and Practical Solutions) Citation 数理解析研究所講究録 (2005), 1461:

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(2005), 1461: 40-46 Issue Date 2005-12 URL http://hdlhandlenet/2433/47968

1 The existence of zeros of monotone Titleoptimization problems(mathematics o and Practical Solutions) Author(s) 松下, 慎也 ; 高橋, 渉 Citation 数理解析研究所講究録 (2005), 1461: Issue Date URL Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

2 The existence of zeros of monotone operators concerning optimization problems ( ) Shin-ya Matsushita ( )*, Wataru Takahashi ( ) \dagger Department of Mathematical and Computing Sciences, Tokyo Institute of Technology ( ) 1 $E$ Banach $f,$ $(\mathrm{i}=1,, m)$ $Earrow \mathbb{r}$ $g_{i}$ : (11) : $f(x)$ $arrow$ (11) : $g_{i}(x)\leq 0(\mathrm{i}=1, \ldots m)7^{\cdot}$ $C=\{x\in E : g_{i}(x)\leq 0(\mathrm{i}=1, \ldots, m)\}$ (11) (1 1) approximate gradient Martinez-Svaiter [7] projection condition Yokoyama-Shiraishi [14] Slater \epsilon \epsilon Karush-Kuhn-Tucker [2], [6], [13] $H$ Hilbert $f$ : $Harrow(-\infty, \infty]$ proper $r_{n}>0$ $x_{0}=x\in H$ $x_{n+1}= \arg\min_{y\in H}\{f(y)+\frac{1}{2r_{n}} y-x_{n} ^{2}\}(n=0, 1, \ldots)$ shinya2@istitech,acjp $\mathrm{e}$-mail: $\dagger_{\mathrm{e}}$-mail wataru@istitechacjp

3 4 $[$ $1$ $\{x_{n}\}$ $f(u)= \arg\min_{x\in H}f(x)$ $u$ $x\in H$ $\partial f(x)=$ { $z\in H$ : $f(y)\geq\langle y-x,$ $z\rangle$ $f(x)$ $(\forall y\in H)$ } $\partial f$ $H$ $H$ $f$ $\langle x-y, x^{*}-y^{*}\rangle\geq 0$ $(x, x^{*}),$ $(y, y^{*})\in G(\partial f)$ $G(\partial f)$ $\partial f$ $G(\partial f)=$ $\partial f$ $\{(x, x^{*}) : x^{*}\in\partial f(x)\}$ $\partial f$ $f(u)= \min_{x\in H}f(x)$ $\mathrm{o}\in\partial f(u)$ $T$ $\mathrm{o}\in Tu$ (12) $u$ (L2) $u\in H$ $T$ $T$ $T^{-1}0$ $[3, 4, 5, 10, 11]_{0}$ 1976 Rockafellar[10] 11(Rockafellar [10]) $H$ $\{x_{n}\}$ Hilbert, : $T$ $Harrow 2^{H}$ $x_{0}=x\in H$ $x_{n+1}=j_{r_{n}}x_{n}(n=0,1, \ldots)$, $=(I+r_{n}T)^{-1},$ $\{r_{n}\}\subseteq(0, \infty)$ $\lim\inf_{narrow\infty}r_{n}>0$ $T$ $\{x_{n}\}$ Banach - [5] Banach 2 $E$ $E^{*}$ Banach $x\in E$ $\langle x, x^{*}\rangle$ $x^{*}\in E^{*}$ $E$, $ x = y =1$ $x\neq y$

4 42 1 $E$ $ \frac{x+y}{2} $, $x,$ $y\in U=\{x\in E : x =1\}$, $\lim_{tarrow 0}\frac{ x+ty - x}{t}\underline{ }$ $E$ $x$, $E$ $2^{E^{*}}$ $J$ $J(x)=\{x^{*}\in E^{*}$ : $\{x, x^{*}\rangle= x ^{2}= x^{*} ^{2}\}$, $J$ $E$ $T$ $Earrow 2^{E^{*}}$ : $(x, x^{*}),$ $(y, y^{*})\in G(T)$ $\langle x-y, x^{*}-y^{*}\rangle\geq 0$ $T$ $T$ $E$ Banach $T:Earrow 2^{E^{*}}$ $r>0$ $x\in E$ $Q_{r}x=\{z\in E:Jx\in Jz+rTz\}$ (21) $Q_{r}$ $E$ $D(T)$ $([5, 11])_{\text{ }}$ $Q_{r}$ $T$ $Q_{r}$ $Q_{r}=(J+rT)^{-1}J$ $x\in E$ $\phi$ : $\frac{jx-jq_{r}x}{r}\in TQ_{r}x$ (22) $E\mathrm{x}Earrow[0, \infty)$ $\phi(x, y)= x ^{2}-2\langle x, Jy\rangle+ y ^{2}(\forall x, y\in E)$ $\phi$ 21 $E$ Banach $a,$ $b,$ $c\in E$ $2\langle c-a$, Jb-Ja) $=\phi(c, a)+\phi(a, b)-\phi(c, b)$ 3,

