUndHRPlHB0eqc=">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</latexit><latexit sha1_base64="MUFKaMlcnO9DvwUE4TDryxocwHc=">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</latexit>(I) 和・スカラー倍を定義して、性質①から⑧を満たすとする。

<latexit sha1_base64="JJ0z7lDMaBlmerSZAzeeyPifyoo=">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</latexit><latexit sha1_base64="JJ0z7lDMaBlmerSZAzeeyPifyoo=">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</latexit>(II) 内積(II) 内積hx, yi := a1 b1 + a2 b2 + a3 b3 hx, yi 2 R<latexit sha1_base64="1vPLj3q0th1mBHqZzAjcVoFYHJ0=">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</latexit><latexit sha1_base64="oV6TIhQiQWv0Xn4JUDX8blw5AWw=">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</latexit>を定義すると、性質①から④を満たす。

<latexit sha1_base64="QGhALYIxX19v1BGuaCw87GktLeU=">AAAClHichVHLSsNAFD3Gd320KojgplgU3ZRJERURKRTBbsRXH1ClJHFaQ9MkJNOilv6AP+CiK0UX4gf4AW78ARd+grhUcOPCmzQgKuoNmTlz5p6bHI5qG7orGHvskDq7unt6+/pDA4NDw+HIyGjWtWqOxjOaZVhOXlVcbugmzwhdGDxvO1ypqgbPqZWUd5+rc8fVLXNXHNt8v6qUTb2ka4ogqhgJ7wl+JBqz6XR6LtqMhoqRGIszv6I/gRyAGILatCK32MMBLGiooQoOE4KwAQUuPQXIYLCJ20eDOIeQ7t9zNBEibY26OHUoxFZoLdOpELAmnb2Zrq/W6CsGvQ4po5hmD+yavbB7dsOe2Puvsxr+DO9fjmlX21puF8OnEztv/6qqtAscfqr+UKjU/bcngRKWfC86ebN9xnOptefXT85edpa3pxsz7II9k79z9sjuyKFZf9Wutvh2C15A8vc4foJsIi4vxBNb87HkahBVHyYxhVnKYxFJrGMTGT+zFi5xJY1LK1JKWmu3Sh2BZgxfStr4ADTelMY=</latexit>を定義して、性質①から④を満たすとする。
<latexit sha1_base64="QGhALYIxX19v1BGuaCw87GktLeU=">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</latexit>(III) ノルム p
kxk := hx, xi(III) ノルム p
kxk := hx, xi

定理（参考文献57頁; （C1））：(xn ) ⇢ R3 : bounded=) 9(xni ) ⇢ (xn ) : xni ! x 2 R3(xn ) ⇢ H : bounded=) 9(xni ) ⇢ (xn ) : xni * x 2 H<latexit sha1_base64="OiMzl5pncR8ugMt8nYy5CMwo5Ls=">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</latexit>を定義すると、性質①から③を満たす。

<latexit sha1_base64="WNZT/CAWfnmLiCTP1ZEezthUtsQ=">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</latexit><latexit sha1_base64="sqzAs6mbBKBM+mbxJBOomN5CNkg=">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</latexit><latexit sha1_base64="OiMzl5pncR8ugMt8nYy5CMwo5Ls=">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</latexit>を定義すると性質①から③を満たす。
<latexit sha1_base64="MkzWKPxnci3V559hURA/4RZxccA=">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</latexit>(IV)

以下を満たすとする：<latexit sha1_base64="HeYSNQl4XGs/qodRalsh5RBaRws=">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</latexit><latexit sha1_base64="K26MqDNU/Yk5v0DceDAtgaXydr4=">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</latexit>

H : ヒルベルト空間<latexit sha1_base64="eQeP6+Orkd7qFOUndHRPlHB0eqc=">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</latexit>

ヒルベルト空間

H: 内積空間 【※ (I), (II) を満たす空間】
H: ヒルベルト空間()<latexit sha1_base64="2davGnKYefrTIjmWCyMdKOQFBd8=">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</latexit>

def（IV）H 上の有界な点列が弱収束する部分列をもつ。

9(xni ) ⇢ (xn ) : xni *x 2 H<latexit sha1_base64="KbcONvH8LaXKlp5B4K2t4XwuDnU=">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</latexit><latexit sha1_base64="2davGnKYefrTIjmWCyMdKOQFBd8=">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</latexit>()
def<latexit sha1_base64="GHOa1l5v4V3emwO8yBM8deISuxU=">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</latexit>© Meiji University,All rights reserved.

内積の意味で強収束する

8y 2 H, hxni , yi ! hx, yiよく⾒るヒルベルト空間の定義との関係

H: 内積空間 【※ (I), (II) を満たす空間】
H: ヒルベルト空間()<latexit sha1_base64="2davGnKYefrTIjmWCyMdKOQFBd8=">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</latexit>def

H 上の有界な点列が弱収束する部分列をもつ()<latexit sha1_base64="yu4FVW7qWhlrO+RcOs2rZhMSvWU=">AAACm3ichVG7SgNBFD1Z3/GRqCCCCMGgWIWJiIpV0EbEQqNRQSXsrpNkyGZ3mZ1EYvQH/AELKwUL0d4PsPEHLPwEsVSwsfDuZkFU1DvMzJkz99yZwzVcS3iKsceI1tLa1t7R2RXt7unti8X7BzY9pypNnjMdy5Hbhu5xS9g8p4Sy+LYruV4xLL5llBf9+60al55w7A1Vd/leRS/aoiBMXRGVjw9Fd1ccu2jxgpKiWFK6lM5BIpqPJ1mKBZH4CdIhSCKMVSd+i13sw4GJKirgsKEIW9Dh0dhBGgwucXtoECcJieCe4xhR0lYpi1OGTmyZ1iKddkLWprNf0wvUJr1i0ZSkTGCcPbAr9sLu2TV7Yu+/1moENfy/1Gk3mlru5mMnw+tv/6oqtCuUPlV/KAzK/tuTQgFzgRdB3tyA8V2azfq1w9OX9fnseGOCXbBn8nfOHtkdObRrr+blGs+ewW9Q+ns7foLNqVR6JjW1Np3MLISt6sQIxjBJ/ZhFBktYRY7ePcIFrnGjjWqL2rK20kzVIqFmEF9Cy30A6J6YlQ==</latexit>

H: 完備（complete）※ 参考⽂献99⾴© Meiji University,All rights reserved.

強収束と弱収束

q強収束（ノルムの意味で収束する）

xn ! x () lim kxn<latexit sha1_base64="cSUal2uaST0y9g3QDuFCkohQcOI=">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</latexit>

def n!1

xk = 0q弱収束（内積の意味で収束する）

xn * x () 8y 2 H, lim hxn , yi = hx, yi<latexit sha1_base64="PnfI2vKnc/bH1qBBrgKuRvb2/aI=">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</latexit>

def

n!1

() 8y 2 H, lim |hxn , yi<latexit sha1_base64="GHz5B3ggi9hzl8q8f7EywkcdRnI=">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</latexit>

n!1© Meiji University,All rights reserved.

hx, yi| = 0<latexit sha1_base64="ghUzy0tlVw+oJCutf1oucp1ycTE=">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</latexit>

強収束 ⇒ 弱収束

xn ! x =) xn * x i.e.,<latexit sha1_base64="F74HIqbTca5aVN872G7WESwuUm8=">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</latexit><latexit sha1_base64="Pd+Qi/z3mxUC7lK3GRdLFXHt/Hk=">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</latexit>

lim kxn

xk = 0 =) lim |hxn , yi

n!1*<latexit sha1_base64="cfA/f+Vtb1ZJCaMB2a1mFt6/08s=">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</latexit><latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit>n!1

8y 2 H,|hxn , yi<latexit sha1_base64="8LMRgSB9LgJG6UjHKUZzLRisHSs=">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</latexit>

③, ②

②

hx, yi|x, yi|

 kxn

xkkyk ! 0

Cauchy-Schwarz<latexit sha1_base64="HOFBuOKo1S85hj3YGOQVstPpwfE=">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</latexit>© Meiji University,All rights reserved.

hx + y, zi = hx, zi + hy, zi<latexit sha1_base64="Xqu2/m5hYNLZ8AVY13A70ZlLKzo=">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</latexit>= |hxn<latexit sha1_base64="JuBAKcyZcAdq2HAFV9mMdX8yVJE=">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</latexit>

hx, yi| = 0<latexit sha1_base64="3NCQ1WviPReVlMvT1uqtMAFodRA=">AAACiXichVG7SgNBFD1ZXzG+ojaCTTBErMJdEZVUQRtLNSYGooTdddTVze6yOwnE4A9Y2YlaKViIH+AH2PgDFvkEsYxgY+HNZkFU1DvMzJkz99yZw9Vdy/QlUTOidHX39PZF+2MDg0PDI/HRsYLvVD1D5A3HcryirvnCMm2Rl6a0RNH1hFbRLbGpHy637zdrwvNNx96QdVdsV7Q929w1DU0yVdiSToIS5XiS0hRE4idQQ5BEGKtO/B5b2IEDA1VUIGBDMragwedRggqCy9w2Gsx5jMzgXuAYMdZWOUtwhsbsIa97fCqFrM3ndk0/UBv8isXTY2UCKXqiW2rRI93RM73/WqsR1Gj/pc673tEKtzxyMpF7+1dV4V1i/1P1h0Ln7L89SexiMfBisjc3YNoujU792tFZK5dZTzWm6Zpe2N8VNemBHdq1V+NmTaxfIsYNUr+34ycozKbV+fTs2lwyuxS2KopJTGGG+7GALFawijy/e4BTnONCGVBUZVHJdFKVSKgZx5dQlj8AHA2RXw==</latexit>③h↵x, yi = ↵hx, yi<latexit sha1_base64="HogNhvjJDcshYIUbqEJHiJnLsco=">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</latexit>３次元ユークリッド空間

R<latexit sha1_base64="A4EZRzZgRrxrc56SGLvxNyPp4sA=">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</latexit><latexit sha1_base64="MUFKaMlcnO9DvwUE4TDryxocwHc=">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</latexit>

ヒルベルト空間

H : 集合3(I) 和・スカラー倍を定義すると、

性質①から⑧を満たす。

<latexit sha1_base64="eQeP6+Orkd7qFOUndHRPlHB0eqc=">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</latexit><latexit sha1_base64="MUFKaMlcnO9DvwUE4TDryxocwHc=">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</latexit>(I) 和・スカラー倍を定義して、性質①から⑧を満たすとする。

<latexit sha1_base64="JJ0z7lDMaBlmerSZAzeeyPifyoo=">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</latexit><latexit sha1_base64="JJ0z7lDMaBlmerSZAzeeyPifyoo=">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</latexit>(II) 内積(II) 内積hx, yi := a1 b1 + a2 b2 + a3 b3 hx, yi 2 R<latexit sha1_base64="1vPLj3q0th1mBHqZzAjcVoFYHJ0=">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</latexit><latexit sha1_base64="oV6TIhQiQWv0Xn4JUDX8blw5AWw=">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</latexit>を定義すると、性質①から④を満たす。

<latexit sha1_base64="QGhALYIxX19v1BGuaCw87GktLeU=">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</latexit>を定義して、性質①から④を満たすとする。
<latexit sha1_base64="QGhALYIxX19v1BGuaCw87GktLeU=">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</latexit>(III) ノルム p
kxk := hx, xi(III) ノルム p
kxk := hx, xi

定理：(xn ) ⇢ R3 : bounded=) 9(xni ) ⇢ (xn ) : xni ! x 2 R3(xn ) ⇢ H : bounded=) 9(xni ) ⇢ (xn ) : xni * x 2 H<latexit sha1_base64="OiMzl5pncR8ugMt8nYy5CMwo5Ls=">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</latexit>を定義すると、性質①から③を満たす。

<latexit sha1_base64="WNZT/CAWfnmLiCTP1ZEezthUtsQ=">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</latexit><latexit sha1_base64="sqzAs6mbBKBM+mbxJBOomN5CNkg=">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</latexit><latexit sha1_base64="OiMzl5pncR8ugMt8nYy5CMwo5Ls=">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</latexit>を定義すると性質①から③を満たす。
<latexit sha1_base64="MkzWKPxnci3V559hURA/4RZxccA=">AAACkXichVHLSsNAFD2Nr1ofrXYjuCkWRTdlWsTXQopuFDdttVXwRRKnNTRNQjIt1tIf8AcUXLXgQvwAP8CNP+DCTxCXCm5ceJMGREW9w8ycOXPPnTlcxdI1RzD2GJC6unt6+4L9oYHBoeFwZGS04JhVW+V51dRNe0eRHa5rBs8LTeh8x7K5XFF0vq2UV9377Rq3Hc00tkTd4vsVuWRoRU2VBVEHe4KfiMb0emEm1oyFDiNxlmBexH6CpA/i8CNjRm6xhyOYUFFFBRwGBGEdMhwau0iCwSJuHw3ibEKad8/RRIi0VcrilCETW6a1RKddnzXo7NZ0PLVKr+g0bVLGMMke2DV7Yffshj2x919rNbwa7l/qtCsdLbcOw2djm2//qiq0Cxx/qv5QKJT9tyeBIhY8Lxp5szzGdal26tdOz182l3KTjSnWZs/kr8Ue2R05NGqv6lWW5y7hNij5vR0/QSGVSM4lUtnZeHrZb1UQ45jANPVjHmmsIYM8vWvjAi20pai0KKWllU6qFPA1UXwJaeMDAJqUTw==</latexit>(IV) 以下を満たすとする：<latexit sha1_base64="HeYSNQl4XGs/qodRalsh5RBaRws=">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</latexit><latexit sha1_base64="K26MqDNU/Yk5v0DceDAtgaXydr4=">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</latexit>

H : ヒルベルト空間<latexit sha1_base64="eQeP6+Orkd7qFOUndHRPlHB0eqc=">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</latexit>

ヒルベルト空間の例（有限次元ユークリッド空間）

q有限次元ユークリッド空間
q有限次元ユークリッド空間では

強収束 ⇔ 弱収束
m強収束 ⇒ 弱収束（証明済）
m弱収束 ⇒ 強収束（演習問題）© Meiji University,All rights reserved.

ヒルベルト空間の例（l2 空間）

q２乗総和可能な数列全体からなる空間<latexit sha1_base64="lvC6OV7ulCcTiTxyhCvQvDeORIs=">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</latexit>

2

` :=

(

1
x = (xi )i=1 :

1
X
i=1

2

|xi | < 1x, y 2 `2 ,(I) x + y := (x1 + y1 , x2 + y2 , . . .)↵x := (↵x1 , ↵x2 , . . .)1Xpxi yi (III) kxk := hx, xi(II) hx, yi :=<latexit sha1_base64="Rjx3PViwmaZWAOWLrMEON878Nsk=">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</latexit>)<latexit 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sha1_base64="bs9VQnXj47g3qeoPiYiLVQ8vOWg=">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</latexit><latexit sha1_base64="hDV+xNS8yaSpb63koexw8q2oEfo=">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</latexit><latexit sha1_base64="yYEOxKSv1CpvRPA/6oTcSYm92Ko=">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</latexit><latexit sha1_base64="kiIWPLXXtXpqC0/ZeT9EN/6tGP4=">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</latexit><latexit sha1_base64="Plvabxw3tCGsmfDJIZE6Y8aOXew=">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</latexit>

i=1© Meiji University,All rights reserved.

