# Theoretical Aspect of Diagonal Bregman Proximal Methods

Authors: S. Kabbadj
Journal: Journal of Applied Mathematics
 (2020)

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## Abstract

In this paper, we propose and study a diagonal inexact version of Bregman proximal methods, to solve convex optimization problems with and without constraints. The proposed method forms a unified framework for existing algorithms by providing others.


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## Body

## 1. Introduction

Letfi:Rd⟶R i=0,1,…,m convex functions and C the nonempty subset of Rd are defined by(1)C=x∈Rd:fix≤0,i=1,…,m.Let us consider the problem of convex optimization:(2)P:minf0x,x∈C.To solveP, many authors [1–7] have combined the exterior penalty methods with the proximal method (PM) defined by(3)xk∈εk−Argminf⋅+12λk⋅−xk−12,where f∈Γ0Rd is set of proper closed convex functions on Rd. PM and its variants have been studied by several authors [6, 8–13]. In this labor, we generalize this process by introducing Bregman’s distance Dh.,. defined by(4)Dhx,y≔hx−hy−x−y,∇hy,where h is Bregman’s function [14].In order to solveP, we study the coupling of the methods of the exterior penalty with the diagonal inexact version of the Bregman proximal methods defined by(5)xk∈εk−Argminf⋅+λk−1Dh⋅,xk−1.The exact version PMD is defined by(6)xk=argminf⋅+λk−1Dh⋅,xk−1,has been studied by several authors [15–18].We propose and study a diagonal inexact version of the Bregman proximal method, which we call DBPM, defined by(7)xk∈εk−Argminfk⋅+λk−1Dh⋅,xk−1,where the sequence fkk⊂Γ0Rd is given and approaches f.By introducing the penalty functions in DBPM, we deduce a solution ofP.Iffk=f∀k, the proposed method appears as an inexact version of (6) and solves the problem of convex optimization without constraints:(8)P′:minfx,x∈Rd.Forh⋅=1/2⋅2, DBPM coincides with diagonal proximal method of Alart and Lemaire [1] as well as the penalization method given by Auslender [2].