5 $4\theta$ 31 Banach $E$ $T:Earrow 2^{E^{*}}$ $\{x_{n}\}$ $x_{0}=x\in E$ $x_{n+1}=q_{r_{n}}x_{n}(n=0,1, \ldots)$ $\{r_{n}\}$ $\subset(0, \infty)$ $\inf_{narrow\varpi}r_{n}>0$ Jim $\{x_{n}\}$ $T$ - [5] $(u, v)\in G(T)$ 21 $a=x_{k+1\text{ }}b=x_{k\backslash }c=u$ $2r_{k}\langle u-x_{k+1}, (Jx_{k}-Jx_{k+1}\mathrm{I}/rk\rangle=2\{u-x_{k+1,k}Jx-Jx_{k+1}\rangle$ $=\phi(u, x_{k+1})+\phi(x_{k+1}, x_{k})-\phi(u, x_{k})$ (31) (22) $T$ $\langle x_{k+1}-u, (Jx_{k}-Jx_{k+1})/r_{k}-v\rangle\geq 0$ (31) $2r_{k}\langle u-x_{k+1}, v\rangle\geq\phi(u, x_{k+1})+\phi(x_{k+1}, x_{k})-\phi(u, x_{k})$ $2r_{k}\langle x_{k+1}-u, v\rangle\leq\phi(u, x_{k})-\phi(u, x_{k+\mathrm{i}})-\phi(x_{k+1}, x_{k})$ $\leq\phi(u, x_{k})-\phi(u, x_{k+1})$ 2 $\sum_{k=0}^{n}r_{k}\langle x_{k+1}-u, v\rangle\leq\phi(u, x_{0})-\phi(u_{\mathrm{j}}x_{1})$ $+\phi(u, x_{1})-\phi(u, x_{2})$ $\cdot$ $+\phi$ (, $u$ $\iota$ x ) $-\phi(u, x_{n})$ $+\phi(u, x_{n})-\phi(u, x_{n+1})$ $=\phi(u, x_{0})-\phi(u, x_{n+1})$ 2 $\sum_{k=0}^{n}$ $r_{k}$ $\langle z_{n}-u, v\rangle\leq\frac{\phi(u,x_{0})-\phi(u,x_{n+1})}{2\sum_{k=0}^{n}r_{k}}$ $\leq\frac{\phi(u,x_{0})}{2\sum_{k=0}^{n}r_{k}}$ (32)

6 $\not\in)$ $\bullet$ 44 $z_{n}= \frac{\sigma_{k-}^{n}-r_{k}x_{k+1}}{\sigma_{k=0}^{\mathrm{n}}r_{k}}$ $z_{n_{i}}arrow\hat{z}(\mathrm{i}arrow\infty)$ $\lim\inf_{narrow\infty}r_{n}>0$ $\mathrm{f}_{1}5\backslash P^{1}\mathrm{J}\{x_{n}\}$, lxfi $\{z_{n}\}$ g $\{z_{n_{i}}\}\subset$ $\{z_{n}\}$ $\mathit{2}\in E$ $\sum_{k=0}^{n}r_{k}arrow\infty(narrow\infty)$ (32) $\langle z_{n_{i}}-u, v\rangle\leq\frac{\phi(u,x_{0})}{2\sum_{k=0}^{n_{i}}r_{k}}$ $\mathrm{i}arrow\infty$ $\langle\hat{z}-u, v\rangle\leq 0$ $(u, v)$ $G(T)$ $T$ $(\hat{z}, \mathrm{o})\in G(T)$ $0\in Tz^{\mathrm{A}}$ 4 31 $f$ $Earrow \mathbb{r}$ : $x,$ $y\in E$ $\lambda\in(0,1)$ $f(\lambda x+(1-\lambda)y)\leq\lambda f(x)+(1-\lambda)f(y)$ $f$ $x^{*}\in E^{*}$ $x\in E$ $y\in E$ Gateaux $\lim_{tarrow 0}\frac{f(x+ty)-f(x)}{t}=\langle y,$ $x^{*}\rangle$ (4 1) Gateaux $f$ $x\in E$ $x^{*}$ (41) $\nabla f$ $\nabla f$ $f$ $C$ $E$ $z\in C$ $C$ normal cane $N_{C}(z)$ $Nc(z)=\{z^{*}\in E^{*} : \langle z-u, z^{*}\rangle\geq 0(\forall u\in C)\}$ 41 $E$ Banach $f,$ $g_{i}$ : $Earrow \mathbb{r}(\mathrm{i}=$ $1,2,$ $\ldots,$ $m)$ $C=\{x\in E : g_{i}(x)\leq 0(\mathrm{i}=1,2, \ldots, m)\}$ $f$ Gateaux $C$ $r_{n}>0,$ $\lim\inf_{narrow\varpi}r_{n}>0$ $\{x_{n}\}\subset E$ $x_{0}=x\in E$ $x_{n+1}=y_{n}$ $(n=0$,1, $)$