例題2.3

不動点近似法の証明でよく使う不等式・等式
q Cauchy-Schwarz の不等式（証明済）|hx, yi|  kxkkyk<latexit sha1_base64="ZZk/WKRTaNpqoPv4J00HSlx2FTI=">AAACr3ichVFNS+NQFD3Gr9oZbXU2gptgcRCR8iKi4qowm1n6VSsYrUl81oevSUzSYm38A65m58KVggtxr0sFN/4BF/4EcemAGxfepgFRUW9I3rnn3nNfDtd0pfADxu5alNa29o7ORFfyx8/unlS6t2/RdyqexfOWIx1vyTR8LoXN84EIJF9yPW6UTckL5tafRr1Q5Z4vHHshqLl8pWyUbLEhLCMgqpgeCXVp2CXJ1Z1RtabqXjMJVV3ybVXVwx2dcEiVsJjOsCyLQv0ItBhkEMeMk76AjnU4sFBBGRw2AsISBnx6lqGBwSVuBXXiPEIiqnPsIUnaCnVx6jCI3aJvibLlmLUpb8z0I7VFt0h6PVKqGGK37JQ9sht2xu7Z86ez6tGMxr/U6DSbWu4WU/v980/fqsp0Bth8VX2hMKn7a08BNjAVeRHkzY2YhkurOb+6e/A4Pz03VP/NjtkD+Ttid+yaHNrV/9bJLJ87RJIWpL1fx0ewOJbVJrJjs+OZ3GS8qgQGMIhh2sckcviLGeTp3n84xyWuFE0pKKvKWrNVaYk1v/AmFPEC9XugBg==</latexit>

q 三⾓不等式（演習問題）

kx + yk  kxk + kyk<latexit sha1_base64="bJQRPRHdW1nQ0VEQLLcLryaJWk4=">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</latexit>

q ２乗ノルム展開（証明済）kx ± yk2 = kxk2 ± 2hx, yi + kyk2
<latexit sha1_base64="HNHwOLgY0aRbrBDUr/jr8g5aOIM=">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</latexit>

q 平⾏四辺形公式（２乗ノルム展開から）

kx + yk2 + kx
<latexit sha1_base64="qnLannlG55+YPyLNhDjJSGeP/ko=">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</latexit>© Meiji University,All rights reserved.

yk2 = 2kxk2 + 2kyk2

不動点近似法の証明でよく使う不等式・等式
q Cauchy-Schwarz の不等式（証明済）|hx, yi|  kxkkyk<latexit sha1_base64="ZZk/WKRTaNpqoPv4J00HSlx2FTI=">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</latexit>

q 三⾓不等式（演習問題）

kx + yk  kxk + kyk<latexit sha1_base64="bJQRPRHdW1nQ0VEQLLcLryaJWk4=">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</latexit>

q ２乗ノルム展開（証明済）kx ± yk2 = kxk2 ± 2hx, yi + kyk2
q 本当によく使う等式 ↵ 2 R
k↵x + (1 ↵)yk2<latexit sha1_base64="HNHwOLgY0aRbrBDUr/jr8g5aOIM=">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</latexit><latexit sha1_base64="84qV56UOP8KiztscpF/XJLMyW0Y=">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</latexit><latexit sha1_base64="55YyNATWrMgCT92gEyAYcVU4TJc=">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</latexit>2= ↵kxk + (1© Meiji University,All rights reserved.

2

↵)kyk

↵(1

↵)kx

yk

2

本当によく使う等式の証明
2<latexit sha1_base64="zfExNPh03X9+78aGEP2rb7QCLOY=">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</latexit>k↵x + (1↵)yk

= k↵xk2 + k(122= ↵ kxk + (12<latexit sha1_base64="Ec35nrFqjZgt2JtGSCgqn3FqWd8=">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</latexit>kx

2

yk = kxk↵)yk2 + 2h↵x, (122↵) kyk + 2↵(12hx, yi + kyk

2hx, yi = kxk2 + kyk2
<latexit sha1_base64="+hYg2qtEhxG/zDouNf8ultjz5fs=">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</latexit><latexit sha1_base64="55YyNATWrMgCT92gEyAYcVU4TJc=">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</latexit>k↵x + (1↵)hx, yi

2

yk2

2

↵)yk

= ↵kxk2 + (1© Meiji University,All rights reserved.

kx

↵)yi

↵)kyk2

↵(1

↵)kx

yk2

ヒルベルト空間上の近似法の収束解析の流れ① 近似法で⽣成される点列が有界である

ことを⽰す（よく使う不等式・等式）② 弱収束する部分列が存在する（ヒルベ
ルト空間の定義）③ 弱収束先が解になることを⽰す（オピアルの定理）④ 全体点列の弱収束先も解になることを⽰す（オピアルの定理）© Meiji University,All rights reserved.

オピアル (Opial) の定理
H: ヒルベルト空間

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※証明で使⽤する事実:①弱収束する点列は有界 (参考⽂献: 定理5.4.3)②ε-n 論法③下極限の性質 (参考⽂献: 1.4 章)
参考⽂献: 定理5.4.4（111⾴）※付録© Meiji University,All rights reserved.

ここまでのまとめ

qヒルベルト空間m有界な点列が弱収束する部分列をもつ内積空間（完備な内積空間）
m有限次元ユークリッド空間、l2 空間
qヒルベルト空間上で成り⽴つ便利な不等式・等式© Meiji University,All rights reserved.

講義２

q不動点
q⾮拡⼤写像q⾮拡⼤写像の不動点
q例© Meiji University,All rights reserved.

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不動点とは︖<latexit sha1_base64="s/1umOOw1nU9yZo9oGeZ6MUnfbI=">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</latexit>

T: H !H<latexit sha1_base64="jhN5SX55jZo10vPfcoFHO1jeuOo=">AAADX3icpVI9TxtBEH32hUAIARMaIhoLi4g01jqJIKJCoaHky4AECN0di7PivnR3Nh+W/wBNyiSiAikF4mfQ5A+koEwZpSRSmhS8XR9CBAGR2NPdzL6dN7NvbpzIU0kqxFkubz3qeNzZ9aT7ac+z3r5C//PFJKzHrqy6oRfGy46dSE8Fspqq1JPLUSxt3/HkkrM1pc+XGjJOVBgspLuRXPPtWqA2lWunhKoLxdGdV+uFkigLs4o3nUrmlJCtmbA/N4xVbCCEizp8SARI6XuwkfBZQQUCEbE1NInF9JQ5l2ihm9w6oyQjbKJb/Na4W8nQgHudMzFsl1U8vjGZRYyI7+JYnItv4kT8FH9vzdU0OfRddmmdNldG6337g/N/7mX5tCk+XLHuYDiMvltTik28M1oUtUUG0Srddv7G3qfz+Ym5keZLcSR+Ud+hOBOnVBg0frtfZ+XcgckekLNtuuGb+wXsf5O47m7NIDvMp5GQNSRRrUH37bKf26yvVXm0vvGT7O9c8Vv/VUvreEitNr/Fuav8O2U3ncXX5cpY+c3s29Lk+2wCuzCEYYxyysYxiWnMoMrbKnzEZ3zJ/7A6rV6r0A7N5zLOAK4t68UF+ty9jg==</latexit>

T (x) = x<latexit sha1_base64="20fg71Bcphv8Z/6+vko0agGh+Bc=">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</latexit><latexit sha1_base64="hhqqn11Hmuu8x+TWkUgiY51+Z7s=">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</latexit>

T

x = T (x)<latexit sha1_base64="14JMhrNlfaqrK+3j+Jj+DIK7G0s=">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</latexit><latexit sha1_base64="w8YHX7wUbMLVvpugWc50WH0vhno=">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</latexit>© Meiji University,All rights reserved.

T を作用させても動かない点のこと
T の不動点不動点とは︖<latexit sha1_base64="s/1umOOw1nU9yZo9oGeZ6MUnfbI=">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</latexit>

T: H !H<latexit sha1_base64="jhN5SX55jZo10vPfcoFHO1jeuOo=">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</latexit>

T (x) = x<latexit sha1_base64="20fg71Bcphv8Z/6+vko0agGh+Bc=">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</latexit>

T を作用させても動かない点のこと
T の不動点T の不動点集合

Fix(T ) := {x 2 H : T (x) = x}<latexit sha1_base64="CHSttsxqbh8UDh2lDSqYzO0lrCk=">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</latexit>© Meiji University,All rights reserved.

⾊々な不動点の存在（不動点定理）

Dr. L.E.J. Brouwer（1881--1966）
コンパクト凸集合上の連続写像は不動点をもつ（1909）

⾊々な不動点の存在（不動点定理）

完備距離空間上の縮小写像の不動点は一意に存在する(1922)<latexit sha1_base64="Sna03fpE5y6sRu3kJ/dr3kNHbmc=">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</latexit>

xn+1 = T (xn )xn ! x<latexit sha1_base64="Hj4er484+DmlueT31qa9QhFPcHY=">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</latexit>

Dr. Stefan Banach（1892--1945）?

縮小写像 T の不動点に収束する

⾊々な不動点の存在（不動点定理）

角谷 静夫 先生
[Dr. Shizuo Kakutani]（1911--2004）

位相空間上のある集合値写像には不動点が存在する （Nash均衡の存在が言える）（1941）

⾊々な不動点の存在（不動点定理）

非拡大写像（半群）の不動点の存在性を明らかにした

高橋 渉 先生[Dr. Wataru Takahashi]（1944--2020）

⾮拡⼤写像
H: ヒルベルト空間

T: H → H, ⾮拡⼤（nonexpansive）() 8x, y 2 H,<latexit sha1_base64="2davGnKYefrTIjmWCyMdKOQFBd8=">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</latexit><latexit sha1_base64="A5/R15ZWiW4qFYZ7CjrPvApl4EM=">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</latexit>defkT (x)© Meiji University,All rights reserved.

T (y)k  kxyk

⾮拡⼤写像と例
H: ヒルベルト空間

T: H → H, ⾮拡⼤（nonexpansive）() 8x, y 2 H,<latexit sha1_base64="A5/R15ZWiW4qFYZ7CjrPvApl4EM=">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</latexit><latexit sha1_base64="2davGnKYefrTIjmWCyMdKOQFBd8=">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</latexit>defkT (x)

T (y)k  kx

yk

n ⾮拡⼤写像ー不動点集合
n 例1: 射影ー射影先n 例2: 射影の積ー射影先の共通部分
n 例3: 最急降下ー凸最適解の集合© Meiji University,All rights reserved.

例1: 射影（Projection, Projector）
H: ヒルベルト空間

C: H の空でない閉凸集合
PC: H → C, 射影() PC (x) 2 C, kPC (x)<latexit sha1_base64="oYw2w4htScBfLsZfFNolC0les58=">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</latexit><latexit sha1_base64="2davGnKYefrTIjmWCyMdKOQFBd8=">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</latexit>def

xk = min ky
y2C

xk

x
y

PC (x)Fix(PC ) = C<latexit sha1_base64="VWzVHejilljx+L4A5wV3K/90Ibw=">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</latexit><latexit sha1_base64="uJ53DmnvS8ONzvwIXB0q6mYSPvA=">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</latexit>C© Meiji University,All rights reserved.

例1: 射影（Projection, Projector）の性質

C: H の空でない閉凸集合
PC: H → C, 射影
1. 射影の不動点集合＝射影先
Fix(PC ) = C<latexit sha1_base64="U7udiu00p+ZKwthVheVs8dwAthU=">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</latexit>

2. 射影と内積の関係

8y 2 C, hx<latexit sha1_base64="2bT+8FyJsBKSHOchZNxII2Zhr5s=">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</latexit><latexit sha1_base64="MqArKxGmHnVrK3F3hIrckh6tWLY=">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</latexit>

PC (x), y3. 射影は（堅）⾮拡⼤

8x, y 2 H,
2kPC (x) PC (y)k  hx

PC (x)i  0<latexit sha1_base64="y5A7D8BwfpFwW5648FJCaJ/xg5U=">AAAClnichVHLSsNAFD3Gd31V3RTcFIviQspERMWFiCJ2WR+1hbaUJE7r4DQJSVqtxR/wB1yICwUV8QP8ADf+gAs/QVwquHHhbRoQFfUOM3PmzD135nB1WwrXY+yxRWlta+/o7OoO9fT29Q+EB4e2XKviGDxlWNJyMrrmcilMnvKEJ3nGdrhW1iVP67vLjft0lTuusMxNr2bzfFkrmaIoDM0jqhAezBUtR5Myuj9ZywkzmpgshGMszvyI/gRqAGIIImmFb5HDNiwYqKAMDhMeYQkNLo0sVDDYxOVRJ84hJPx7jkOESFuhLE4ZGrG7tJbolA1Yk86Nmq6vNugVSdMhZRRj7IFdsxd2z27YE3v/tVbdr9H4S412vanldmHgKLLx9q+qTLuHnU/VHwqdsv/25KGIOd+LIG+2zzRcGs361YPjl4359bH6ODtnz+TvjD2yO3JoVl+NizW+foIQNUj93o6fYGsqrs7Ep9emY4tLQau6MIJRTFA/ZrGIBJJI0bt7OMUlrpSIsqCsKKvNVKUl0AzjSyjJD1t6ljE=</latexit><latexit sha1_base64="0EpgmOj3yCWwvAW7DT5Y9EoAeqg=">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</latexit>© Meiji University,All rights reserved.

y, PC (x)PC (y)i例1 性質 2.

hx<latexit sha1_base64="2bT+8FyJsBKSHOchZNxII2Zhr5s=">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</latexit>

PC (x), yx<latexit sha1_base64="easzNvb3QZVS757zW5/nldhEoqg=">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</latexit>

PC (x)i  0<latexit sha1_base64="YkFbmwQZw5jxxccHlu48/1Ml0iI=">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</latexit>

PC (x)y<latexit sha1_base64="ENo/EKPc+GbvAtmSNdi9XG5oPxE=">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</latexit>© Meiji University,All rights reserved.

参考文献
補助定理5.2.2（102頁）

例1 性質 3. の証明

8x, y 2 H,
2kPC (x) PC (y)k  hx
* 8z 2 C, 例1 性質 2.
<latexit sha1_base64="y5A7D8BwfpFwW5648FJCaJ/xg5U=">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</latexit><latexit sha1_base64="0EpgmOj3yCWwvAW7DT5Y9EoAeqg=">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</latexit>y, PC (x)<latexit sha1_base64="seloXbw+30P9k9Y5PZnPgRZKygo=">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</latexit><latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit>hxhy

PC (x), z
PC (y), zhx

PC (x), PC (y)<latexit sha1_base64="35rFHp4kiTO+zynyt0Qyfot4Pko=">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</latexit><latexit sha1_base64="j3z6UXG9G7FzbNhIqGCdQf5i6Q0=">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</latexit>

PC (x)i  0
PC (y)i  0<latexit sha1_base64="P73fXp/d+DZfq4EC1XEwKT9f9K0=">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</latexit>() hPC (x)<latexit sha1_base64="MQgRZHR+1mjqADp7CjfURzR900g=">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</latexit><latexit sha1_base64="79Fx02afTkh9bIQ/hTPkCBS4ppI=">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</latexit>hy

PC (x)i  0x, PC (x)<latexit sha1_base64="h01EdXON/WFjV5xMT3D69my5tQ8=">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</latexit>

PC (y), PC (x)© Meiji University,All rights reserved.