## 2. Preliminary

In this section, we remind some theoretical properties of the approximations called entropic studied by Kabbadj in [17]. This study covers the properties of regularity and approximations of the Moreau–Yosida approximations [19]. These results are necessary for the analysis of the methods proposed in Section 3.Let S be an convex open subset ofRd andh
:
S
¯
⟶
R
. We define Dh.,. by(9)∀x∈S¯,∀y∈S:Dhx,y≔hx−hy−x−y,∇hy.Let us consider the following hypotheses:H1: h is continuously differentiable on S.H2: h is continuous and strictly convex on S¯.H3: ∀r≥0,∀x∈S¯,∀y∈S, the sets L1x,r and L2y,r are bounded where(10)L1x,r=y∈S/Dhx,y≤r,L2y,r=x∈S¯/Dhx,y≤r.(i)
H4: if ykk⊂S is such that yk⟶y∗∈S¯, then,(11)Dhy∗,yk⟶0.H5: if xkk and ykk are two sequences of S such that Dhxk,yk⟶0 and xk⟶x∗∈S, then(12)yk⟶x∗.Definition 1.
(i)
h:S¯⟶R is a Bregman type function on S or “D-function” if h verifies H1,H2,H3,H4, and H5.
(ii)
D
h
.,. is called entropic distance if h is a Bregman function.
We put
A (S) = {h:S¯⟶R verifying H1 and H2}
B (S) = {h:S¯⟶R verifying H1,H2,H3,H4, and H5}.Theorem 1 (see [17]).
Letf∈Γ0Rd and h∈AS such that domf∩S¯≠ϕ.
If one of the two following conditions is verified,(i)
infS¯f>−∞ and h verifies H3(ii)
Im∇h=Rd,
then for allx∈S and for all λ>0, the function u⟶fu+λ−1Dhu,v has a unique minimum point on S¯.Definition 2.
f and h verify the hypothesis of Theorem 1.(i)
The entropic approximation off compared to h, of parameter λλ>0, is the function defined by(13)fhλx≔infy∈S¯fy+λ−1Dhy,x,∀x∈S.(ii)
The application entropic proximal off comparing to h, of parameter λ, is the operator defined by(14)hλfx≔proxλfhx≔argminy∈S¯fy+λ−1Dhy,x,∀x∈S.Proposition 1 (see [17]).
Leth∈AS and f∈Γ0Rd such that(a)
ri (domf) ∩S≠ϕ(b)
Im∇h=Rd
Then,∀x∈S,∀λ>0.(15)hλfx∈S,(16)infSfhλ=infSf,(17)∇hx−∇hhλfxλ∈∂fhλfx,(18)fhλx≤fhμx≤fx,∀μ:0<μ≤λ.Proposition 2 (see [17]).
We suppose thath and f verify the conditions of Proposition 1.Ifinff>−∞ and h verify H3, then hλf:S⟶S is a continuous application.Proposition 3 (see [17]).
We suppose thath and f verify the hypothesis of Proposition 2.Ifh is twice continuously differentiable on S and Dh.,. and jointly convex, then fhλ is continually differentiable and convex such that ∀x∈S:(19)∇fhλx=λ−1Hxx−hλfx,where  H=∇2h.Proposition 4.
We suppose thath and f verify the hypothesis of the Proposition 3. If H is defined positive, then(20)ArgminSf=ArgminSfhλ.Proof.
Letu∗∈ArgminSfhλ.(21)fhλu∗=infSfhλ⟺0∈∂fhλu∗⟺0=∇fhλu∗⟺Hu∗u∗−hλfu∗.
SinceH is defined positive, we deduct then that u∗=hλfu∗. From (17), we have(22)u∗=hλfu∗⟹0∈∂fu∗⟹u∗∈argminSf.
We get thenArgminSfhλ⊂ArgminSf.
Reciprocally, letx∗ such that fx∗=infSf. From (16) and (18), we have(23)fx∗=infSfhλ≤fhλx∗≤fx∗;thus, we have fx∗=infSfhλ=fhλx∗, which completes the demonstration.
Some examples of Bregman functions are given below.Example 1.
IfS0=Rd and h0x=1/2x2, then(24)Dh0x,y=12x−y2.Example 2.
IfS1=R++d≔x∈Rd/xi>0,i=1,…,d and(25)h1x=∑i=1i=dxilogxi−xi;∀x∈S¯1,with the convention 0log0=0, then(26)Dh1x,y=∑i=1dxilogxiyi+yi−xi,∀x,y∈S¯1XS1.Example 3.
IfS2=−1,1d and h2x  = −∑i=1i=d1−xi2, then(27)Dh2x,y=h2x+∑i=1d1−xiyi1−yi2,∀x,y∈S2¯XS2.
We easily verify thathi∈BSi,i=0,1,2.