7 $\langle$ 45 $\{y_{n}\}$ $n\in \mathrm{n}\cup\{0\}$ $y_{n}\in C$ $u-y_{n}$, $Jx_{n}+r_{n}\nabla f(y_{n})\}\geq 0(\forall u\in C)$ Jy (11) $\{x_{n}\}$ $g_{i}$ : $Earrow \mathbb{r}(i=1,2, \ldots, m)$ $C$ $E$ $f$ $Earrow \mathbb{r}$ : Gateaux $C$ $\nabla f$ $C$ $[\mathrm{s}]_{\text{ }}$ $\nabla f+nc$ : $Earrow 2^{E^{*}}$ $[9]_{\text{ }}$ yn=qrnx $Q_{r_{n}}$ # $\nabla f+nc$ N $y_{n}\in C$ $\langle u-y_{n}, Jy_{n}-Jx_{n}+r_{n}\nabla f(y_{n})\rangle\geq 0(\forall u\in C)$ $y_{n}\in C$ $N_{C}(y_{n})\ni-Jy_{n}+Jx_{n}-r_{n}\nabla f(y_{n})$ $r_{n}>0$ $Nc(y_{n})=r_{n}Nc(y_{n})$ $y_{n}\in C$ $Jx_{n}\in Jy_{n}+r_{n}(\nabla f+n_{c})(y_{n})$ $y_{n}=q_{r_{n}}x_{n}$ { (11) $z\in E:z$ $f$ $Earrow \mathbb{r}$ : $C$ $z\in C$ $f(x)\geq f(z)(\forall x\in C)$ } $=(\nabla f+nc)^{-1}0$ $z\in C$ $\nabla f(z)\cap(-n_{c}(z))\neq\emptyset$ $[1, 11]_{0}$ $x^{*}=\nabla f(z),$ $x^{*}\in-n_{c}(z)$ $x^{*}\in E^{*}$ $0\in(\nabla f+n_{c})(z)$ { $z\in E$ $z$ : (11) } $=(\nabla f+nc)^{-1}0$ $\blacksquare$ 31

8 48 [1] J M Borwein and Q J Zhu, Techniques of variational analysis, Springer-Verlag, New York, 2005, 2001 [2],, [3] 0 G\"uler, Ergodic convergence in proximal point algorithms with Bregman functions, in Advances in optimization and approximation, , Nonconvex Optim Appl, 1, Kluwer Acad Publ, Dordrecht, 1994 [4] S Kamimura and W Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, J Approx Theory 106 (2000), [5] S Kamimura, F Kohsaka and W Takahashi, Weak and strong convergence theorems for maxim al monotone operators in a Banach space, Set-Valued Anal 12 (2004), [6],,, 2004 [7] J M Martinez and B F Svaiter, A practical optimality condition without constraint qualifications for nonlinear programming, J Optim Theory Appl 118 (2003), , [8] D Pascali and S Sburlan, Nonlinear mappings of monotone type, Noordhoff, Leiden, 1978 [9] R T Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans Amer, Math Soc 149 (1970), [10] R T RockafelJar, Monotone operators and the proximal point algorithm, SIAM J Control Optim 14 (1976), [11],,, 2000 [12] W Takahashi, Nonlinear $Funct\mathrm{i}ona_{\iota}^{7}$ A nalysis, Yokohama-Publishers, 2000 [13],,, 1994 [14] K Yokoyama and S Shiraishi, An c-kkt condition and its illustrative examples (in Japanese), RIMS Kokyuroku, 1409 (2005),

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