数理解析研究所講究録
第 1544 巻 2007 年 206-211

206

OBSERVATION ON VARIOUS CONJUGATES OFQUASICONVEX FUNCTIONS

島根大学大学院総合理工学研究科 1 鈴木 聡 (Satoshi Suzuki)
島根大学大学院総合理工学研究科 1 黒川真史 (Masafumi Kurokawa)
島根大学総合理工学部 2 黒岩大史 (Daishi Kuroiwa)
1 Interdisciplinary Graduate School of Science and Engineering, Shimane University
2 Interdisciplinary Faculty of Science and Engineering, Shimane University

Abstract

We observe various conjugates and their biconjugacies of quasiconvex
functions. Especially, we give a sufficient condition which assures biconjugacy is satisfied for O-quasiconjugate.

1Introduction], and assume
to.
Throughout this paper, let be a function fromis proper, that is, its domain dom $f=\{x\in \mathbb{R}^{n}|f(x)<\infty\}$ is not empty.
$x_{2}\in domf$ and $\alpha\in(0,1)$ ,Remember that is said to be convex if for all$\mathbb{R}^{n}$$f$$(-\infty, \infty$

$f$

$f$

$x_{1},$$f((1-\alpha)x_{1}+\alpha x_{2})\leq(1-\alpha)f(x_{1})+\alpha f(x_{2})$.

and its Fenchel conjugate function$f^{*}$,

is defined as follows: for any$f^{*}( \xi)=\sup\{\langle\xi, x\rangle-f(x)|x\in domf\}$$\xi\in \mathbb{R}^{n}$,.

] is proper convex lower semicontinuous, and:We know that
also if is lower semicontinuous, then we have$f^{*}$$\mathbb{R}^{n}arrow(-\infty, \infty$$f$$f=f^{**}$,

that is, lower semicontinuity and the biconjugacy are equivalent for any proper
convex function. It is well-known that this property plays very important roles
to consider dual problems of convex minimization problem.
Similar researches of conjugates of quasiconvex functions, have been observed,
see [1, 2, 3, 4, 5]. Various types of conjugates are introduced, and biconjugacies
of functions are investigated. In this paper, we give a sufficient condition whichassures biconjugacy is satisfied for a notion of conjugate, called O-quasiconjugate.

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2Conjugates of quasiconvex functions
Remember that$f$is said to be quasiconvex if, for all$x_{1},$$x_{2}\in domf$$f((1- \alpha)x_{1}+\alpha x_{2})\leq\max\{f(x_{1}), f(x_{2})\}$

or equivalently, for any$\alpha\in \mathbb{R}$and $\alpha\in(0,1)$ ,,, its level set$L_{\alpha}(f)=\{x\in \mathbb{R}^{n}|f(x)\leq\alpha\}$is a convex set. Clearly, the notion of quasiconvexity is a generalization of convexity. For quasiconvex functions, various conjugates have been defined. In thispaper, we treat -quasiconjugate, quasiconjugate, and R-quasiconjugate. At first,
we mention about -quasiconjugate.
$\lambda$$\lambda$, the -quasiconjugate ofDefinition 1. For any,defined as follows: for any$\lambda\in \mathbb{R}$is the functional$f$$\lambda$$f_{\lambda}^{\nu}$:$\xi\in R^{n}$$\mathbb{R}^{n}arrow(\infty, \infty]$$f_{\lambda}^{\nu}(x^{*})= \lambda-\inf\{f(x)|\langle x^{*},x\rangle\geq\lambda\}$.

By Greenberg and Pierskalle, the normalized second quasiconjugate is introduced. Note that this notion is not given by two-times iteration of the same
operation.

Deflnition 2. The normalized second quasiconjugate ofdefined as follows: for any$\mathbb{R}^{n}arrow(-\infty, \infty]$$f$is the functional .

$f^{\nu\nu}$:$x\in \mathbb{R}_{f}^{n}$$f^{\nu\nu}(x)= \sup_{\lambda\in R}(f_{\lambda}^{\nu})_{\lambda}^{\nu}(x)$.

Evenly quasiconvexity, defined as follows, assures biconjugacy.
is evenly convex if there exists a family of open
is equal to the intersection of the family of open half space.

Definition, 3. A subset $A$ ofhalf space such that$A$

Definition 4. A function
is evenly convex.

$f$$\mathbb{R}^{n}$is evenly quasiconvex if for all$\alpha\in(-\infty, \infty$],$L_{\alpha}(f)$Theorem 1. If a function $f$ is evenly quasiconvex, then $f^{\nu\nu}=f$ .

Theorem 2. The followzng formula holds:.