## 3. Analysis of the Diagonal Bregman Proximal Method

In this paragraph, we assume the following:(A):h∈BS:Im∇h=Rd and domf¯⊂S(B):f,fk∈Γ0Rd:domfk¯⊂S, k=1,2,…(C): lim infinffk>−∞From (15), we can then construct the sequence xkk defined by (Algorithm 1):Algorithm 1:DPMD.
(1)
Input:x0∈S
(2)
Chooseλ¯≥λk≥λ¯>0 and εk≥0, and find xk∈S, such that xk∈εk−Argminfk⋅+λk−1Dh⋅,xk−1.
(3)
Setk⟵k+1 and go to step 2In what follows, we will derive a convergence result (Theorem2) for the DPMD framework. First, we need to establish a few technical results.Lemma 1 (see [20]).
Letf1,f2 be two functions of Γ0Rd if there exists x¯∈domf1 in which f2 is finite and continuous, then for ε>0, for all y∈domf1∩domf2,(28)∂εf1+f2y=∪ε1+ε2=ε,ε1≥0,ε2≥0∂ε1f1y+∂ε2f2y.Definition 3.
The sequenceλk;ak;bk;ck;dkk∈R+∗XS4 verifies the K-property only if the following properties are verified:K1:∃λ¯>0,∀k,λk≥λ¯.K2:ak is bounded and Adhak⊂S.K3:Dhak,bk⟶0.K4:Dhak,ck⟶0.K5:dk=∇hbk−∇hck/λk.Lemma 2.
If the sequenceλk;ak;bk;ck;dkk verifies the K-property, then dk⟶0.Proof.
If the sequencedk does not tend to zero, then it exists that M>0 and the subsequence dki of dk such that(29)∀ki,dki>M.
The sequenceaki is bounded and Adh aki⊂S; it exists that the subsequence aki of aki and u∗∈S such that akj⟶u∗. Dhakj,bkj⟶0 and Dhakj,ckj⟶0 allow to write, from H5, bkj⟶u∗ and ckj⟶u∗. On the other hand,(30)0≤dkj=∇hbkj−∇hckjλkj≤λ¯−1∇hbkj−∇hckj,∇h is continuous on S, then ∇hbkj−∇hckj⟶0. It follows that dkj⟶0. dki is a subsequence of dki, from with the entropic proximal method (29), we have 0≥M>0, so dk⟶0.
Lets consider now the functionhu,λ defined by hu,λ:S¯⟶R, ∀λ>0,∀u∈S.(31)hu,λx=λ−1Dhx,u,∀x∈S¯.Proposition 5.
∀
ε
>
0
,
∀
λ
>
0
,
∀
u
∈
S
,
∀
x
∗
∈
S
¯
.
(32)
∂
ε
h
u
,
λ
x
∗
=
z
z
=
∇
h
x
¯
−
∇
h
u
λ
with
x
¯
∈
S
and
D
h
x
∗
,
x
¯
≤
λ
ε
.Proof.
z
∈
∂
ε
h
u
,
λ
x
∗.(33)⟺hu,λx−hu,λx∗≥z,x−x∗−ε,∀x∈S¯,⟺λ−1hx−hu−x−u,∇hu−hx∗+hu+x∗−u,∇hu≥z,x−x∗−ε,∀x∈S¯,⟺hx∗−hx−x∗−x,∇hu≤λz,x∗−x+λε,∀x∈S¯,which is equivalent to(34)hx∗−hx−x∗−x,∇hu+λz≤λε.
According toA, it exists that x¯∈S such that(35)∇hu+λz=∇hx¯,which means(36)z=∇hx¯−∇huλ∃x¯∈S.
Replacing in (34) x by x¯, we get(37)Dhx∗,x¯≤λε.
Finally(38)∂εhu,λx∗⊂zz=∇hx¯−∇huλwithx¯∈SandDhx∗,x¯≤λε.
Conversely, letz such as(39)z=∇hx¯−∇huλ,Dhx∗,x¯≤λε,Dhx∗,x¯≤λε⟹hx∗−hx¯−x∗−x¯,∇hx¯≤λε≤λε+Dhx,x¯,⟹hx∗−hx¯−x∗−x¯,∇hx¯−hx+hx¯+x−x¯,∇hx¯≤λε,⟹hx∗−hx−x∗−x,∇hx¯≤λε.
Replacing∇hx¯ by ∇hu+λz, we get (34). According to what precedes,(40)34⟺z∈∂εhu,λx∗,which establishes the desired equality.Definition 4.
(41)
∀
λ
>
0
,
∀
ρ
≥
0
,
∀
f
,
g
∈
Γ
0
R
d
,
d
h
,
λ
ρ
f
,
g
≔
sup
x
≤
ρ
,
x
∈
S
f
h
λ
x
−
g
h
λ
x
.Theorem 2.