PC (y)i  0

PC (y)i  0

PC (y)i例1 性質 3. の証明

hx, zi + hy, zi = hx + y, zi
8x, y 2 H,
2kPC (x) PC (y)k  hx y, PC (x) PC (y)i
* hPC (x) x, PC (x) PC (y)i  0
hy PC (y), PC (x) PC (y)i  0h(PC (x) x) + (y PC (y)), PC (x) PC (y)i  0<latexit sha1_base64="e0oW1fPgxEmiIVLYULA5NLlM8YM=">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</latexit><latexit sha1_base64="y5A7D8BwfpFwW5648FJCaJ/xg5U=">AAAClnichVHLSsNAFD3Gd31V3RTcFIviQspERMWFiCJ2WR+1hbaUJE7r4DQJSVqtxR/wB1yICwUV8QP8ADf+gAs/QVwquHHhbRoQFfUOM3PmzD135nB1WwrXY+yxRWlta+/o7OoO9fT29Q+EB4e2XKviGDxlWNJyMrrmcilMnvKEJ3nGdrhW1iVP67vLjft0lTuusMxNr2bzfFkrmaIoDM0jqhAezBUtR5Myuj9ZywkzmpgshGMszvyI/gRqAGIIImmFb5HDNiwYqKAMDhMeYQkNLo0sVDDYxOVRJ84hJPx7jkOESFuhLE4ZGrG7tJbolA1Yk86Nmq6vNugVSdMhZRRj7IFdsxd2z27YE3v/tVbdr9H4S412vanldmHgKLLx9q+qTLuHnU/VHwqdsv/25KGIOd+LIG+2zzRcGs361YPjl4359bH6ODtnz+TvjD2yO3JoVl+NizW+foIQNUj93o6fYGsqrs7Ep9emY4tLQau6MIJRTFA/ZrGIBJJI0bt7OMUlrpSIsqCsKKvNVKUl0AzjSyjJD1t6ljE=</latexit><latexit 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sha1_base64="MQgRZHR+1mjqADp7CjfURzR900g=">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</latexit><latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">AAACi3ichVHLSsNAFD2N7/po1Y3gRiwVV2VaRUVciCK4rK19QFtKMo51aJqEJC1o8QdcunGhGwUX4gf4AW78ARd+gris4MaFt2lAtFhvyMyZM/fc5HA0S5eOy9hLQOnrHxgcGh4Jjo6NT4TCk1NZx6zbXGS4qZt2XlMdoUtDZFzp6iJv2UKtabrIadXt9n2uIWxHmsa+e2yJUk2tGPJQctUlKl/UBFfrjiiHIyzGvJrrBnEfROBX0gw/oIgDmOCoowYBAy5hHSocegqIg8EiroQmcTYh6d0LnCJI2jp1CepQia3SWqFTwWcNOrdnOp6a01d0em1SziHKntkda7Ends9e2eefs5rejPa/HNOudbTCKofOZtIf/6pqtLs4+lb1UGjU3duTi0OseV4kebM8pu2Sd+Y3Ti5a6fVUtLnAbtgb+btmL+yRHBqNd367J1KXCFJA8d9xdINsIhZfiSX2liObW35Uw5jFPBYpj1VsYhdJZLwcznGJK2VcWVLWlY1OqxLwNdP4UcrOF6dskvg=</latexit><latexit sha1_base64="h01EdXON/WFjV5xMT3D69my5tQ8=">AAACt3ichVHPLytRFP4Mj6qHPjYSm4mGkNCcijy8lbCxrB9FojKZGbftxO3MdGba6Gv8A/4BEisveQvxB9jZWPAPWPgTxJLExsLpdEIQnMmdc+53v+/c++UYrrT8gOimRWlt+9HeEeuMd/3s7ulN/Opb852KZ4qs6UjH2zB0X0jLFtnACqTYcD2hlwwp1o2dhcb5elV4vuXYq0HNFVslvWBbecvUA4a0xGRO6nZBCrWmTqgZbWG0NjYe5t2xF0DNeU1OToqySloiSSkKQ/1YpKMiiSgyTuIMOWzDgYkKShCwEXAtocPnbxNpEFzGtlBnzOPKCs8F9hBnbYVZghk6ozv8L/BuM0Jt3jd6+qHa5FskL4+VKobpmk7onq7olG7p6dNe9bBH4y01zkZTK1ytd39g5fFbVYlzgOKr6guFweyvPQXIYyb0YrE3N0QaLs1m/+rfg/uVP8vD9RH6R3fs75hu6IId2tUH8/+SWD5CnAeUfj+Oj8XaZCr9OzW1NJWcm49GFcMghjDK85jGHBaRQZbvPcQ5LnGlzCqakleKTarSEmn68SaU8jNaFKBf</latexit><latexit sha1_base64="fvTGtk+HRgsTOH6xh30UX7T8l8A=">AAACyXichVE9TxtBEH0+8gGGBAcapDSrWCBbSaxxhAKiQpAiUhobMCBhy7o7FufE+u64O1sYK1U6/kCKVImUIuEH8ANoKNOkcP4BogSJhoK5DwURC5jV7b59M2/2nsZwleUHRL2UNvDg4aPHg0Pp4ZEnT0czz8ZWfaflmbJiOsrx1g3dl8qyZSWwAiXXXU/qTUPJNWN7McyvtaXnW469EnRcWWvqDdvaskw9YKqeeVdVut1QUuRK9cXcbl68Fry9FLkOo5Dq5PPilfiXjClR9WJVVckdQfVMlgoUhegHxQRkkUTJyRyiik04MNFCExI2AsYKOnxeGyiC4DJXQ5c5j5EV5SU+Ic3aFldJrtCZ3ea9wbeNhLX5Hvb0I7XJryj+PFYKTNIf+klndEwHdEKXt/bqRj3Cf+nwacRa6dZH9yeWL+5VNfkM8PFadYfC4Oq7PQXYwmzkxWJvbsSELs24f3vvy9ny3NJkd4q+0yn7+0Y9OmKHdvvc/FGWS1+R5gEV/x9HP1h9Uyi+LUyXp7PzC8moBvEcL5DjecxgHu9RQoXf/YXf6OGv9kHb0Xa1vbhUSyWacdwI7fMVFwyk4A==</latexit>h(PC (x)<latexit sha1_base64="aMrSsraQz1Vej3Mmy+4iXIWG/6M=">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</latexit>hPC (x)PC (y))PC (y), PC (x)<latexit sha1_base64="eaEfBbP/1FlB5TdRXZq2+S8ZTrg=">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</latexit>kPC (x)PC (y)k

© Meiji University,All rights reserved.

2(x

PC (y)ihxy), PC (x)hx

PC (y)i  0y, PC (x)y, PC (x)PC (y)i  0

PC (y)i  0

例1 性質3. 射影は堅⾮拡⼤ ⇒ ⾮拡⼤
q 射影は堅⾮拡⼤2<latexit sha1_base64="0EpgmOj3yCWwvAW7DT5Y9EoAeqg=">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</latexit>kPC (x)PC (y)k  hx

y, PC (x)PC (y)i

Cauchy-Schwarz

 kx<latexit sha1_base64="0uc37U7wXCmI4CatHkt3haSUqGk=">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</latexit>

ykkPC (x)PC (y)kq 射影は⾮拡⼤でもある

kPC (x)<latexit sha1_base64="EgVBN47ZmCEL5/Hjk9vOO6R1oGo=">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</latexit>

PC (y)k  kx
x<latexit sha1_base64="5mj4zC1GyHvW7GIi4JYNrqF3ew4=">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</latexit><latexit sha1_base64="ApRHi4w2cPH3tvnG166B89eU0/Q=">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</latexit>

PC (x)y<latexit sha1_base64="1tb3JWuXhBiKHuhlP1a0FRd8F3c=">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</latexit><latexit sha1_base64="4wDE46InEN0DgPo4Ik0EA6m8F5s=">AAACiXichVG7SgNBFD1ZXzFRE7URbIIhok2YFVGxEtNY5mEeEEPYXUddsy92N4EY/AErO1ErBQvxA/wAG3/AIp8glhFsLLy7WRAV9Q4zc+bMPXfmcGVLUx2XsW5IGBgcGh4Jj0aiY+MTsfjkVMkxm7bCi4qpmXZFlhyuqQYvuqqr8Yplc0mXNV6WGxnvvtzitqOaxrbbtnhNl/YNdU9VJJeoUraeWWgv1uNJlmZ+JH4CMQBJBJE14/fYwS5MKGhCB4cBl7AGCQ6NKkQwWMTV0CHOJqT69xzHiJC2SVmcMiRiG7Tu06kasAadvZqOr1boFY2mTcoEUuyJ3bIee2R37Jm9/1qr49fw/tKmXe5ruVWPncwU3v5V6bS7OPhU/aGQKftvTy72sOZ7Ucmb5TOeS6Vfv3V01ius51OdeXbNXsjfFeuyB3JotF6VmxzPXyJCDRK/t+MnKC2lxZX0cm45ubEZtCqMWcxhgfqxig1sIYsivXuIU5zjQogKorAmrPdThVCgmcaXEDIfN5aRbg==</latexit>

PC (y)© Meiji University,All rights reserved.

yk閉球への射影nB(c, r) := {x 2 R : kxx
PB (x)cck  r}

閉球への射影nB(c, r) := {x 2 R : kx
kx

ck  r}PB (x) = c+ck
PB (x)

x

r

c

0

r
kx

ck

(x

c)

閉球への射影nB(c, r) := {x 2 R : kxxck  r}PB (x)c
PB (x) =

8
<c +
:x

r
kx

ck

(x

c)

(x 2
/ B)
(x 2 B)半空間への射影

a 6= 0, r 2 R<latexit sha1_base64="8jBjykcXcdmvc2DjXwsMq9IgD78=">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</latexit>nH(a, r) := {x 2 R : ha, xi  r}<latexit sha1_base64="T01tf7SFfURZyH4U1xwm80mSwVI=">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</latexit><latexit sha1_base64="t42UX23W4gydxAjttKQhnWSI0EI=">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</latexit>

PH (x) =?? 演習問題

© Meiji University,All rights reserved.

⾮拡⼤写像と例
H: ヒルベルト空間

T: H → H, ⾮拡⼤（nonexpansive）() 8x, y 2 H,<latexit sha1_base64="A5/R15ZWiW4qFYZ7CjrPvApl4EM=">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</latexit><latexit sha1_base64="2davGnKYefrTIjmWCyMdKOQFBd8=">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</latexit>defkT (x)

T (y)k  kx

yk

n ⾮拡⼤写像ー不動点集合
n 例1: 射影ー射影先n 例2: 射影の積ー射影先の共通部分
n 例3: 最急降下ー凸最適解の集合© Meiji University,All rights reserved.

例2: 射影の積
H: ヒルベルト空間

Ci: H の空でない閉凸集合 (i = 1,2,…,m)
P i = PC i
1 2
m
x
C3

T = P P ···P<latexit sha1_base64="tVLbibNoWupT6Lr9cZ68r94xpCI=">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</latexit><latexit sha1_base64="C8HMm1lZijccsQITPyXVXvPwgvE=">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</latexit>

T (x) = P1 P2 P3 (x)P3 (x)P2 P3 (x)C1
© Meiji University,All rights reserved.

C2

例2: 射影の積の性質
H: ヒルベルト空間

Ci: H の空でない閉凸集合 (i = 1,2,…,m)
P i = PC i
1 2m
1. 射影積の不動点集合＝射影先の共通部分

T = P P ···P<latexit sha1_base64="tVLbibNoWupT6Lr9cZ68r94xpCI=">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</latexit><latexit sha1_base64="C8HMm1lZijccsQITPyXVXvPwgvE=">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</latexit><latexit sha1_base64="zA96SZZGbsad3iBYGlwW6S8UdAA=">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</latexit>

Fix(T ) =

m
\

i=1

Ci 6= ; 凸実行可能問題

2. 射影積は⾮拡⼤<latexit sha1_base64="OiOjLugPSm+kxssN8beLjAfrwxY=">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</latexit>

8x, y 2 H,

kT (x)<latexit sha1_base64="ZSiE0vsST91tdbK55J9MNK/z/Js=">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</latexit>© Meiji University,All rights reserved.

T (y)k  kxyk

例2 性質 2. の証明

T = P1 P2 · · · Pm
<latexit sha1_base64="tVLbibNoWupT6Lr9cZ68r94xpCI=">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</latexit>kPi (x)<latexit sha1_base64="Ob89s1Mvg0HeSlFmfnKwkfXSOp0=">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</latexit>kT (x)<latexit sha1_base64="HGb+zwYd4FnTE07Z7uwDSRLqLNE=">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</latexit>

Pi (y)k  kx

T (y)k = kP1 ((P2 · · · Pm )(x))
 k(P2 · · · Pm )(x)
 k(P3 · · · Pm )(x)
..
.
 kx

© Meiji University,All rights reserved.

ykyk
例1 性質 3.

P1 ((P2 · · · Pm )(y))k(P2 · · · Pm )(y)k(P3 · · · Pm )(y)k例2 性質 1. の証明<latexit sha1_base64="uQf+7RQbHmjy0Mti5KE6dWoPvpI=">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</latexit><latexit sha1_base64="zA96SZZGbsad3iBYGlwW6S8UdAA=">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</latexit><latexit sha1_base64="gdJ0BazzI8lqx+7/IFhCQZgRzSE=">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</latexit>m\

T = P1 P2 · · · Pm
Fix(T
)
⇢
C
=
6
;
i
m
\
i=1
Fix(T ) =Ci 6= ;
演習問題
<latexit sha1_base64="tVLbibNoWupT6Lr9cZ68r94xpCI=">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</latexit>

Fix(T )

i=1
m
\

i=1

Ci 6= ;

<latexit sha1_base64="Po4lFkWhJMJCo5h1CmfIW9SCRJ8=">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</latexit>

x
=
P
(x)
i
*
x2
Ci () 8i, x 2 Ci = Fix(Pi )i=1=) T (x) = P1 P2 · · · Pm 1 Pm (x)= P1 P2 · · · Pm 1 (x)= ··· = x<latexit sha1_base64="oWuLWw4nFpg89wbcaRcYTa4ySy0=">AAAC3XichVHPS9xAGP1Mf6hrrWt7EXoZXCwWyjIR0VIQpIL00MOqXV0wNkzibHZwkgmT2a027NGLSK8eemqhh9I/oDcvvfQf6MG7F5H2YqGXHvolGygq2i8k8+bNe9/k8XmxFImh9KjPunHz1u3+gcHS0J3huyPl0XuriWprn9d9JZVueCzhUkS8boSRvBFrzkJP8jVvayE7X+twnQgVvTQ7Md8IWRCJpvCZQcotN7aJIyLieCLwWeymYs7uvgrJgitIyXmhokDyptEiaBmmtXqNZFNpJiURj0nPmknniBMy09Jhuii2u5M1VzxyyxVapXmRy8AuQAWKqqnyF3BgExT40IYQOERgEEtgkOCzDjZQiJHbgBQ5jUjk5xy6UEJvG1UcFQzZLfwGuFsv2Aj3Wc8kd/t4i8RXo5PABP1OP9Ez+o1+pif0z5W90rxH9i87uHo9L4/dkb2xld//dYW4Gmj9c13j8FB9fSYDTXiSZxGYLc6ZLKXf6995c3C28nR5In1IP9BTzPeeHtGvmDDq/PI/LvHld1DCAdkXx3EZrE5V7Znq9NJ0Zf5ZMaoBeADjMInzmIV5eA41qOO9h3AMP+Cn5Vq71r71tie1+grPfThX1sFfdzmxrQ==</latexit><latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit>m\

例1 性質 1.