$f^{\nu\nu}= \max\{(f_{-1}^{\nu})_{-1}^{\nu}, (f_{0}^{\nu})_{0}^{\nu}, (f_{1}^{\nu})_{1}^{\nu}\}$Next, we define notions of quasiconjugate and R-quasiconjugate, which areclosely concerned withand we state sufficient conditions to obtainthat each biconjugates are equal to .
$(f_{-1}^{\nu})_{-1}^{\nu},$$(f_{1}^{\nu})_{1)}^{\nu}$

$f$

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Definition 5 ([4]). Quasiconjugate of $f$ is the functional
defined by:$f^{H}$$\mathbb{R}^{n}arrow(-\infty, \infty$]$f^{H}(\xi)=\{\begin{array}{ll}-\inf\{f(\xi)|\langle x,\xi\rangle\geq 1\} if \xi\neq 0-\sup\{f(x)|.x\in \mathbb{R}^{n}\} if \xi=0.\end{array}$

The quasiconjugate of the function $f^{H}is$ called the biquasiconjugate ofdenoted by $f^{HH}$ .

Note that $f_{1}^{\nu}=1-f^{H}$ on$\mathbb{R}^{n}\backslash \{0\}$Definition 6. We say that$f(x_{n}) arrow\sup\{f(x)|x\in \mathbb{R}^{n}\}$Theorem 3. Let$f$$fachie_{\backslash }ves$and.

achieves the maximum value at the infinite if
such that 1.
for any sequence$f$$\{x_{n}\}$$x_{k}\Vertarrow\infty$

be a lower semicontinuous quasiconvex function satishing$f( O)=\inf\{f(x)|x\in\cdot \mathbb{R}^{n}\backslash \{0\}\}$

If

$f$.

the maximum value at the infinite, then $f^{HH}=f$ .

Definition 7. R-quasiconjugate ofdefined by$f$is the $fi_{4}nctionalf^{R}$ :$f^{R}( \xi)=-\inf\{f(x)|\langle\xi,x\rangle\geq-1\}$

The R-quasiconjugate of the function
and denoted by.

$f^{R}$$\mathbb{R}^{n}arrow(-\infty, \infty$].

is called the R-biquasiconjugate of$f$$f^{RR}$

Note that $f_{-1}^{\nu}=-1-f^{R}$ on$\mathbb{R}^{n}$.

Definition 8. A subset ofis R-evenly convex if the intersection of a familyof open half spaces which closure do not contain .
$A$$\mathbb{R}^{n}$$0$

Deflnition 9. A function].
for all$f$

is R-evenly quasiconvex if,$\cdot$$L_{\alpha}(f)$is R-evenly convex$\alpha\in(-\infty, \infty$

Theorem 4. If a function

3$f$is R-evenly quasiconvex, then $f^{RR}=f$ .

Main theorem

Motivated by Theorems 1, 2, 3, and 4 in the previous section, we consider biconjugacy for O-quasiconjugate.
Example 1. Let :calculate conjugate$f$$\mathbb{R}^{2}arrow \mathbb{R}$defined by $f(x,y)=(x-1)^{2}+y^{2}$ . Then we can$f_{0}^{\nu}(a, b)=\{\begin{array}{ll}-\frac{a^{2}}{a^{2}+b^{2}} \dot{j}fa<00 lfa\geq 0.\end{array}$

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Figure; graph and contour graph of functionLet$g=f_{0}^{\nu}$, then we have the conjugate$g_{0}^{\nu}$$f_{0}^{\nu}$on$(-\infty, 0)\cross \mathbb{R}$on$(0, \infty).x\mathbb{R}$.

as follows:$g_{0}^{\nu}(x, y)=\{\begin{array}{ll}\frac{y^{2}}{x^{2}+y^{2}} if x>01 if x\leq 0.\end{array}$Figure: graph and contour graph of jfunction$fi\}vm$

this, we have$(f_{0}^{\nu})_{0}^{\nu}\neq f$. However, we can show

$g_{0}^{\nu}$$(g_{0}^{\nu})_{0}^{\nu}=.g$.

.

Inspired the example, we give a sufficient condition for biconjugacy. To thepurpose, we show the following properties concerned with convex cone.

Lemma 1. If $K$ be a nonempty closed convex pointed cone in
not empty, where.

$\mathbb{R}^{n}$, then intK“$is$$K^{*}=\{x^{*}\in \mathbb{R}^{n}|\langle x^{*}, x\rangle\leq 0,\forall x\in K\}$This proof is omitted. The following lemma is important to show our main result.

210.
Lemma 2. Let $K$ be a nonempty closed convex pointed cone insuch thatthen there exists
for all$\mathbb{R}^{n}$$x_{0}\not\in$$x\in K\backslash \{0\}$$\langle a, x_{0}\rangle\geq 0>\langle a, x\rangle$$a\in \mathbb{R}^{n}$$K\backslash \{0\}_{f}$If.