We assume that(i)
∑kεk+2dk,ρ<+∞,∀ρ≥0, where(42)dk,ρ≔dh,λ¯ρfk,f.(ii)
The sequencexk generated by DPMD is bounded.Then(a)
fkxk⟶inff(b)
Moreover, iff and h verify the conditions of Proposition 4, then(43)Adhxk⊂Argminf.Proof.
(44)
x
k
∈
ε
k
−
Arg
min
f
k
u
+
λ
k
−
1
D
h
u
,
x
k
−
1
⟹
f
k
x
k
+
λ
k
−
1
D
h
x
k
,
x
k
−
1
≤
f
h
λ
k
k
x
k
−
1
+
ε
k
,according to (18), we can write(45)fhλ¯kxk+λk−1Dhxk,xk−1≤fhλ¯kxk−1+εk.
The sequencexk is bounded; let ρ≥0 such that(46)∀k,xk≤ρ.
Considering (45),(47)λk−1Dhxk,xk−1+fhλ¯xk≤fhλ¯xk−1+εk+2dk,ρ.
Therefore,(48)fhλ¯xk≤fhλ¯xk−1+εk+2dk,ρ.
So, from (i), we have(49)limfhλ¯xk=l≥inffhλ¯=inff≥−∞.On one hand,(50)fkxk≤fhλ¯kxk−1+εk⇓fkxk≤fhλ¯xk−1+dk,ρ+εk,on the other hand, we have(51)fhλ¯xk−dk,ρ≤fkxk;finally, the two previous inequalities make it possible to write(52)fhλ¯xk−dk,ρ≤fkxk≤fhλ¯xk−1+dk,ρ+εk.
Ifl=−∞, then inff=−∞ and fkxk⟶−∞. So,(53)fkxk⟶inff.
Ifl>−∞, then, from (52),(54)limfkxk=limfhλ¯xk=l.
Let us show thatl=inff, from (45),
λ
k
−
1
D
h
x
k
,
x
k
−
1
⟶
0 when k⟶+∞.
Asλ¯≤λk≤λ¯, we have(55)Dhxk,xk−1⟶0.
On the other hand,(56)xk∈εk−Argminfku+λk−1Dhu,xk−1⟺0∈∂εkfk⋅+λk−1Dh⋅,xk−1xk.
From Lemma1, there exists εk1,εk2≥0 such that εk1+εk2=εk and(57)0∈∂εk1fkxk+∂εk2λk−1Dh⋅,xk−1xk.
Since∂εfk increases with ε, we have(58)0∈∂εkfkxk+∂εkλk−1Dh⋅,xk−1xk.
Therefore, there existszk∈∂εkfkxk such that(59)−zk∈∂εkλk−1Dh⋅,xk−1xk.
From Proposition5, there exits x¯k∈S such that(60)−zk=∇hx¯k−∇hxk−1λk,Dhxk,x¯k≤λkεk.
Finally, there existsx¯k such that(61)zk=∇hxk−1−∇hx¯kλk∈∂εkfkxkwithDhxk,x¯k≤λkεk.
From (55) and (61), we have(62)zk=∇hxk−1−∇hx¯kλk∈∂εkfkxk,Dhxk,x¯k⟶0,Dhxk,xk−1⟶0.
Since Adhxk⊂∪domfk¯⊂S, the sequence λk;xk;x¯k;xk−1;zkk verifies then the K-property. From Lemma 2, zk⟶0. On the other hand, for all y∈S,(63)fky≥fkxk+zk,y−xk−εk⟹fky+λ¯−1Dhy,x≥fkxk+zk,y−xk−εk,there exists x^k such that for all k∈N,(64)inffky+λ¯−1Dhy,x=fhλ¯kx=fkx^k+λ¯−1Dhx^k,x.
By replacingy by x^k in (63), we get(65)fkx^k+λ¯−1Dhx^k,x≥fkxk+zk,x^k−xk−εk.
It is still(66)fhλ¯kx≥fkxk+zk,x^k−xk−εk.x^k is bounded. Indeed,(67)−∞<infkinfSfk≤fkx^k,so it exists K1∈R such that K1≤fkx^k.
From (i),(68)dk,ρ⟶0⟹fhλ¯kxis convergent,so(69)∃K2∈R:K2≥K1,fhλ¯kx≤K2.
From (64), we have(70)Dhx^k,x≤λ¯K2−K1.
FromH3, x^k is bounded. Going to the limit in (66), we have(71)fhλ¯x≥limfkxk=l,then,(72)fx≥l,∀x⟹inff≥l,52⟹fhλ¯xk≤fkxk+dk,ρ,⟹inff=inffhλ¯≤fkxk+dk,ρ,⟹inff≤limfkxk=l.
Finally, we have(73)limfkxk=inff.
(b) Letx∗∈Adhxk, there exists then the subsequence xki of xk such that xki⟶x∗, we have(74)inffhλ¯≤fhλ¯x∗≤limfhλ¯xki≤limfkixki=inff=inffhλ¯⟹inffhλ¯=fhλ¯x∗.
From (20), we have(75)x∗∈Argminfhλ¯=Argminf.


