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Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 692325, 9 pages doi:10.1155/2012/692325 Research Article Sufficient Optimality and Sensitivity Analysis of a Parameterized Min-Max Programming Huijuan Xiong,1 Yu Xiao,2 and Chaohong Song1 1 2 College of Science, Huazhong Agricultural University, Wuhan 430070, China School of Basic Science, East China Jiaotong University, Nanchang 330000, China Correspondence should be addressed to Huijuan Xiong, [email protected] Received 4 June 2012; Accepted 17 July 2012 Academic Editor: Jian-Wen Peng Copyright q 2012 Huijuan Xiong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Suﬃcient optimality and sensitivity of a parameterized min-max programming with fixed feasible set are analyzed. Based on Clarke’s subdiﬀerential and Chaney’s second-order directional derivative, suﬃcient optimality of the parameterized min-max programming is discussed first. Moreover, under a convex assumption on the objective function, a subdiﬀerential computation formula of the marginal function is obtained. The assumptions are satisfied naturally for some application problems. Moreover, the formulae based on these assumptions are concise and convenient for algorithmic purpose to solve the applications. 1. Introduction In this paper, suﬃcient optimality and sensitivity analysis of a parameterized min-max programming are given. The paper is triggered by a local reduction algorithmic strategy for solving following nonsmooth semi-infinite min-max-min programming SIM3 P, see 1, 2, etc. for related applications reference: min fx s.t. gx max min gi x, y ≤ 0. x y∈Y 1≤i≤q 1.1 2 Journal of Applied Mathematics With the local reduction technique, the SIM3 P can be rewritten as a bilevel programming first, where the lower problem is the following parameterized min-max programming Px see 3– 5 for related reference of local reduction strategy: min y s.t. g x, y max −gi x, y 1≤i≤q 1.2 y ∈ Y. To make the bilevel strategy applicable to SIM3 P, it is essential to discuss the second-order suﬃcient optimality of Px and give sensitivity analysis of the parameterized minimum yx and corresponding marginal function gx, yx. Sensitivity analysis of optimization problems is an important aspect in the field of operation and optimization research. Based on diﬀerent assumptions, many results on kinds of parametric programming have been obtained 6–9, etc.. Among these, some conclusions on parameterized min-max programming like 1.2 have also been given. For example, based on variation analysis, parameterized continuous programming with fixed constraint was discussed in 7. Problem like 1.2 can be seen as a special case. Under the inf-compactness condition and the condition objective function is concave with respect to the parameter, directional derivative computational formula of marginal function for 1.2 can be obtained directly. However, concave condition cannot be satisfied for many problems. Recently, Fréchet subgradients computation formula of marginal functions for nondiﬀerentiable programming in Asplund spaces was given 9. By using Fréchet subgradients computation formula in 9, subgradient formula of marginal function for 1.2 is direct. But the formula is tedious, if utilizing the formula to construct optimality system of 1.1, the system is so complex that it is diﬃcult to solve the obtained optimality system. For more convenient computational purpose, the focus of this paper is to establish suﬃcient optimality and simple computation formula of marginal function for 1.2. Based on Clarke’s subdiﬀerential and Chaney’s second-order directional derivative, suﬃcient optimality of the parameterized programming Px is given first. And then Lipschitzian continuousness of the parameterized isolated minimizer yx and the marginal function gyx, x is discussed; moreover, subdiﬀerential computation formula of the marginal function is obtained. 