Lemma 1 and $K=K^{**}$ .
. By continuity of thesuch that, there exists
Since.
inner product, we can choose $r>0$ such that $y^{*}\in B(x^{*}, r)$ implies“, and let
Choose $z^{*}\in intK$ such that

Proof. From the assumption, we have intK*$x^{*}\in K^{*}$$x_{0}\not\in K^{**}$$\neq\emptyset.from$$\langle x_{0},x^{*}\rangle>0$$\langle x_{0}, y^{*}\rangle>0$$z^{*}\neq x^{*}$$a= \frac{||x^{*}-z^{*}\Vert}{\Vert x^{*}-z^{*}\Vert+r}x^{*}+\frac{r}{\Vert x^{*}-z^{*}\Vert+r}z^{*}$,

then we can check $a\in B(x^{*},r)$ and $a\in intK$“. Hence we have (;for all$\langle a,x\rangle<0$$a,x_{0}\rangle$$>0$ and口$x\in K\backslash \{0\}$$\cdot$Lemma 3. The following formula holds$f_{0}^{\nu}(0)= \max\{f_{0}^{\nu}(x^{*}).|.x^{*}\in \mathbb{R}^{n}\}$Proof. For any$x^{*}\in \mathbb{R}^{n}$.

,$f_{0}^{\nu}(x^{*})=- \inf\{f(x)|\langle x^{*}, x\rangle\geq 0\}$,and also$f_{0}^{\nu}(0)=- \inf\{f(x)|\langle 0,x\rangle\geq 0\}=-\inf\{f(x)|\in x\in \mathbb{R}\})$

then we have$f_{0}^{\nu}(x^{*})\leq f_{0}^{\nu}(0)$.

口,is a closed convex pointed cone, orTheorem 5. Assume that.
. If $f( O)=\sup\{f(x)|x\in \mathbb{R}^{n}\backslash \{0\}\}$ , thenfor all
such that. Assume that there exists
Proof. It is clear that
. By using Lemma 3 and $f( O)=\sup\{f(x)|x\in \mathbb{R}^{n}\backslash \{0\}\}$ , wesatisfying. Choose
may assume$L_{\alpha}(f)\cup\{0\}$$\mathbb{R}^{n}$$\alpha\in \mathbb{R}$$f=(f_{0}^{\nu})_{0}^{\nu}$$x_{0}\in \mathbb{R}^{n}$$f\geq(f_{0}^{\nu})_{0}^{\nu}$$f(x_{0})>(f_{0}^{\nu})_{0}^{\nu}(x_{0})$$\alpha\in \mathbb{R}^{n}$$x_{0}\neq 0$$f(x_{0})>\alpha>(f_{0}^{\nu})_{0}^{\nu}(x_{0})$Since$x_{0}\not\in L_{\alpha}(f)\cup\{0\}$, and$\exists a\in \mathbb{R}^{n}$$L_{\alpha}(f)\cup\{0\}is$s.t.

a closed convex pointed cone, then$\langle a, x_{0}\rangle\geq 0>\langle a,x\rangle,\forall x\in L_{\alpha}(f)$by using Lemma 2. This shows$x\in L_{\alpha}(f)$$\Rightarrow$$\langle a, x\rangle<0$$\langle a,x\rangle\geq 0$$\Rightarrow$$f(x)>\alpha$or equivalently,,.

Hence.
,$(f_{0}^{\nu})_{0}^{\nu}(x_{0})=- \inf\{f_{0}^{\nu}(x^{*})|\langle x^{*}, x_{0}\rangle\geq 0\}\geq-f_{0}^{\nu}(a)=\inf\{f(x)|\langle x, a\rangle\geq 0\}\geq\alpha$this is contraction.

$\square$211

References[1] H. J. Greenberg and W. P. Pierskalle, Quasi-conjugate functions and surrogate duality, $Cah$ . Cent. Etud. Rech. Oper 15 (1973) 437-448.

[2] J.-E. Martinez-Legaz, A generalized concept of conjugation, Optimization:
theory and algorthms (Confolant, 1981), 45-59, Lecture Notes in Pure andAppl. Math., 86, Dekker, New York, 1983.
[3] I. Singer, The lower semi-continuous quasi-convex hull as a normalized

second conjugate, Nonlinear Analysis, Theow Methods(1983) 1115-1121.

$\emptyset$

Applications 7[4] P. T. Thach, Quasiconjugates of functions, duality relationship between
quasiconvex minimization under a reverse convex constraint and quasiconvex maximization under a convex constraint, and aPplications, JMAA 159(1991) 299-322.
[5] P. T. Thach, Diewert-Crouzeix conjugation for general quasiconvex duality
and applications, $JO$ TA 86 (1995) 719-743.