## 4. Exterior Penalty Coupled with Bregman Proximal Method

Letfi:Rd⟶R,i=1,…,m, be the convex function and let C the set of constraints given by(76)C=x∈Rd:fix≤0,i=1,…,m.We suppose that C verifies the condition of Slater:(77)∃x∗∈Rd:fix∗<0,i=1,…,m.Let us consider the functions of the linear penalty defined by(78)φ1nx=rn∑i=1mfi+x,∀x∈Rd,∀n∈N∗,and the quadratic exterior penalty defined by(79)φ2nx=rn∑i=1mfi+x2,∀x∈Rd,∀n∈N∗,where a+ = max 0,a and rnn is an increasing sequence of strictly positive real numbers which tends to +∞.Let us put∀n∈N∗:(80)f=f0+ΨC,f1n=f0+φ1n,f2n=f0+φ2n.In what follows, we assume(A′): h∈BRd:Im∇h=Rd(B′): infRdf>−∞,so conditions (A), (B), and (C) of Section 3 are verified for f and fjn, j = 1, 2; n∈N.We give below an estimate ofdλ,ρhfjn,f, j = 1, 2.Proposition 6.
(a)
∀
λ
¯
>
0
,
∀
ρ
≥
0
,
∃
r
h
≥
0
,
(81)
d
λ
,
ρ
h
f
1
n
,
f
=
0
,
∀
n
:
r
n
≥
r
h
,
∀
λ
≥
λ
¯
.
(b)
∀
λ
¯
>
0
,
∀
ρ
≥
0
,
∃
μ
h
≥
0
,
(82)
d
λ
,
ρ
h
f
2
n
,
f
≤
μ
h
r
n
,
∀
n
≥
1
,
∀
λ
≥
λ
¯
.Proof.
Letλ>0 and x∈Rd.(83)fhλx=inff+λ−1Dh⋅,x=infCf0+λ−1Dh⋅,x.
Since Slater’s condition is verified, there exists from Ekeland-Temam [21] (chap 3, Theorem 5.2) multiplicators of Lagrange(84)pjλ,x≥0;j=1,…,m,
such that∀y∈Rd.(85)fhλx≤f0y+λ−1Dhy,x+∑j=1mpjλ,xfjy.
From (18), we have fhλx≥infCf0, by replacing y with x∗, we obtain(86)infCf0≤f0x∗+λ−1Dhx∗,x+∑j=1mpjλ,xfjx∗,where x∗ verifies (3). On the other hand(87)∑j=1mpjλ,xfjx∗≤∑j=1mpjλ,xsupjfjx∗.
Let us put(88)supjfjx∗=−c,wherec>0.
It follows that(89)∑j=1mpjλ,xfjx∗≤−cpλ,x1,where pλ,x=p1λ,x,…,pmλ,x. Therefore,(90)Dhx∗,x≤Dhx∗,x+Dhx,x∗=x∗−x,∇hx∗−∇hx≤x∗−x⋅∇hx−∇hx∗,which leads to(91)Dhx∗,x≤x∗+x∇hx∗+∇hx.
From (86), (89), and (91), we obtain(92)pλ,x1≤1cf0x∗+λ¯−1x∗+x∇hx∗+∇hx−infCf0.(a)
From (85),(93)fhλx≤f0y+λ¯−1Dhy,x+pλ,x1∑i=1mfi+y.Forx∈Bρ, from (92), we have(94)pλ,x1≤1cf0x∗+λ¯−1x∗+ρ∇hx∗+supx∈Bρ∇hx−infCf0≕rh.Thus, forn such as rn≥rh,(95)fhλx≤f1ny+λ¯−1Dhy,x,∀y∈Rp⇓fhλx≤f1nhλx,∀x∈Bρ.Conversely,(96)f1nhλx=infy∈Rpf0y+rn∑i=1mfi+y+λ−1Dhy,x≤infy∈Cf0y+λ−1Dhy,x=inffy+λ−1Dhy,x.Therefore,(97)f1nhλx≤fhλx.(b)
If⋅m indicates the Euclidean norm on Rd, so(98)∑j=1mpjλ,xfjx≤12rnpλ,xm2+rn2∑i=1mfi+y2;by proceeding as below, we deduce the result.
Let us consider now the methods of the coupled exterior penalties with the entropic proximal method (Algorithm2):Algorithm 2: (DBPM)j, j = 1,2.
(1)
Input:xj0∈Rd
(2)
Chooseλ¯≥λn≥λ¯>0 and εn≥0, and find xjn∈S, such that xjn∈εn−Argminfjn⋅+λn−1Dh⋅,xjn−1.
(3)
Setn⟵n+1 and go to step 2Theorem 3.
Let us suppose(i)
∑n=1∞εn<+∞, and ∑n=1∞1/rn<+∞(ii)
∃i0≥1:fi0 is coercive
Then the sequencexjnn generated by DBPMj j = 1, 2 is bounded, Adhxjn⊂Argminf, and(99)fjnxjn⟶inff,j=1,2….Proof.
Let us show thatx1nn is bounded.(100)f1nx1n+λn−1Dhx1n,x1n−1≤f1nu+λn−1Dhu,x1n−1+εn.
By replacingu by x1n−1 in (100), we obtain(101)f0x1n+rn∑i=1mfi+x1n≤f0x1n−1+rn∑i=1mfi+x1n−1+εn.
LetR =  infRdf0, we have(102)f0x1n−Rrn+∑i=1mfi+x1n≤f0x1n−1−Rrn+∑i=1mfi+x1n−1+εnrn,rn is increasing, so we put(103)An=f0x1n−Rrn+∑i=1mfi+x1n,we deduce(104)An≤An−1+εnrn.which leads to(105)An≤A0+∑k=1nεkrk.
Letε such as εk≤ε, since f0x1n−R/rn>0, we have(106)fi0x1n≤A0+∑k=1∞εrk.
From (i) and (ii), we deduce that the sequencex1nn is bounded, so by application of Theorem 2 and Proposition 6, the result is immediate.
In a similar way, we deduce the result forj = 2.