<latexit sha1_base64="dOsCVJG0ot2VOCPdt7RJMaKRYXg=">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</latexit><latexit sha1_base64="RqBx1MoSm0T4OVbJaJmsXVVkwyk=">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</latexit><latexit sha1_base64="qdmTl5BWHde4q9hI/B13ErzFNcY=">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</latexit>© Meiji University,All rights reserved.

例2 の応⽤例（ネットワーク資源割当）

Ci: H の空でない閉凸集合 (i = 1,2,…,m)

T = P1 P2 · · · Pm
<latexit sha1_base64="tVLbibNoWupT6Lr9cZ68r94xpCI=">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</latexit><latexit sha1_base64="zA96SZZGbsad3iBYGlwW6S8UdAA=">AAACunichVHLShxBFD12Ho6Th2OyEdw0GQxmM1SLjxAyMDggLn3MQ7BN092WYzH9srtmyKSZH/AHXLhScCH5gKxDNiFZZzGfEFwquHGR2z0NYiTmFlV16tQ9t+pwrcARkWRsMKI8ePjo8WhuLP/k6bPn44WJF43I74Q2r9u+44eblhlxR3i8LoV0+GYQctO1HN602tXkvtnlYSR8ryZ7Ad92zZYndoVtSqKMwrzumnIvdONl8bE/U3ujllXdEi3bDIxYlLX+B1etGkLVPb6v6twNZC/iMm8UiqzE0lDvAi0DRWSx6he+QMcOfNjowAWHB0nYgYmIxhY0MATEbSMmLiQk0nuOPvKk7VAWpwyT2DatLTptZaxH56RmlKptesWhGZJSxTT7xc7YBfvOPrPf7PqfteK0RvKXHu3WUMsDY/xgcuPqvyqXdom9G9U9Couy7/cksYu3qRdB3oKUSVzaw/rdT4cXG+/Wp+PX7ISdk79jNmDfyKHXvbRP1/j6EZIGaX+34y5ozJa0hdLc2lyxspS1KocpvMIM9WMRFaxgFXV69whf8QM/lfeKpQilPUxVRjLNS9wKRf4B4imkhg==</latexit>

Fix(T ) =

m
\

i=1

Ci 6= ;l ネットワーク上のユーザ i が所望する条件
Cim<latexit sha1_base64="h/XYhC2o8el2WxqStTjpZ3MXveo=">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</latexit>l ネットワーク全体の条件

\

Ci

i=1
H.I.: Distributed Optimization for Network Resource Allocation With Nonsmooth Utility
Functions, IEEE Transactions on Control of Network Systems 6 (4): 1354-1365 (2019)© Meiji University,All rights reserved.

例2 の応⽤例（機械学習）

Ci: H の空でない閉凸集合 (i = 1,2,…,m)

T = P1 P2 · · · Pm
<latexit sha1_base64="tVLbibNoWupT6Lr9cZ68r94xpCI=">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</latexit><latexit sha1_base64="zA96SZZGbsad3iBYGlwW6S8UdAA=">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</latexit>

Fix(T ) =Ci 6= ;

i=1 (<latexit sha1_base64="7bE/AlLSToJhWV5msDpEIM5276g=">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</latexit>l スパース学習:

m
\

Cs =

x : kxk1 =

n
X
i=1

|xi |  sl ダイバーシティ学習: Cd = {x : fd (x)  d}<latexit sha1_base64="0d0QiFPYLciIEMkkyYCaEo0sEpc=">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</latexit>

Cs \ Cd<latexit sha1_base64="i5yhkNmOIKfYO1P87o/9/1IgqGo=">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</latexit>l スパース&ダイバーシティ学習:

H.I.: Stochastic Fixed Point Optimization Algorithm for Classifier
Ensemble, IEEE Transactions on Cybernetics 50 (10): 4370–4380 (2020)© Meiji University,All rights reserved.

)

例2 の応⽤例（機械学習; もう少し丁寧に）
nq 制約集合: R+ \ Cs \ Cd ← とても複雑<latexit sha1_base64="e/3KAOuXY9zWTt0YbUr4rdbroBs=">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</latexit>

計算可能な非拡大写像

T = P+ Ps Pdn=) Fix(T ) = R+ \ Cs \ Cd<latexit sha1_base64="MFMZM5LPTaicOzOO6wdfZuqySNE=">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</latexit><latexit sha1_base64="ALAlALsOwcTTr7iYx2bbcPxYqNY=">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</latexit>

q ⽬的関数: データ i に関する損失関数 fi の総和<latexit sha1_base64="gSSTBn5b5NeZGM/jn29bvK9PV2U=">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</latexit>

f (x) =NX

fi (x)

i=1

q ⾃然に思いつくアルゴリズム ← 計算困難
<latexit sha1_base64="59V6qPi9+gcxN/XvPv4iT56R0ww=">AAAD03icpVLPa9RAFH5p/FHX1m71IngZulRaisukFFsEodiLx+3WbQtNDTPp7G7YySQk2d3WIRfxJHj24EnBg/hnePEf8NCDf4B4rODFgy+z0aKLVXBCMm++97735nt5PJZBmlF6bE3Y585fuDh5qXJ5avrKTHX26nYa9RNftPxIRskuZ6mQgRKtLMik2I0TwUIuxQ7vbRT+nYFI0iBSD7KjWOyHrKOCduCzDCGvOnR5qA9zT/eWnJzcJQ1PEzdkWZdz3cy9pYeKuD6LyYanDZyEOs3zMewAsbyyUCbrkVvEZTLuMjRdxbhkpO0F5Kd/cdGr1midmkXGDac0alCuRjRrzYELBxCBD30IQYCCDG0JDFJ89sABCjFi+6ARS9AKjF9ADhXk9jFKYARDtIffDp72SlThuciZGraPVSS+CTIJzNMP9A09oe/pW/qJfvtjLm1yFHc5wp2PuCL2Zp5e3/r6V1aIewbdU9YZDI7RZ2vKoA1rRkuA2mKDFCr9Uf7Bo+cnW3ea8/omfUU/o76X9Ji+Q4Vq8MV/vSmaL0x2hZyh6UZo7qew/xrxorsdgxxivgKJsIZAtNBQ9O1HP4dYv1AlcQ+NnZZ/55Sf/1OtQsf/1Brxc5w75/cpGze2l+vO7frK5kpt/V45gZNwA+ZgAadsFdbhPjSghbf9aNnWlDVtt2xtP7afjEInrJJzDX5Z9rPvytTn3g==</latexit>

xk+1 = PRn+ \Cs \Cd (xk© Meiji University,All rights reserved.

↵k rfi (xk ))例2 の応⽤例（機械学習; もう少し丁寧に）
nq 制約集合: R+ \ Cs \ Cd ← とても複雑<latexit sha1_base64="e/3KAOuXY9zWTt0YbUr4rdbroBs=">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</latexit>

計算可能な非拡大写像

T = P+ Ps Pdn=) Fix(T ) = R+ \ Cs \ Cd<latexit sha1_base64="MFMZM5LPTaicOzOO6wdfZuqySNE=">AAADf3icpVLLahRBFD2T9hHjI4luBDdDhqggDHck+AIhoAuXk8ckgSQM3Z3KpEm/6K6ZGJv+AP0BF64UXIgLt7p24w+4yCeIywhuXHiqpkPQYBSsprvuPXXPvXVuXy8Ng1yL7NVGnBMnT50ePTN29tz5C+MTkxeX8qSf+arjJ2GSrXhursIgVh0d6FCtpJlyIy9Uy972A3O+PFBZHiTxot5N1Xrk9uJgM/BdTag70Vis36+3uzf4rkWu3sqiIi/pFAfeRlkySppiV/2o0aqMBqrVTiZrU1jDBhL46COCQgxNO4SLnM8qWhCkxNZREMtoBfZcocQYuX1GKUa4RLf57dFbrdCYvsmZW7bPKiHfjMw6puWzvJF9+SRv5Yv8+GOuwuYwd9nl7g25Ku2OP7u88P2vrIi7xtYh6xiGx+jjNWls4o7VElBbahGj0h/mHzx5vr9wb366uCqv5Cv1vZQ9+UiF8eCb/3pOzb+w2WNydmw3Inu/mP0viJvu9izymPkMkrCGImo0mL4d9HOH9Y2qkHtk7bz6O4f88p9qGR3/U2vIN3PX+n3KjhpLN5utW82ZuZnG7MNqAkdxBVO4zim7jVk8Qhsd3vYp3uE9Pjg155rTdGQYOlKrOJfwy3Lu/gQXE8mE</latexit><latexit sha1_base64="ALAlALsOwcTTr7iYx2bbcPxYqNY=">AAADsHicpVLNThRBEK5l/AH8YZGLiZcJGwzGZNNLiBoTExKM8eABll3YhMVJz9DMtsx0T3p6d4HJvAAv4MGTJh6Mb+DViy/ggZtX9IiJFw9W9wwhQkQTezJT1V/XV9VfTflJxFNNyEFlxLlw8dLl0bHxK1evXZ+oTt5YTWVfBawdyEiqjk9TFnHB2prriHUSxWjsR2zN314052sDplIuRUvvJmwjpqHgWzygGiGv2uo+kyJUPOxpqpQcut2Y6p6Ksyd8J59t3XEfFYjvZ83cu/tcuN2AJu6idxyX5qeRzdyr1kid2OWedRqlU4NyLcnJyjR0YRMkBNCHGBgI0OhHQCHFZx0aQCBBbAMyxBR63J4zyGEcuX2MYhhBEd3Gb4i79RIVuDc5U8sOsEqEr0KmCzPkM3lHjsgn8p4ckp9/zJXZHOYuu2j9gssSb2L/5sqPv7JitBp6J6xzGD5Gn69JwxY8sFo4akssYlQGRf7B3sujlYfNmew2eUO+ob7X5IB8RIVi8D14u8yar2x2gZyh7UZs7yew/xnipruhRXYwn0Ek1mCIGg2mb8f9HGJ9oypCG1s/Lf/OCT//p1pGx//UKvhm7hqnp+ysszpXb9yrzy/P1xYelxM4CrdgGmZxyu7DAjyFJWjjbT/AFziEr86c03E8hxahI5WSMwW/LefFL9Je3fY=</latexit>

q ⽬的関数: データ i に関する損失関数 fi の総和<latexit sha1_base64="gSSTBn5b5NeZGM/jn29bvK9PV2U=">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</latexit>

f (x) =NX

fi (x)

i=1

q 不動点最適化アルゴリズム ← 計算可能
<latexit sha1_base64="EKW6nVBuJX5oobJv3uuD0mXGJFE=">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</latexit>

xk+1 = T (xk )↵k rfi (T (xk ))H.I.: Stochastic Fixed Point Optimization Algorithm for Classifier
Ensemble, IEEE Transactions on Cybernetics 50 (10): 4370–4380 (2020)© Meiji University,All rights reserved.

例3: 最急降下
H: ヒルベルト空間
C: H の空でない閉凸集合
f : H ! R, 凸
rf : H ! H, リプシッツ連続<latexit sha1_base64="8KwDGZU/CWsTx6NnObZbz6i3FIU=">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</latexit><latexit sha1_base64="PhfNLfaNVrHKeVNGvnuxwoiZlwY=">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</latexit>() 9L > 0 : 8x, y 2 H,<latexit sha1_base64="2davGnKYefrTIjmWCyMdKOQFBd8=">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</latexit><latexit sha1_base64="kwxWfQ3/+LaGG3f2GHT5KaOQ8ps=">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</latexit>def

krf (x)

例: A 2 S<latexit sha1_base64="MdR4sDHjDqzCpYFnrHtWqPHQxPg=">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</latexit>

n

;0 

rf (y)k  Lkx
min 

yk

b 2 Rn<latexit sha1_base64="3bYyPuwAoYNXdWv14lLZomEDoy4=">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</latexit>max

1
f (x) = hx, Axi hb, xi
2=) rf (x) = Ax b : max ー リプシッツ連続<latexit sha1_base64="hodaCPArdiWoOxr0B+AznQ2ao6U=">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</latexit><latexit sha1_base64="fCs7IZECQqHjA7Di6HcV7Y1qrEw=">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</latexit>© Meiji University,All rights reserved.

例3: 最急降下
H: ヒルベルト空間
最急降下法
C: H の空でない閉凸集合xk+1 = xk
rf (xk )f : H ! R, 凸✓
2
rf : H ! H, L-リプシッツ連続
2 0,<latexit sha1_base64="9760XkBBktXojXx/UJuCERSIpIk=">AAADgXicpVLPT9RAFH5LRRFQFr2YeGnYrIEYN1NC1GhMCF488sMFEpY00zK7NjudNm13WWz2H9A/wAMnTTgQT1716MV/wAN/gvEICRcOfjNbQ5SIJEzTvjffvO+9+V6fF8sgzRg7KA1ZV4avXhu5Pjo2fuPmRHny1moadRJf1P1IRsm6x1MhAyXqWZBJsR4ngoeeFGte+7k+X+uKJA0i9TLbicVmyFsqaAY+zwC55WrPzdv3nb79zO65bfuB3ZAgb3G7obgnud2cBjzjliusxsyyzzpO4VSoWIvRZGmKGrRFEfnUoZAEKcrgS+KU4tkghxjFwDYpB5bAC8y5oD6NgttBlEAEB9rGt4XdRoEq7HXO1LB9VJF4EzBtqrLvbJ8dsm/sI/vBTv6ZKzc59F12YL0BV8TuxJs7K8f/ZYWwGb06ZZ3D8BB9vqaMmvTYaAmgLTaIVukP8ndfvztcebJcze+xD+wn9L1nB+wrFKrukb+3JJZ3TXYFzrbpRmjup9D/HLjubssgPeTTSIQaAqjWoPv2u5/bqK9VSdjQ+Gnxd075/QvV0jouU2vA72PunL+n7KyzOltzHtbmluYq8wvFBI7QXZqiaUzZI5qnF7RIddz2LX2iz/TFsqwZi1mzg9ChUsG5TX8s6+kvQZvI1g==</latexit><latexit sha1_base64="8KwDGZU/CWsTx6NnObZbz6i3FIU=">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</latexit><latexit sha1_base64="22V16NSkhgtEi9ekf+gI8GRgq1M=">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</latexit><latexit sha1_base64="PhfNLfaNVrHKeVNGvnuxwoiZlwY=">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</latexit>

T : H ! H,
<latexit sha1_base64="G8iMK3mBUS8MdPHgivZ0x1+OiRw=">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</latexit><latexit sha1_base64="IhnIVzozC+1kUXjq11xRbt4fyYs=">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</latexit>

T (x) = PC (x© Meiji University,All rights reserved.