2. Main Results Let Y in 1.2 be defined as Y {y ∈ Rm : hi y ≤ 0, i 1, . . . , l}, where hi · and i 1, . . . , l, are twice continuously diﬀerentiable functions on Rm , and gi ·, · in 1.2 are twice continuously diﬀerentiable functions on Rn×m . In the following, we first give the suﬃcient optimality condition of 1.2 based on Clarke’s subdiﬀerential and Chaney’s second-order directional derivative, and then make sensitivity analysis of the parameterized problem Px . Journal of Applied Mathematics 3 2.1. Sufficient Optimality Conditions of Px Definition 2.1 see 10. For a given parameter x, a point y∗ ∈ Y is said to be an local minimum of problem Px if there exists a neighborhood U of y∗ such that g x, y ≥ g x, y∗ , ∗ ∀y ∈ U ∩ Y, y /y . 2.1 Assumption 2.2. For a given parameter x, suppose that Px satisfying the following constraint qualification: T d ∈ Rm : ∇hi y d < 0, ∀i ∈ Ih y , y ∈ Y / ∅, 2.2 where Ih y {i {1, . . . , l} : hi y 0}. For a given parameter x, denote the Lagrange function of Px as Lx, y, λ gx, y l i1 λi hi y, then the following holds. Theorem 2.3. For a given parameter x, if y∗ is a minimum of Px , Assumption 2.2 holds, then there exists a λ∗ ∈ Rl such that 0 ∈ ∂y Lx, y∗ , λ, where ∂y Lx, y∗ , λ∗ denotes the Clarke’s subdiﬀerential of Lx, y∗ , λ∗ . Specifically, the following system holds: l 0 ∈ ∂y g x, y∗ λi ∇hi y∗ , 2.3 i1 where ∂y gx, y denotes Clarke’s subdiﬀerential of gx, y with respect to y, it can be computed as co{∇y gi x, y∗ : i ∈ Ix, y∗ }, co{·} is an operation of making convex hull of the elements, Ix, y∗ {i ∈ {1, . . . , q} : gx, y∗ gx, y∗ }. Proof. The conclusion is direct from Theorem 3.2.6 and Corollary 5.1.8 in 11. Since gx, y max1≤i≤p {gi x, y} is a directional diﬀerentiable function Theorem 3.2.13 in 11, the directional derivative of gx, y with respect to y in direction d can be computed as follows: gy x, y; d max ξT d : ∀ξ ∈ ∂y g x, y , ∀d ∈ Rm . 2.4 Definition 2.4 see 10. Let fx is a locally Lipschitzian function on Rn , u be a nonzero vector in Rn . Suppose that x d ∈ ∂u fx υ ∈ Rn : ∃{xk }, {υk }, s. t. xk −→, υk −→ υ, υk ∈ ∂fxk for each k 2.5 define Chaney’s lower second-order directional derivative as follows: f− x, υ, u lim inf fxk − fx − υT xk − x t2k , 2.6 4 Journal of Applied Mathematics taking over all triples of sequences {xk }, {υk }, and {tk } for which a tk > 0 for each k and {xk } → x; b tk → 0 and xk − x∗ /tk converges to u; c {υk } → υ with υk ∈ ∂fxk for each k. Similarly, Chaney’s upper second-order directional derivative can be defined as f lim sup fxk − fx − υT xk − x t2k 2.7 , taking over all triples of sequences {xk }, {υk }, and {tk } for which a, b, and c above hold. For parameterized max-type function gx, y max1≤i≤p {−gi x, y}, where x is a given parameter, its Chaney’s lower and upper second-order directional derivatives can be computed as follows. Proposition 2.5 see 12. For any given parameter x, Chaney’s lower and upper second-order directional derivatives of gx, y with respect to y exist; moreover, for any given 0 / u ∈ Rq , υ ∈ ∂u gx, y, it has g− q 1 T 2 aj u ∇y gi x, y u : a ∈ Tu g, y, υ , x, y; d min 2 i1 g x, y; d max q 1 2 2.8 aj uT ∇2y gi x, y u : a ∈ Tu g, y, υ , i1 where ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨a ∈ Rq : Tu g, y, υ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∃ yk , ak , υk , such that ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k ⎪ 1 y −→ y in direction u, ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ k k k , k 1, 2, . . . , , 2 υ −→ υ, and υ ∈ ∂y g x, y ⎪ ⎪ p ⎪ ⎪ ⎪ k k k k k ⎪ ,⎪ 3 a −→ a, a ∈ Eq , υ ai ∇y gi x, y ⎪ ⎪ ⎪ i1 ⎪ ⎪ ⎪ k ⎭ k /K y 4 a 0, for j ∈ j 2.9 g q where Kg yk {i ∈ Q : gi x, yn gx, yn , ∃yn ∈ By, 1/n, ∀n ∈ N}, Eq {a ∈ R : p i1 ai 1}, Q {1, . . . , q}, and By, 1/n denotes the ball centered in y with radius 1/n. Theorem 2.6 suﬃciency theorem. For a given parameter x ∈ Rn , Assumption 2.2 holds, then there exists y∗ ∈ Rm such that 2.3 holds. Moreover, for any feasible direction d ∈ Rm of Y , that is, max{∇hi y∗ T d : 1 ≤ i ≤ l} ≤ 0, if d satisfying one of the following conditions: Journal of Applied Mathematics 5 0; 1 gy x, y∗ ; d / 2 gy x, y∗ ; d 0, l i1 λi ∇hi yT d 0, that is, L y x, y; d 0, and 1 ai dT ∇2y gi x, y∗ d : a ∈ Eq min 2 i1 q l λi dT ∇2 hi y∗ d > 0, 2.10 i1 then y∗ is a local minimum of Px . Proof. 