## 5. Example

Let us consider the following optimization problem:(107)P1:mina,x,x2≤b,where a∈Rd and b∈R+∗.TheDBPM2 algorithm can be applied to solve (P1). We take(108)∀n,εn=0,rn=2n,λ¯≥λn≥λ¯>0.Let us consider the functionh: Rd⟶R defined by(109)hx=1p∑i=1i=dxip;p≥2;∀x∈Rd.We easily show thath∈BRd and which checks (A′)Let us put∀x∈Rd:(110)f0x=a,x,f1x=x2−b,φnx=φ2nx=rnf1+x2,∀n∈N∗.C=x∈Rd:f1x≤0 is compact and f0 is continuous, so(111)Argminf≠∅,infRdf=infCf0>−∞.We have(112)f=f0+ΨC,fn≔f2n=f0+φn.The sequencexn generated by the DBPM2 algorithm is defined by x0∈Rd and(113)xn∈Argminfn⋅+λn−1Dh⋅,xn−1,n≥1.By writing the condition of optimality, we have(114)∇fnxn+λn−1∇hxn−∇hxn−1=0.On the other hand,(115)∇f1+2x=2f1+x∇f1x=4f1+xx.Then,(116)2n+2f1+xnxn+λn−1∇hxn=λn−1∇hxn−1−a,where(117)∇hxni=signxinxinp−1,i=1,…,d,f1 is coercive, so by applying Theorem 3, we have(118)Adhxn⊂Argminf=x∈Rd:f0x=infCf0,fnxn⟶infCf0.Remarks 1.
(i)
The convergence performance of thexn can be discussed according to the parameter p. We take note that for p=2,
(119)
h
x
=
1
2
x
2
.
(ii)
Let
(120)
f
0
x
=
1
2
<
A
x
,
x
>
−
<
c
,
x
>
,
∀
x
∈
R
d
,where A is matrix symmetric definite positive and c∈Rd. Previously developed methods can solve optimization problems of the type(121)P2:min12Ax,x−c,x,x2≤b.


## 6. Conclusion

The class of the methods studied in this work constitutes a unified framework for several existing methods that solve convex optimization problems with and without constraints while providing others, more precisely.(i)
Forh⋅=1/2⋅2, DBPM coincides with the diagonal proximal method DPM studied by Alart and Lemaire [1].(ii)
Ifh⋅=1/2⋅2 in DBPM, j = 1, 2, we find then the methods of penalization studied by Auslender [2].(iii)
Iffk=f∀k, DBPM appears as an inexact version of BPM and solves the problem of convex optimization without constraints:(122)P′:minfx,x∈Rd,the convergence of this version is included in our analysis and responds to the question asked by Eckstein in [15].(i)
Iffk=f and εk=0,∀k in DBPM, we find then BPM studied by [15–18].(ii)
Iffk=f and h⋅=1/2⋅2 in DBPM, we find then PM studied by [6, 8–13].(iii)
Iffk=f and h1x=∑i=1i=dxilogxi−xi; ∀x∈S1¯, DBPM allows to minimize f on(123)S1=R++d≔x∈Rd/xi>0,i=1,…,d.



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*Source: 3108056-2020-01-17.xml*