L

rf (x))

例3: 最急降下の性質<latexit sha1_base64="22V16NSkhgtEi9ekf+gI8GRgq1M=">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</latexit>✓

2
H: ヒルベルト空間
2 0,
C: H の空でない閉凸集合Lf : H ! R , 凸, rf : H ! H, L-リプシッツ連続<latexit sha1_base64="8KwDGZU/CWsTx6NnObZbz6i3FIU=">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</latexit><latexit sha1_base64="PhfNLfaNVrHKeVNGvnuxwoiZlwY=">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</latexit><latexit sha1_base64="IhnIVzozC+1kUXjq11xRbt4fyYs=">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</latexit>

T (x) = PC (xrf (x))

1. T の不動点集合＝C 上の f の最⼩点集合<latexit sha1_base64="lsXiqbi0OgRboLpbb8eOX15QfB4=">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</latexit>

Fix(T ) = argmin f (x)x2C

2. T は⾮拡⼤<latexit sha1_base64="OiOjLugPSm+kxssN8beLjAfrwxY=">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</latexit>

8x, y 2 H,

kT (x)<latexit sha1_base64="ZSiE0vsST91tdbK55J9MNK/z/Js=">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</latexit>© Meiji University,All rights reserved.

T (y)k  kxyk

例3 性質 2. の証明

rf (x))<latexit sha1_base64="IhnIVzozC+1kUXjq11xRbt4fyYs=">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</latexit>

T (x) = PC (x=) kT (x)T (y)k  kx

<latexit sha1_base64="Twnl8wZfshLf6PL0DOiHwL57x54=">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</latexit>*kT (x)

yk<latexit sha1_base64="22V16NSkhgtEi9ekf+gI8GRgq1M=">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</latexit>

2

✓

2
0,
L

<latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit><latexit sha1_base64="TAL+2vD9RP8RC/E1hH6A23kfU0o=">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</latexit>

T (y)k= kPC (x
 k(x
= k(x

rf (x))

rf (x))y)© Meiji University,All rights reserved.

2(y(rf (x)PC (y2

rf (y))k1 性質 3.
rf (y))k2 例
P は非拡大Crf (y))k2

例3 性質 2. の証明

rf (x))<latexit sha1_base64="IhnIVzozC+1kUXjq11xRbt4fyYs=">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</latexit>

T (x) = PC (x=) kT (x)<latexit sha1_base64="Twnl8wZfshLf6PL0DOiHwL57x54=">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</latexit>*kT (x)

T (y)k  kx

<latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit><latexit sha1_base64="jDG0q3Ss2Osmo3dKbJTBechEfo8=">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</latexit>

T (y)k k(x= kx<latexit sha1_base64="n+Wz5zR6c9YlZ2Kb61PAiFnhiK0=">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</latexit>

+

2

2

y)
2

yk

krf (x)

2

✓

2
0,
L

Theorem18.15(i), (v) 322頁

rf (y))k2(rf (x)2 hx

yk<latexit sha1_base64="22V16NSkhgtEi9ekf+gI8GRgq1M=">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</latexit>y, rf (x)2

rf (y)k<latexit sha1_base64="0a4p1NQVSjHxHPmHHGLqzo//tl4=">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</latexit>rf (y)i1
krf (x)Lrf (y)k2

参考文献: H.H. Bauschke, P.L. Combettes: Convex Analysis and Monotone Operator
Theory in Hilbert Spaces, Springer International Publishing (2017)© Meiji University,All rights reserved.

例3 性質 2. の証明<latexit sha1_base64="IhnIVzozC+1kUXjq11xRbt4fyYs=">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</latexit>

T (x) = PC (x=) kT (x)<latexit sha1_base64="Twnl8wZfshLf6PL0DOiHwL57x54=">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</latexit>*kT (x)

rf (x))

T (y)k  kx

yk<latexit sha1_base64="22V16NSkhgtEi9ekf+gI8GRgq1M=">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</latexit>

2

✓

2
0,
L

<latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit><latexit sha1_base64="jDG0q3Ss2Osmo3dKbJTBechEfo8=">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</latexit>

T (y)k

 k(x
= kx

y)
2

yk

2

(rf (x)
2 hx

rf (y))k2

y, rf (x)rf (y)i1
krf (x) ✓rf (y)k◆krf (x) rf (y)k2
L2
22
 kx ykkrf (x) rf (y)kL<latexit sha1_base64="n+Wz5zR6c9YlZ2Kb61PAiFnhiK0=">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</latexit>+

2<latexit sha1_base64="2h44Y2vkrbyezuNC+dUG6JuIX3A=">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</latexit>© Meiji University,All rights reserved.

2<latexit sha1_base64="0a4p1NQVSjHxHPmHHGLqzo//tl4=">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</latexit>

例3 性質 2. の証明

rf (x))<latexit sha1_base64="IhnIVzozC+1kUXjq11xRbt4fyYs=">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</latexit>

T (x) = PC (x=) kT (x)T (y)k  kx

<latexit sha1_base64="Twnl8wZfshLf6PL0DOiHwL57x54=">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</latexit>*kT (x)

yk<latexit sha1_base64="22V16NSkhgtEi9ekf+gI8GRgq1M=">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</latexit>

2

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<latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit><latexit sha1_base64="PaOFVAzTg/lXfSrG9NBO7FHqWTA=">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</latexit>

T (y)k
2

 kx

yk

 kx

2

yk

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krf (x)rf (y)k

H. I. : Iterative Algorithm for Solving Triple-hierarchical Constrained Optimization
Problem, Journal of Optimization Theory and Applications, 148 (3): 580-592 (2011)© Meiji University,All rights reserved.

2

例3 性質 1. の証明（付録）<latexit sha1_base64="IhnIVzozC+1kUXjq11xRbt4fyYs=">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</latexit>

T (x) = PC (xrf (x))<latexit sha1_base64="lsXiqbi0OgRboLpbb8eOX15QfB4=">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</latexit>

Fix(T ) = argmin f (x)x2C© Meiji University,All rights reserved.

ここまでのまとめ（⾮拡⼤写像と例）
H: ヒルベルト空間

T: H → H, ⾮拡⼤（nonexpansive）() 8x, y 2 H,<latexit sha1_base64="A5/R15ZWiW4qFYZ7CjrPvApl4EM=">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</latexit><latexit sha1_base64="2davGnKYefrTIjmWCyMdKOQFBd8=">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</latexit>defkT (x)

T (y)k  kx

n 例1: 射影ー射影先 PC

yk<latexit sha1_base64="ujUDWBBV4YP/L2tyE1yZzRH9MRM=">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</latexit>

Fix(PC ) = C

n 例2: 射影の積ー射影先の共通部分\<latexit sha1_base64="tNk7XMwtfm8oJIIkcvA5dGHVoJg=">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</latexit>

T = P1 · · · Pm

Fix(T ) =Cin 例3: 最急降下ー凸最適解の集合<latexit sha1_base64="RL6RCbJYnQLIBH1DiyJFlpsCGeY=">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</latexit>

T = PC (I© Meiji University,All rights reserved.

rf )Fix(T ) = argmin f (x)
x2C

講義３

q不動点近似法
m不動点を⾒つけるための⼿法
mKrasnoselʼskiĭ-Mannq最近の研究動向© Meiji University,All rights reserved.

Krasnoselʼskiĭ-Mann 不動点近似法

Dr. Mark Krasnosel’skiĭ（1920--1997）

Dr. William Mann
（1920--2006）

Krasnosel’skiĭ-Mann 不動点近似法（1955）（1953）

Krasnoselʼskiĭ-Mann 不動点近似法
H: ヒルベルト空間
T: H → H, ⾮拡⼤写像, Fix(T)≠∅↵ 2 (0, 1)<latexit sha1_base64="XbVR60xxDab6FDQxzAUzGKDbmMQ=">AAADbHicpVLNahRBEK7dyZ8xPxv1IITAkiUhQlhqJKh4CnrJMT9uEsiGZWbsbJr09Awzsxvjsi/gC3jQi4EIIY+Riy/gIY8QBC8RvHjw696RoMFEsIeZqv66vqr+asqPlUwz5rNC0enrHxgcujV8e2R0bLw0cWc9jVpJIGpBpKJk0/dSoaQWtUxmSmzGifBCX4kNf++5Od9oiySVkX6RHcRiO/SaWu7IwMsANUqluqfiXa9cl7o8x/Pug0apwlW2q3zVcXOnQvlajiYK01SnlxRRQC0KSZCmDL4ij1I8W+QSUwxsmzrAEnjSngvq0jC4LUQJRHhA9/BtYreVoxp7kzO17ABVFN4EzDLN8Gc+5gv+xCd8zj/+mqtjc5i7HMD6Pa6IG+Nv7q99v5EVwma0e8m6huEj+npNGe3QE6tFQltsEaMy6OVvv357sfZ0daYzy4f8Bfo+8BmfQqFufwuOVsTqO5tdg7NvuxHa+2n0vwPcdLdpkVfIZ5AINQRQo8H07Vc/91HfqFKwofXT/O9c8rv/VMvo+J9aPX4Xc+f+OWVXnfWHVfdRdWFlobL4LJ/AIZqkaZrDlD2mRVqiZarhtm16T0f0sfjVuedMOlO90GIh59yl35Yz+xOxE8Gk</latexit>x0 2 H,<latexit sha1_base64="A0Soh5j3Eq1HLFV9oatZVgRYc14=">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</latexit>

xk+1 = ↵xk + (1↵)T (xk )<latexit sha1_base64="/gLJvIff4m/Yv3hVJGnYaREYQwc=">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</latexit>?

(k = 0, 1, . . .)=) xk * x 2 Fix(T )<latexit sha1_base64="y0ht0FMTPJCNGnzx4zRRqI6qlVc=">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</latexit>© Meiji University,All rights reserved.

回転写像

0 はTの不動点

Krasnoselʼskiĭ-Mann 不動点近似法の例

11xk+1 = xk + T (xk )22<latexit sha1_base64="S3L6SrznAkpt5CsUTJfHyzquwN0=">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</latexit><latexit sha1_base64="C78ogSZtaAHsf3vFAEK508N3Ia4=">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</latexit>

xk+1 = T (xk ) は 0 に収束しない数値例（30°回転）
150

1/2
1/3
1/5

100

50

0

-50

-100

-150
-150

-100

-50

0

50

100

150

ヒルベルト空間上の近似法の収束解析の流れ① 近似法で⽣成される点列が有界である

ことを⽰す（よく使う不等式・等式）② 弱収束する部分列が存在する（ヒルベ
ルト空間の定義）③ 弱収束先が解になることを⽰す（オピアルの定理）④ 全体点列の弱収束先も解になることを⽰す（オピアルの定理）© Meiji University,All rights reserved.

証明① 近似法で⽣成される点列が有界であることを⽰す（よく使う不等式・等式）

x0 2 H,<latexit sha1_base64="A0Soh5j3Eq1HLFV9oatZVgRYc14=">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</latexit>

xk+1 = ↵xk + (1x 2 Fix(T ),↵)T (xk )<latexit sha1_base64="QI/NeSI1MNlX6kTMUDoGuABhDh8=">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</latexit>

kxk+1

xk  …  kxk
2

本当によく使う等式k↵x + (1 ↵)yk2<latexit sha1_base64="dLVCzJ25Etj2k6PfPeE5B1TqAFs=">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</latexit>

xk2<latexit sha1_base64="55YyNATWrMgCT92gEyAYcVU4TJc=">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</latexit>= ↵kxk2 + (1© Meiji University,All rights reserved.

↵)kyk2

↵(1

↵)kx

yk2

証明
① 近似法で⽣成される点列が有界であることを⽰す（よく使う不等式・等式）

x0 2 H,<latexit sha1_base64="A0Soh5j3Eq1HLFV9oatZVgRYc14=">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</latexit>

xk+1 = ↵xk + (1x 2 Fix(T ),↵)T (xk )<latexit sha1_base64="uJLj6YhXElAnLhFMcbJs1/Cio40=">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</latexit>

kxk+1<latexit sha1_base64="55YyNATWrMgCT92gEyAYcVU4TJc=">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</latexit>

xk2 = k↵xk + (1= k↵(xk x) + (1<latexit sha1_base64="i3FD2lbMtHZlazei06t3ur0rHas=">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</latexit>k↵x + (1↵)yk2

= ↵kxk2 + (1© Meiji University,All rights reserved.

↵)T (xk ) xk2
2↵)(T (xk ) x)k↵)kyk2

↵(1

↵)kx

yk2

証明
① 近似法で⽣成される点列が有界であることを⽰す（よく使う不等式・等式）

x 2 Fix(T ),<latexit sha1_base64="uJLj6YhXElAnLhFMcbJs1/Cio40=">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</latexit>

kxk+1

2

xk = k↵xk + (1= k↵(xk x) + (1<latexit sha1_base64="i3FD2lbMtHZlazei06t3ur0rHas=">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</latexit><latexit sha1_base64="d4ShaLtRjzOKhJMP+LJj51caILE=">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</latexit>= ↵kxk↵(1<latexit sha1_base64="55YyNATWrMgCT92gEyAYcVU4TJc=">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</latexit>k↵x + (1↵)T (xk ) xk↵)(T (xk ) x)k2

xk2 + (1↵)kT (xk )

↵)kxk

2

T (xk )k↵)kyk2

↵(1

xk2

↵)yk2

= ↵kxk2 + (1© Meiji University,All rights reserved.

2

↵)kx

yk2

証明
① 近似法で⽣成される点列が有界であることを⽰す（よく使う不等式・等式）

x 2 Fix(T ),<latexit sha1_base64="l7dFd/H+/XS8SLJTtFzyv5Ve0MA=">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</latexit>

kxk+1

2

2

xk + (1↵)kT (xk )xk= kT (xk )T (x)k2  kxk

xk2

xk = ↵kxk<latexit sha1_base64="bERLBUIerYEiUyBY8xWH9Gm0iIw=">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</latexit>↵)kxk T (xk )k2↵(1 ↵)kxk T (xk )k2<latexit sha1_base64="EQr4UGAniJ6o7jpML9Duczge8JY=">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</latexit><latexit sha1_base64="6w23yA2yYi/LgXyA2DMuHS2vHis=">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</latexit>

 kxk

↵(1
xk2

 kxk xk2
) 9 lim kxk xk ) (xk ) : bounded<latexit sha1_base64="/kkVMdczvfofGPDNj3LJVqHPQRc=">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</latexit><latexit sha1_base64="In7KCdl+N68Z0lUDDS8b7Njg+tM=">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</latexit><latexit sha1_base64="3lIMkO/yV2hwwtW7FB/gyvxfYTU=">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</latexit>

k!1© Meiji University,All rights reserved.

2

証明① 近似法で⽣成される点列が有界であることを⽰す（よく使う不等式・等式）<latexit sha1_base64="yaNhUjZbbTNgTc/3k+RsJ8h1z5g=">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</latexit>

kxk+1

xk2  kxk

<latexit sha1_base64="X21zTovTzwVGU0ocOok8S2KLg2E=">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</latexit>

↵(1

↵)kxk

xk2

↵(1

T (xk )k2  kxk

↵)kxk
xk2
k!1

) lim kxk

T (xk )k = 0) lim kxk

T (xk )k = 0<latexit sha1_base64="F26kFYvHUOoBWjr8HoDd5kGyGI0=">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</latexit>

k!1

2<latexit sha1_base64="IsVv/V2zJctZIaBvyFcPfLixGsA=">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</latexit>

k!1© Meiji University,All rights reserved.

kxk+1

9 lim kxk
<latexit sha1_base64="R/l+KlYfTl8Sij1CvFV/3It7d2c=">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</latexit>

T (xk )k2

xk

xk2

証明
① 近似法で⽣成される点列が有界であることを⽰す（よく使う不等式・等式）② 弱収束する部分列が存在する（ヒルベルト空間の定義）

9(xki ) ⇢ (xk ) : xki * x 2 H<latexit sha1_base64="SBrq/seSdcnIdGjmS+IJLLVGHmo=">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</latexit>③ 収束先が不動点になることを⽰す（オピアルの定理）: 背理法 x 6= T (x)<latexit sha1_base64="GizBkA3ion9U5OlZ/2VdlKkWmWU=">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</latexit><latexit sha1_base64="mHpXOL7J7s0CAMlRkQEQVyCcq2c=">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</latexit>

lim kxki

xk < lim kxki

lim kxk

T (xk )k = 0

i!1<latexit sha1_base64="nN44mBELhTl9BstK7y+M1ILeQdY=">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</latexit>

k!1© Meiji University,All rights reserved.

i!1

T (x)k証明 ③

9(xki ) ⇢ (xk ) : xki * x 2 H<latexit sha1_base64="SBrq/seSdcnIdGjmS+IJLLVGHmo=">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</latexit>

x 6= T (x)<latexit sha1_base64="GizBkA3ion9U5OlZ/2VdlKkWmWU=">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</latexit><latexit sha1_base64="mHpXOL7J7s0CAMlRkQEQVyCcq2c=">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</latexit>

lim kxkii!1

xk < lim kxki

T (x)ki!1= lim kxki

T (xki ) + T (xki ) lim {kxkiT (xki )k + kT (xki )<latexit sha1_base64="QsQPmXsDg7WbMRvoG6YAT56yQ/0=">AAADqHicpVLLahRBFL2T9hHHRya6EdwUGSKJ4lAjQUUQgm5cuJg8JhNJh6a7UzMW0y+6ayYZ2/4Bf8CFKwUX4s5fcOMPCGapO3EZwY0LT9W0RA1GwWq669Spe+6tc7u8JJCZ4ny3MmEdOXrs+OSJ6slTp89M1abPrmXxIPVF24+DOF333EwEMhJtJVUg1pNUuKEXiI7Xv6P3O0ORZjKOVtUoEZuh24tkV/quAuXU7t1iVXvopoEMZdR1cslsFTMbWI0KZj9iO07ed2TBrrDVuRLPs8s/L8zOPEOwU6vzBjeDHQTNEtSpHK14ujJDNm1RTD4NKCRBESnggFzK8GxQkzgl4DYpB5cCSbMvqKAqtANECUS4YPv49rDaKNkIa50zM2ofVQK8KZSMZvk7/pLv8bf8Ff/Ev/0xV25y6LOMMHtjrUicqcfnV77+VRViVvRgX3WIwkP04Z4UdemG8SLhLTGMdumP8w8fPtlbubk8m1/kz/ln+HvGd/kbOIyGX/wXS2L5qckeQbNtuhGa80Xofw5ed7dnmB3k00yMGgKs9qD79qOf26ivXQWYQ4Oz8u/s64t/qqV9/E+tsb7AvWv+fssOgrWrjea1xsLSQn3xdnkDJ+kCzdAcbtl1WqS71KI2Tvua3tMH+mhdslpWx7o/Dp2olJpz9MuwvO8gDNd8</latexit>

i!1<latexit sha1_base64="43862YaYggVHPP1uWGsTgWY+nd4=">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</latexit>

i!1

三角不等式
© Meiji University,All rights reserved.

T (x)kT (x)k}証明 ③

9(xki ) ⇢ (xk ) : xki * x 2 H<latexit sha1_base64="SBrq/seSdcnIdGjmS+IJLLVGHmo=">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</latexit>

x 6= T (x)<latexit sha1_base64="GizBkA3ion9U5OlZ/2VdlKkWmWU=">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</latexit>

lim kxki<latexit sha1_base64="mHpXOL7J7s0CAMlRkQEQVyCcq2c=">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</latexit>

xk < lim kxki

i!1

i!1

 lim {kxki<latexit sha1_base64="43862YaYggVHPP1uWGsTgWY+nd4=">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</latexit>

i!1<latexit sha1_base64="nN44mBELhTl9BstK7y+M1ILeQdY=">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</latexit>

lim kxk

k!1

T (xki )k + kT (xki )T (x)k}T (xk )k = 0= lim kxki
<latexit sha1_base64="xq7FhCX+ria8vl41b1SsnaVIN7g=">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</latexit>

i!1<latexit sha1_base64="bfsBSAxQIOJPEezWK7bs4wn3Awo=">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</latexit>

T (x)kT (xki )k + lim kT (xki )T (x)ki!1lim (an + bn ) = lim an + lim bn 参考文献
問題1.4.4 (2)n!1© Meiji University,All rights reserved.

n!1

n!1

証明 ③

9(xki ) ⇢ (xk ) : xki * x 2 H<latexit sha1_base64="SBrq/seSdcnIdGjmS+IJLLVGHmo=">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</latexit>

x 6= T (x)<latexit sha1_base64="GizBkA3ion9U5OlZ/2VdlKkWmWU=">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</latexit><latexit sha1_base64="mHpXOL7J7s0CAMlRkQEQVyCcq2c=">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</latexit>

lim kxkii!1

xk < lim kxki

T (x)ki!1

 lim {kxkiT (xki )k + kT (xki )= lim kxki

T (xki )k + lim kT (xki )<latexit sha1_base64="43862YaYggVHPP1uWGsTgWY+nd4=">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</latexit>

i!1<latexit sha1_base64="xq7FhCX+ria8vl41b1SsnaVIN7g=">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</latexit>

i!1= lim kT (xki )<latexit sha1_base64="98D+CoZwWWc4msbZTtG1XtCN0xY=">AAADjnicpVLLThRBFL1D+0B8MOjGxE2HCQYWTmoIQUJCJLBhyWuAhCad7rZmqEz1I901A2PbP+APYOIKExfGf3DjQn7ABZ9gXGLixoWnatoQJaKJ1em+t07dc2+d29dPpMgUY6eVIevK1WvXh2+M3Lx1+85odezuVhZ304A3g1jG6Y7vZVyKiDeVUJLvJCn3Ql/ybb+zrM+3ezzNRBxtqn7C90KvHYmWCDwFyK1OL9hOz0ulCEXUcnNhOyq2HfiqX9jO883JQzfvuKKYsh/Z2EwBc6s1Vmdm2RedRunUqFyr8VhlnBx6SjEF1KWQOEWk4EvyKMOzSw1ilADboxxYCk+Yc04FjYDbRRRHhAe0g28bu90SjbDXOTPDDlBF4k3BtGmCfWJv2Rk7Ye/YZ/b9j7lyk0PfpQ/rD7g8cUdf3N/49ldWCKto/5x1CcNH9OWaFLVozmgR0JYYRKsMBvl7z47ONubXJ/KH7DX7An3H7JR9gMKo9zV4s8bXX5nsETgHphuhuV+E/ufAdXfbBjlEPo3EqMGBag26bz/7eYD6WpWEDY2flX/nnF/8Uy2t439qDfgF5q7x+5RddLam643Z+szaTG1xqZzAYXpA4zSJKXtMi7RCq9TEbV/Se/pIJ1bVmrUWrCeD0KFKyblHvyxr5QfvXs6t</latexit>

i!1© Meiji University,All rights reserved.

T (x)k}T (x)ki!1

T (x)k  lim kxki xk
i!1
T : 非拡大
矛盾<latexit sha1_base64="IMU1sUtIAXT17KE3yPiB1O44b0A=">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</latexit>

証明① 近似法で⽣成される点列が有界であることを⽰す（よく使う不等式・等式）② 弱収束する部分列が存在する（ヒルベルト空間の定義）

9(xki ) ⇢ (xk ) : xki * x 2 H<latexit sha1_base64="SBrq/seSdcnIdGjmS+IJLLVGHmo=">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</latexit>③ 弱収束先が不動点になることを⽰す (オピアルのx = T (x)定理) 背理法 x 6= T (x)④ 全体点列の弱収束先も不動点になることを⽰す (オピアルの定理)<latexit sha1_base64="GizBkA3ion9U5OlZ/2VdlKkWmWU=">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</latexit>© Meiji University,All rights reserved.

<latexit sha1_base64="B5iRi8shMacn7MpsZntKppY64ws=">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</latexit>証明 ④

全体点列が x に弱収束する⇔ すべての弱収束する部分列の収束先が x（参考⽂献 定理 5.4.1）
9(xki ) ⇢ (xk ) : xki * x 2 Fix(T )<latexit sha1_base64="QfJ4dhVAT0Hd+bNI0nYgiaSu8ss=">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</latexit>

証明③の議論9(xkj ) ⇢ (xk ) : xkj * z 2 Fix(T )<latexit sha1_base64="q8HRC4bHWGKqUOJUFICVIIfklhY=">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</latexit><latexit sha1_base64="CPF7THOHSIR5A3AQvNkwo0iXZ80=">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</latexit>

lim kxk
背理法 x 6= z 9 k!1<latexit sha1_base64="T4unDTNyZuyPQ1gH5pwAbOPF2ZM=">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</latexit><latexit sha1_base64="yBWYw8ZtUbACWjsKdAfGdHi+KZM=">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</latexit>

オピアルの定理<latexit sha1_base64="8FK07EtVLSFJJ7rYOGJjgFHuoeI=">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</latexit>

lim kxki

i!1© Meiji University,All rights reserved.

xk (x 2 Fix(T ))xk < lim kxki
i!1=) x = z<latexit sha1_base64="2ppQ/Z4WJ4pltRoECrBApt9klhI=">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</latexit>

zk

証明 ④

全体点列が x に弱収束する⇔ すべての弱収束する部分列の収束先が x（参考⽂献 定理 5.4.1）
9(xki ) ⇢ (xk ) : xki * x 2 Fix(T )<latexit sha1_base64="QfJ4dhVAT0Hd+bNI0nYgiaSu8ss=">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</latexit>

9(xkj ) ⇢ (xk ) : xkj * z 2 Fix(T )<latexit sha1_base64="q8HRC4bHWGKqUOJUFICVIIfklhY=">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</latexit><latexit sha1_base64="CPF7THOHSIR5A3AQvNkwo0iXZ80=">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</latexit>

lim kxk
背理法 x 6= z 9 k!1<latexit sha1_base64="T4unDTNyZuyPQ1gH5pwAbOPF2ZM=">AAADqHicpVLLahRBFL2T9hHjIxPdCG6KDJGJ4FAjQcVVUBAXLiaPyUTSYehuaybF9IvumkmPbf+AP+DClYILcecvuPEHBLPUnbiM4MaFp2paggajYDXddevce+6tc/u6sS9TxfleZco6dvzEyelTM6fPnD03W507v5FGw8QTbS/yo2TTdVLhy1C0lVS+2IwT4QSuLzru4I72d0YiSWUUrqtxLLYDpx/KnvQcBahbvW+LDFVSZvsy6OYDZquI2TLsqXHB7McZoIKxqyzDAT6RqZwVrJ7pGGYHjtpJgvyuzIr6+uJit1rjDW4WO2w0S6NG5WpFc5V5sukhReTRkAISFJKC7ZNDKZ4tahKnGNg25cASWNL4BRU0A+4QUQIRDtABvn2ctko0xFnnTA3bQxUfbwImowX+nr/i+/wdf80/8+9/zJWbHPouY+zuhCvi7uyTi2vf/soKsCvaOWAdwXARfbQmRT26abRIaIsNolV6k/yjR0/3126tLuSX+Qv+Bfqe8z3+FgrD0Vfv5YpYfWayh+Dsmm4E5n4h+p8D193tGyRDPo1EqCGAag26bz/7uYv6WpWPPTB2Wv6dA37xT7W0jv+pNeEXmLvm71N22Ni41mhebyytLNWWb5cTOE2XaJ7qmLIbtEz3qEVt3PYNfaCP9Mm6YrWsjvVgEjpVKTkX6JdluT8A2r7ZWA==</latexit><latexit sha1_base64="yBWYw8ZtUbACWjsKdAfGdHi+KZM=">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</latexit>

オピアルの定理<latexit sha1_base64="FNxKicQLlf3kOV3VbuXXYoy70LI=">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</latexit>

lim kxk

k!1© Meiji University,All rights reserved.

xk (x 2 Fix(T ))xk < lim kxk
k!1=) x = z<latexit sha1_base64="2ppQ/Z4WJ4pltRoECrBApt9klhI=">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</latexit>

zk

証明 ④

9(xki ) ⇢ (xk ) : xki * x 2 Fix(T )<latexit sha1_base64="QfJ4dhVAT0Hd+bNI0nYgiaSu8ss=">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</latexit>

9(xkj ) ⇢ (xk ) : xkj * z 2 Fix(T )<latexit sha1_base64="Yc6kBiKg/0FY7FISGrr75RsZLkg=">AAADqnicpVLNbtNAEJ7U/JTy0xQuSFxWjYoSCUUbVNGqpwokxAm1adMWxZVlu9tkib1reddpWssvwAtw4AQSB8SVN+DCC3AoV7ggjkXiwoHZjVEFFQWJteyZ/Xa+mf3GEyQRV5rSw8qEc+bsufOTF6YuXrp8Zbo6c3VDySwNWSeUkUy3Al+xiAvW0VxHbCtJmR8HEdsMBvfM+eaQpYpLsa73E7Yd+z3Bd3noa4S86kOXjbCKIvWRlw+8x0WDuCoLFNMGGTSWSIkTN+W9vu77aSKlyBJyQFwuiBv7up/G+X0+KurrDa9ao01qFznptEqnBuVakTOVWXBhBySEkEEMDARo9CPwQeHThRZQSBDbhhyxFD1uzxkUMIXcDKMYRviIDvDbw123RAXuTU5l2SFWifBNkUlgjr6nr+gRfUdf08/0+x9z5TaHucs+2mDMZYk3/eT62re/smK0GvrHrFMYAUafrknDLixaLRy1JRYxKsNx/uHB06O1pfZcfpO+oF9Q33N6SN+iQjH8Gr5cZe1nNrtAzp7tRmzvJ7D/OeKmuz2LjDCfQSTWYIgaDaZvP/u5h/WNqghtbH1V/p1jfvFPtYyO/6k15hc4d63fp+yks3G72brTnF+dry3fLSdwEm7ALNRxyhZgGR7ACnTwtm/gA3yET84tp+08crrj0IlKybkGvyxn5wdordsc</latexit>

9 lim kxk zk
x 6= zk!1
lim kxk xk < lim kxk zk = lim kxkjzk

xk = lim kxk

xk<latexit sha1_base64="yBWYw8ZtUbACWjsKdAfGdHi+KZM=">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</latexit><latexit sha1_base64="R4eWZp+8r7vSIWwAjHs9hZRNOVg=">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</latexit><latexit sha1_base64="ndHfMACIchd9D+/HwMMIH+JQSLA=">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</latexit><latexit sha1_base64="FNxKicQLlf3kOV3VbuXXYoy70LI=">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</latexit>

k!1

k!1
x 6= z : オピアルの定理

j!1<latexit sha1_base64="yBWYw8ZtUbACWjsKdAfGdHi+KZM=">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</latexit>< lim kxkj
<latexit sha1_base64="ns2n44aEE0Sx9YcdDLaIc248ucM=">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</latexit>

j!1

)x=z
) xk * x 2 Fix(T )<latexit sha1_base64="aAGlDz5txZNo6q02XkHzaa2JA+Q=">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</latexit><latexit sha1_base64="v0iPl8cPxlCg9NmbmwjZtl27Vic=">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</latexit>© Meiji University,All rights reserved.

<latexit sha1_base64="6jdVxTOhkVg0YXPavDlVEHr13nQ=">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</latexit>

k!1

矛盾

Krasnoselʼskiĭ-Mann 不動点近似法の収束率
H: ヒルベルト空間
T: H → H, ⾮拡⼤写像, Fix(T)≠∅↵ 2 (0, 1)<latexit sha1_base64="XbVR60xxDab6FDQxzAUzGKDbmMQ=">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</latexit>x0 2 H,<latexit sha1_base64="A0Soh5j3Eq1HLFV9oatZVgRYc14=">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</latexit>

xk+1 = ↵xk + (1<latexit sha1_base64="8gKlZx1I4R3ZsJBVBJFUYRaJneo=">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</latexit>=) kxk© Meiji University,All rights reserved.

T (xk )k = O↵)T (xk )

✓

1
p
k

◆

証明
<latexit sha1_base64="RHkZ4+wlfAOdlDrmoB4gmWuEnME=">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</latexit>

8k 2 N, kxk+1

*
kxk+1

T (xk+1 )k  kxk

T (xk )k<latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit><latexit sha1_base64="/wuiuYmxQN0JaiYkgquVOsKOnAo=">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</latexit>

T (xk+1 )k= k↵xk + (1= k↵(xk↵)T (xk )T (xk+1 )kT (xk+1 )) + (1↵)(T (xk )T (xk+1 ))k ↵kxkT (xk+1 )k + (1↵)kT (xk )T (xk+1 )k ↵kxkT (xk+1 )k + (1<latexit sha1_base64="UAjTZoLGEguxKawv+ky1JFU9CaE=">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</latexit>

三角不等式<latexit sha1_base64="DE+rcOXeBKSlSCUXl/AuQ0xUHYw=">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</latexit>

T: 非拡大© Meiji University,All rights reserved.

↵)kxk

xk+1 k証明<latexit sha1_base64="RHkZ4+wlfAOdlDrmoB4gmWuEnME=">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</latexit>

8k 2 N, kxk+1*

T (xk+1 )k  kxk

T (xk )k<latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit><latexit sha1_base64="7omZ2f4LP6mXc06UszM49oG5EG0=">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</latexit>kxk+1

 ↵kxk ↵kxk<latexit sha1_base64="ezgOd8ee1xW3Nc7I00Dh22q6vzc=">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</latexit>

T (xk+1 )kT (xk+1 )k + (1xk+1 k + ↵kxk+1
xk+1 k

+ (1

↵)kxk

kxk

xk+1 k = kxk<latexit sha1_base64="+BylspYfohnl1YZsajNiUIcHLwM=">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</latexit>↵)kxk© Meiji University,All rights reserved.

= (1xk+1 kT (xk+1 )k

↵xk

(1

↵)T (xk )k↵)kxk

T (xk )k

証明
<latexit sha1_base64="RHkZ4+wlfAOdlDrmoB4gmWuEnME=">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</latexit>

8k 2 N, kxk+1*

T (xk+1 )k  kxk

T (xk )k<latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit><latexit sha1_base64="vh+lXMmrJKvXXDH7xNG+ko+VeC8=">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</latexit>

kxk+1

 kxk
= (1

kxk
<latexit sha1_base64="+BylspYfohnl1YZsajNiUIcHLwM=">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</latexit>

T (xk+1 )kxk+1 k + ↵kxk+1

T (xk+1 )kxk+1 k = kxk

↵xk

(1

↵)kxk

T (xk )k↵)kxk

© Meiji University,All rights reserved.

T (xk )k + ↵kxk+1= (1

T (xk+1 )k↵)T (xk )k証明<latexit sha1_base64="RHkZ4+wlfAOdlDrmoB4gmWuEnME=">AAADq3icpVI9b9RAEJ2L+QjhI5fQINGscgoKgpzWUQQRVQQNFSS5XBIRRyfb7F2sW3/E3rskGP8B/gAFFUgUiJpfQMMfoIhooUCUQaKh4O2e0QkiAhJr2Tv7Zt7MvvF4iQwyxflBZcQ6cfLU6dEzY2fPnb8wXp2YXMviXuqLph/LON3w3EzIIBJNFSgpNpJUuKEnxbrXvaP9632RZkEcrar9RGyFbicK2oHvKkCt6n2nHaeulKzLnCBiTuiqbc/L7xXXmfN4r5V3r9kFm2WrM6V9FTBzpNgp3UMnXMBa1Rqvc7PYUcMujRqVaymeqEyRQw8pJp96FJKgiBRsSS5leDbJJk4JsC3KgaWwAuMXVNAYuD1ECUS4QLv4dnDaLNEIZ50zM2wfVSTeFExG0/w9f8UP+Tv+mn/m3/+YKzc59F32sXsDrkha408uNb79lRViV7Q9ZB3D8BB9vCZFbVowWgJoSwyiVfqD/P1HTw8bt1am8yv8Bf8Cfc/5AX8LhVH/q/9yWaw8M9kjcHZNN0Jzvwj9z4Hr7nYMsod8GolRQwDVGnTffvZzF/W1Kok9NHZW/p0hv/inWlrH/9Qa8AvMnf37lB011ubq9o36/PJ8bfF2OYGjdJmmaAZTdpMW6S4tURO3fUMf6CN9smathvXAcgahI5WSc5F+WZb4AQK02QA=</latexit>

8k 2 N, kxk+1*

T (xk+1 )k  kxk

T (xk )k<latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit>

kxk+1<latexit sha1_base64="vh+lXMmrJKvXXDH7xNG+ko+VeC8=">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</latexit>

 kxk
= (1

xk+1 k + ↵kxk+1↵)kxk) kxk+1<latexit sha1_base64="N2NMfz5NJbla1L9kassvGs3X+7U=">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</latexit>

T (xk+1 )kT (xk )k + ↵kxk+1

T (xk+1 )k  kxk© Meiji University,All rights reserved.

T (xk+1 )kT (xk+1 )kT (xk )k

証明
<latexit sha1_base64="RHkZ4+wlfAOdlDrmoB4gmWuEnME=">AAADq3icpVI9b9RAEJ2L+QjhI5fQINGscgoKgpzWUQQRVQQNFSS5XBIRRyfb7F2sW3/E3rskGP8B/gAFFUgUiJpfQMMfoIhooUCUQaKh4O2e0QkiAhJr2Tv7Zt7MvvF4iQwyxflBZcQ6cfLU6dEzY2fPnb8wXp2YXMviXuqLph/LON3w3EzIIBJNFSgpNpJUuKEnxbrXvaP9632RZkEcrar9RGyFbicK2oHvKkCt6n2nHaeulKzLnCBiTuiqbc/L7xXXmfN4r5V3r9kFm2WrM6V9FTBzpNgp3UMnXMBa1Rqvc7PYUcMujRqVaymeqEyRQw8pJp96FJKgiBRsSS5leDbJJk4JsC3KgaWwAuMXVNAYuD1ECUS4QLv4dnDaLNEIZ50zM2wfVSTeFExG0/w9f8UP+Tv+mn/m3/+YKzc59F32sXsDrkha408uNb79lRViV7Q9ZB3D8BB9vCZFbVowWgJoSwyiVfqD/P1HTw8bt1am8yv8Bf8Cfc/5AX8LhVH/q/9yWaw8M9kjcHZNN0Jzvwj9z4Hr7nYMsod8GolRQwDVGnTffvZzF/W1Kok9NHZW/p0hv/inWlrH/9Qa8AvMnf37lB011ubq9o36/PJ8bfF2OYGjdJmmaAZTdpMW6S4tURO3fUMf6CN9smathvXAcgahI5WSc5F+WZb4AQK02QA=</latexit>

8k 2 N, kxk+1①
x 2 Fix(T ),T (xk+1 )k  kxk

T (xk )k<latexit sha1_base64="Z7/qjbVKB1pJrm5i3/FqBr+0Ay0=">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</latexit>2

xk↵)kT (xk )

2

kxk+1

↵(1

↵(1

xk  kxk

↵)

n
X

k=1

kT (xk )© Meiji University,All rights reserved.

2↵(1

xk k  kxk
xk k2 <latexit sha1_base64="/v6ITBpx/O4ol+eoxlxZyi+xSXM=">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</latexit>

n
X

k=1

= kx1

 kx1

↵)kT (xk )
2

kxk+1

xk

xk2

kxk+1

xk2

xk

kxk

xk2
xk2

xk k

kxn+1

xk2

2

2

証明

8k 2 N, kxk+1 T (xk+1 )k  kxk T (xk )kn
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k=1<latexit sha1_base64="ke0Z2ddg8Elj24eht2GXmJLnrCA=">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</latexit>↵(1<latexit sha1_base64="2aasD2h4fS/2nykimpsAEOaNa9I=">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</latexit><latexit sha1_base64="CJWIJil9dtRFAzkdpRjyl8a8Gac=">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</latexit>

2

kT (xn ) xn k
2↵)nkT (xn ) xn k  kx1

kT (xn )<latexit sha1_base64="VJt3rO0zhINQwgcewDyr2eZjpj4=">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</latexit>) kT (xn )© Meiji University,All rights reserved.

2
kx
xk
1
2
xn k 
↵(1 ↵)nkx1 xkxn k  p↵(1 ↵)nxk

2

【参考】適応⼿法の収束率
q AMSGrad (Adaptive Mean Square Gradient)m凸最適化 (Reddi et al.; ICLR2018):<latexit sha1_base64="M4ydUKHMTFUZCPO2pBrvc+2x3G8=">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</latexit>X

1Tft (xt )

f

t2T

?

!

r

=O

1 + log TT!

<latexit sha1_base64="sJ6dwyCJl37Mgk0ItKqX5X/U4OI=">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</latexit>Om⾮凸最適化 (Chen et al.; ICLR2019):<latexit sha1_base64="QbGLjwNSPbNPUBsT1kuWjbHN/bA=">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</latexit>min

k=1,2,...,n⇥2

E krf (xk )k⇤=O

q Krasnoselʼskiĭ-Mann✓◆<latexit sha1_base64="KZ1+zKZHdOg7boWKk5UXIfpNOs8=">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</latexit>

O

1
p
k

© Meiji University,All rights reserved.

✓

log n
p
n

◆

<latexit sha1_base64="yVt+vjx09hNCVFHMlQBfFUWrVGk=">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</latexit>

O

✓
✓

1
p
T

◆

1
p
n

◆

H.I.: Appropriate Learning Rates of Adaptive
Learning Rate Optimization Algorithms forTraining Deep Neural Networks, IEEE
Transactions on Cybernetics (2021)

ここまでのまとめ（Krasnoselʼskiĭ-Mann）

qKrasnoselʼskiĭ-Mann 不動点近似法<latexit sha1_base64="jdXgVITRx13DzlkmQNks07ql93A=">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</latexit>

xk+1 = ↵xk + (1↵)T (xk )q⾮拡⼤写像のある不動点に弱収束することを証明した?
xk * x 2 Fix(T )q収束率は✓<latexit sha1_base64="qg2cNCJPZ+ZZrim3cuycUIAeVcY=">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</latexit><latexit sha1_base64="OLVjN9/DoPVocV0YarHNzN9TgdU=">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</latexit>

kxk© Meiji University,All rights reserved.

T (xk )k = O

1
p
k

◆

プログラミングの演習問題<latexit sha1_base64="zX6wNahALLtFGEMtcC0rFinb2v8=">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</latexit><latexit sha1_base64="Kcrzfd+JM8vSyc4zstRRixmoqm4=">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</latexit>m\

Bi = B(ci , ri ) (i = 1, 2, . . . , m) B =Bi
T 1 = P1 P2 · · · P m
i=1
!
m
X
m
X
(wi ) ⇢ (0, 1) :wi = 1T 2 = P1
w i Pi<latexit sha1_base64="PG5sBpoMORL+8/yRQgnrXTjx8wg=">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</latexit><latexit sha1_base64="dyzv4nQdAiEMaIooIK6MdyLM6AY=">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</latexit>

i=2

i=2

<latexit sha1_base64="UOnU7nkWuxNsNGblGK/cxn3uEcI=">AAAD2XicpVI7T9xAEB7jEOAg4YAGiWbFCXQI5bRGKEFISIg0lLwOkM7Isp3lsG79kO07HpYLOkSbIkUqkChQfgYNf4CCmgpRgpQmBbN7RrwUIMpa9s5+M9/MfuOxAu5EMaXnSov6rvV9W3tHrrPrw8fufE/vcuTXQ5uVbZ/74aplRow7HivHTszZahAy07U4W7FqX4V/pcHCyPG9pXg7YGuuWfWcdcc2Y4SMfDqsW26ylRpJbVRLyRTRTR5smCRDa2SUFDXyKYNHyJKhkeKdc0TXc//MH3vAN/IFWqJykeeGlhkFyNac36MMgg7fwAcb6uACAw9itDmYEOFTAQ0oBIitQYJYiJYj/QxSyCG3jlEMI0xEa/it4qmSoR6eRc5Ism2swvENkUlgiJ7RY3pNT+kvekn//DVXInOIu2zjbjW5LDC69/sXf7/KcnGPYeOe9QLDwuiXNcWwDhNSi4PaAokIlXYzf2Pnx/Xi5MJQMkwP6RXqO6Dn9AQVeo0b+2ieLfyU2T3kbMpuuPJ+HvY/QVx0tyqRLcwnEB9rMESFBtG3u35uYn2hiuPuSjvK/s49P31TLaHjf2o1+SnOnfZ0yp4by2Ml7XNpfH68MD2TTWA7DMAgFHHKvsA0zMIclPG2F0qH0qv0qRV1V91T95uhLUrG6YNHS/1+C29E5aw=</latexit>

xk+1 = ↵xk + (1↵)T1 (xk )xk+1 = ↵xk + (1↵)T2 (xk )B ≠ ∅ ⇒ 挙動を調べてみましょう。
B = ∅ ⇒ 挙動を調べてみましょう。
© Meiji University,All rights reserved.

最近の研究動向（多様体へ）

q深層ニューラルネットワークは通常

ユークリッド空間 (まっすぐな空間)
q⾃然⾔語処理へ応⽤する時
多様体 (曲がった空間)での最適化が必要

q8/24 13:00ー リーマン多様体上の最適化理論とその周辺
佐藤寛之先⽣© Meiji University,All rights reserved.

最近の研究動向（多様体へ）

q深層ニューラルネットワークは通常

ユークリッド空間 (まっすぐな空間)
q⾃然⾔語処理へ応⽤する時
多様体 (曲がった空間)での最適化が必要
qリーマンAdam・リーマンAMSGradの開発に成功（既存⼿法よりも⾼性能）
Hiroyuki SAKAI, H.I.: Riemannian Adaptive Optimization
Algorithms and Its Applications to Natural Language
Processing, IEEE Transactions on Cybernetics(2021)© Meiji University,All rights reserved.

最近の研究動向（多様体上の不動点問題）

q2015年 6/27: 成島先⽣、佐藤先⽣との研究打ち合わせq多様体上での⾮拡⼤写像の不動点問題
m射影ー射影先m射影積ー射影先の共通部分
mリゾルベントーゼロ点集合
m⾃然な拡張ができる

q不動点最適化アルゴリズムを多様体へ拡張
H.I., Hiroyuki SAKAI : Riemannian Stochastic Fixed Point
Optimization Algorithm, submitted© Meiji University,All rights reserved.

これで講義はお終いです付録

qオピアルの定理の証明
q例3 性質 1. の証明© Meiji University,All rights reserved.

<latexit sha1_base64="Ubs2RjPVtH0iQXgjgrSlmlCfdNo=">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</latexit>

lim kxn

xk < lim kxn

オピアルの定理の証明n!1<latexit sha1_base64="4tFIpYcMEFsX9qhsa7X9fI0D3KQ=">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</latexit>

kxn<latexit sha1_base64="tux9c+epxbYtlREjDmrK1XhZfrQ=">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</latexit>

yk2 = k(xn

= kxn

yk2

yk + kxn

xk)(kxn

x, x

= kxn

yk

= (kxnx, x

yi + kx<latexit sha1_base64="SHosHTaDX7Tthdxs0pf4CX64btA=">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</latexit>

2hxn

n!1

x) + (xxk2 + 2hxn

2kxn

<latexit sha1_base64="8yI957mQWoCRK3UmVyO1G2bxbG8=">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</latexit> M (kxn<latexit sha1_base64="4Co2nLorzIASdJKoEe2BZ8Tb8Zs=">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</latexit>

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オピアルの定理の証明n!1<latexit sha1_base64="vdOcGipxBwMml7U/pHpvNa6Fr+k=">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</latexit><latexit sha1_base64="xWShiQZfGjYV7ei60Xd4GS1Ng40=">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</latexit><latexit sha1_base64="i2kK0544cycz1YhTqZN3o8W1FgA=">AAADinicpVLLThRBFL1D+0BEGWBjwqbDBONqUoNGXhuiLlgRHg6Q0GTS3dYMFaqrm+6a4dGZH3Bt4sKVJi6If+AWF/6ACz6BsMTEjQtP1bQhSkQTq9N9b5265946t2+QSJFpxk5Kfc616zdu9t8auD145+5QeXhkLYvbacjrYSzjdCPwMy6F4nUttOQbScr9KJB8Pdh5as7XOzzNRKye64OEb0V+S4mmCH0NqFGuenwfVTJXNZjrCeV6ka+3gyBf7M66XjNOfSldoC2+a0Ia5QqrMrvcy06tcCpUrKV4uDROHr2gmEJqU0ScFGn4knzK8GxSjRglwLYoB5bCE/acU5cGwG0jiiPCB7qDbwu7zQJV2JucmWWHqCLxpmC6NMG+sCN2zj6zD+yUff9jrtzmMHc5gA16XJ40hl7eW/32V1YEq2n7gnUFI0D01Zo0NWnaahHQlljEqAx7+TuHr89XZ1cm8vvsHTuDvrfshB1Doep8Dd8v85U3NrsCZ892I7L3U+h/Dtx0t2WRfeQzSIwaHKjRYPr2s597qG9USdjI+lnxdy743X+qZXT8T60ev4u5q/0+ZZedtclq7XH14fKjyvyTYgL7aYzG6QGmbIrmaYGWqI7bvqKPdEyfnEFn0plx5nqhfaWCM0q/LOfZD4LLzVM=</latexit>

n!1

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4

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<latexit sha1_base64="jhDve7lRZG+MDIJCWNrxOhUSxYA=">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</latexit><latexit sha1_base64="wLtD7XvybRyp04AT/PXSPkT9YI8=">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</latexit>

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yk

<latexit sha1_base64="Ubs2RjPVtH0iQXgjgrSlmlCfdNo=">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</latexit>

lim kxn

オピアルの定理の証明n!1<latexit sha1_base64="2tuNlBYm+ArVdHce9WwEyIJ54Mc=">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</latexit><latexit sha1_base64="iueGEzoj7Nz/6XsN0BD7nLBRl5g=">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</latexit>

8n

x, x

= (kxn M (kxn

2hxn<latexit sha1_base64="bIAuvWfAnH0qN8pJ1UthLNppBog=">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</latexit>

yk

n!1

n0

0 < 2hxn<latexit sha1_base64="7N8aSE3cdWJhVhTbojWxCsZhksw=">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</latexit>

xk < lim kxn

yi + kx

yk

kxn

xk)

yk + kxn

x, xyk

yi + kx

 M kxn yk
lim 2hxn x, x

n!1

 lim M kxn
n!1© Meiji University,All rights reserved.

2

yk

kxn

yk + M kxn

xkxk)(kxn2

yi + kxyk2

yk + M kxn

xk)

xk

lim kxn<latexit sha1_base64="Ubs2RjPVtH0iQXgjgrSlmlCfdNo=">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</latexit>

xk < lim kxn

オピアルの定理の証明n!1

n!1<latexit sha1_base64="bIAuvWfAnH0qN8pJ1UthLNppBog=">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</latexit>

lim

2hxn

n!1

yi + kxx, x lim M kxn

yk2

yk + M kxn

xk

yk

n!1

yk2 + M kxn<latexit sha1_base64="9w8utufZ4bCKo6LcOlJdeBnOU/8=">AAAErXicpVLNbtNAEB4XAyX8tIELEpdVo1ZIpWETECAkpAouXJDSlrSVum1kuxvXqr127U2aYPwCvAAHTiBxQDwAD8CFF+DQA1cQ4lgkLhyYXRtFbUlaibXs/fx5vpn5xmtHvpdISveMsVPm6TNnx8+Vzl+4eGlisnx5OQk7scObTuiH8aptJdz3BG9KT/p8NYq5Fdg+X7G3H6nvK10eJ14onsp+xNcDyxVe23MsiVSrbDRIaYZ1rdj3Ak+0W6lgMmSIZD8jzOdtydI6Aku4Pie9liBzpHeD9Mhcn7A4Z2cJe64Y0ieINupIPNFUHq1IjPXcLckyxrDcTsfaJMzlO2RY5eNKktLsUO3BZkYEHm5y0NuDfzoaUW+QpNSarNAq1YscBbUCVKBYjbBsTAGDTQjBgQ4EwEGAROyDBQlea1ADChFy65AiFyPy9HcOGZRQ28EojhEWstv4dPFtrWAFvquciVY7WMXHO0YlgWn6mb6j+/QTfU+/099Dc6U6h+qlj7uda3nUmnhxdenXsaoAdwlbA9UIhY3Roz1JaMM97cVDb5FmlEsnz9999nJ/6f7idDpD39Af6O813aMf0aHo/nTeLvDFVzq7QM2unkag+xM4/xR5NV1XMz3Mp5gQa3BklQc1t7/z3MX6ypWPe6BxUvydgT47US3l439q5foMz13t8Ck7Cpbr1dqd6q2F25X5h8UJHIdrMAXX8ZTdhXl4DA1ogmN8ML4YX41v5k2zaTJzIw8dMwrNFTiwTPcPHdcuZw==</latexit>

lim

n!1

2hxn

x, x

lim 2hxn

n!1= kx<latexit sha1_base64="36akpX6RD8hGQLoQEdwBa1ZsEc4=">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</latexit>kx

yi + kx

x, x

yi + lim kx
n!1

yk2 + M lim kxn

xk

yk2 + lim M kxn

xk

n!1

xk

n!1

2

yk + M lim kxn© Meiji University,All rights reserved.

n!1

xk  M lim kxn
n!1

yk<latexit sha1_base64="Ubs2RjPVtH0iQXgjgrSlmlCfdNo=">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</latexit>

lim kxn

xk < lim kxn

オピアルの定理の証明n!1<latexit sha1_base64="36akpX6RD8hGQLoQEdwBa1ZsEc4=">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</latexit>kx

2

yk + M lim kxn
n!1<latexit sha1_base64="XUdbvNY0FmCd85hnYLe7TUoptMA=">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</latexit>

0 < kxyk2  M) lim kxn
<latexit sha1_base64="vnr2uIjyRbYkwrBHGhfV+wTlGr8=">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</latexit>

n!1© Meiji University,All rights reserved.

⇢n!1

xk  M lim kxn

lim kxn

xk < lim kxn

yk

n!1

yk

n!1

n!1

yk

lim kxn

n!1

yk

xk

<latexit sha1_base64="lsXiqbi0OgRboLpbb8eOX15QfB4=">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</latexit>

例3 性質 1. の証明

Fix(T ) = argmin f (x)T (x) = PC (xrf (x))<latexit sha1_base64="IhnIVzozC+1kUXjq11xRbt4fyYs=">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</latexit>

x2C

Fix(T ) ⇢ argmin f (x)<latexit sha1_base64="AcDqhuFxF7ETuGFtm7gHXQhS7WU=">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</latexit>

x2C* x 2 Fix(T ) () x = PC (x<latexit sha1_base64="7uVmP2hlyuKGuMZ9LawfRpBzk2w=">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</latexit><latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit>

例1 性質 2.

=)8y 2 C,

rf (x))<latexit sha1_base64="IEubxZZTg/8+3x/ONHqHLi5/mec=">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</latexit>hx
rf (x) x, y() 8y 2 C, hrf (x), y<latexit sha1_base64="6o28Yf842OMKJBkBUFV9uZ0vpE0=">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</latexit>© Meiji University,All rights reserved.

xi  0xi 0<latexit sha1_base64="lsXiqbi0OgRboLpbb8eOX15QfB4=">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</latexit>

例3 性質 1. の証明

Fix(T ) = argmin f (x)T (x) = PC (xrf (x))<latexit sha1_base64="IhnIVzozC+1kUXjq11xRbt4fyYs=">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</latexit>

x2C

Fix(T ) ⇢ argmin f (x)<latexit sha1_base64="AcDqhuFxF7ETuGFtm7gHXQhS7WU=">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</latexit>

x2C* x 2 Fix(T ) =) 8y 2 C, hrf (x), y<latexit sha1_base64="m621Ne1CG0AG5HAuyrwaIYiuVyY=">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</latexit><latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit>f : convex () 8y 2 H, f (y)<latexit sha1_base64="iESN/fTBz6Zwe7RcDlbhGVIycAg=">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</latexit>0f (x) + hy

x, rf (x)if (x) + hy
f (x)() x 2 argmin f (y)x, rf (x)i=)8y 2 C,<latexit sha1_base64="tQoi0WQJ71ueICSJ1brkxX0hjQU=">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</latexit>f (y)<latexit sha1_base64="2IxnTghi7VtGIeqxzKnXokqLBiI=">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</latexit><latexit sha1_base64="iDvyOsXFFHTGnskCao+Laq7RjqA=">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</latexit>

y2C© Meiji University,All rights reserved.

xi

<latexit sha1_base64="lsXiqbi0OgRboLpbb8eOX15QfB4=">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</latexit>

例3 性質 1. の証明

Fix(T ) = argmin f (x)T (x) = PC (xrf (x))<latexit sha1_base64="IhnIVzozC+1kUXjq11xRbt4fyYs=">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</latexit>

x2C<latexit sha1_base64="vWd/NTcqDrt8Tdk2YPPkA5VH3Mw=">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</latexit>

Fix(T )

argmin f (x)
x2C

f: 凸

* x 2 argmin f (y) =) 8y 2 C,<latexit sha1_base64="k1e4uqIWhOPBbKAVIa0GJKl7HVs=">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</latexit><latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit>

y2C<latexit sha1_base64="pWOz5LA6iyic/CqusIvnYChpXnk=">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</latexit>f (x)f (y) + hx=)8y 2 C,y, rf (y)i<latexit sha1_base64="o0IWnyIJWHPqL54CYHR64QrQuQs=">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</latexit>

hy x, rf (y)i=)8y 2 C,

補助定理6.1.7

0<latexit sha1_base64="F5gKA+s+3YEX9azqRX9n7k6yyd8=">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</latexit>© Meiji University,All rights reserved.

hyx, rf (x)i0

<latexit sha1_base64="lsXiqbi0OgRboLpbb8eOX15QfB4=">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</latexit>

例3 性質 1. の証明

Fix(T ) = argmin f (x)T (x) = PC (xrf (x))<latexit sha1_base64="IhnIVzozC+1kUXjq11xRbt4fyYs=">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</latexit>

x2C<latexit sha1_base64="vWd/NTcqDrt8Tdk2YPPkA5VH3Mw=">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</latexit>

Fix(T )

argmin f (x)
x2C

f: 凸

* x 2 argmin f (y) =) 8y 2 C,
y2C
hy x, rf (x)i=)8y 2 C,<latexit sha1_base64="k1e4uqIWhOPBbKAVIa0GJKl7HVs=">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</latexit><latexit sha1_base64="cAMuVYGuSrdr2t9tBySHrA41VhM=">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</latexit><latexit sha1_base64="S6JiKQYiKbyctfroodoJyIQU5pY=">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</latexit>

0<latexit sha1_base64="IEubxZZTg/8+3x/ONHqHLi5/mec=">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</latexit>

hx
rf
(x)
x,
y
xi

0
例1 性質 2.
() x = PC (xrf (x))<latexit sha1_base64="RlitbOB3nWJ8ylAU9aa9AKf6eG4=">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</latexit>() x = T (x)<latexit sha1_base64="8p6m65XZ9sSvRbRsfWgSBr5ZR/Y=">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</latexit>© Meiji University,All rights reserved.