1 If not, then there exists sequences tk ↓ 0, dk → d, yk y∗ tk dk ∈ Y such that g x, yk < g x, y∗ . 2.11 As a result, gy x, y∗ ; d limt↓0 gx, y∗ td − gx, y∗ /t limk → ∞ gx, y∗ tk dk − gx, y∗ /tk ≤ 0. If gy x, y∗ ; d / 0, then gy x, y∗ ; d < 0. From 2.4, we know that ξT d < ∗ 0 for all ξ ∈ ∂y gx, y . Hence, for the direction d ∈ Rm , we have ξT d l T ∇hi y∗ d < 0, ξ ∈ ∂y g x, y∗ . 2.12 i1 On the other hand, from y∗ satisfying 2.3, we know that there exists a ξ ∈ ∂y gx, y∗ such that ξT d l T ∇hi y∗ d 0, 2.13 i1 which leads to a contradiction to 2.12. 2 From Theorem 4 in 10 and Proposition 2.5, the conclusion is direct. 2.2. Sensitivity Analysis of Parameterized Px In the following, we make sensitivity analysis of parameterized min-max programming Px , that is, study variation of isolated local minimizers and corresponding marginal function under small perturbation of x. For convenience of discussion, for any given parameter x, denote y∗ x as a minimizer of Px , υx min{gx, y : y ∈ Y } as the corresponding marginal function value and make the following assumptions first. Assumption 2.7. For given x ∈ Rn , the parametric problem Px is a convex problem, specifically, gi x, y and i 1, . . . , q are concave functions with respect to that variables y and hj y, j 1, . . . , l are convex functions. Assumption 2.8. Let Ih y {i ∈ L : hi y 0}, {∇hi y : i ∈ Ih y} are linearly independent. 6 Journal of Applied Mathematics Definition 2.9 see Definition 2.1, 13. For a given x, y ∈ Y is said to be an isolated local minimum with order i i 1 or 2 of Px if there exists a real m > 0 and a neighborhood V of y such that i 1 g x, y > g x, y my − y , 2 ∀y ∈ V ∩ Y, y / y. 2.14 Theorem 2.10. For a given x ∈ Rn , Assumptions 2.2–2.8 hold, then the following conclusions hold: 1 if y∗ x with corresponding multiplier λ∗ is the solution of 2.3, then y∗ x is a unique first-order isolated minimizer of Px ; 2 for any minimum y∗ x, it is a locally Lipschitzian function with respect to x, that is, there exists a L1 > 0, δ > 0 such that ∗ k y x − y∗ x ≤ L1 xk − x, ∀xk ∈ Ux, δ, y∗ xk ∈ Y xk , 2.15 where Y xk denotes minima set of Pxk ; 3 for any minimum y∗ x, marginal function υx gx, y∗ x is also a locally Lipschitz function with respect to x, and ∂υx ⊆ Sx, where Sx co ∇x gi x, y∗ x , i ∈ I x, y∗ x , 2.16 and Ix, y∗ x {i ∈ {1, . . . , q} : gi x, y∗ x gx, y∗ x}. As a result, ∂υx ⎧ ⎨ ⎩ i∈I x,y ∗ x λi ∇x gi x, y∗ x : λi ≥ 0, i∈I x,y ∗ x ⎫ ⎬ λi 1 . ⎭ 2.17 Proof. 1 From Assumption 2.7, it is direct that y∗ x is a global minimizer of Px . We only prove y∗ x is a first-order isolated minimizer. If the conclusion does not hold, then there exists a sequence {yk } ∈ Y x converging ∗ y∗ x, and a sequence mk , mk > 0, and mk converges to 0 such that to y x, yk / 1 g x, yk ≤ g x, y∗ x mk yk − y∗ , 2 yk ∈ Y. 2.18 Take dk yk − y∗ x/yk − y∗ x, for simplicity, we suppose dk → d, with d 1. Let tk yk − y∗ x, then from yk ∈ Y , dk → d and Y is compact, we have y∗ x tk d ∈ Y, tk −→ 0, 2.19 that is, T ∇hi y∗ x d ≤ 0, ∀i ∈ I x, y∗ x . 2.20 Journal of Applied Mathematics 7 0. As a result, we have From Assumption 2.8, we know that i∈Ix,y∗ x ∇hi y∗ xT d / T ∗ i∈Ix,y∗ x ∇hi y x d < 0. From the first equation of 2.3, we know that there exists a z ∈ ∂y gx, y∗ x such that for any feasible direction d, zT d − i∈Ix,y∗ x λi ∇hi y∗ xT d > 0. Hence, gy x, y∗ x; d max ξT d : ξ ∈ ∂y g x, y∗ x ≥ zT d > 0. 2.21 On the other hand, from y∗ x is a minimizer, we know that gy x, y∗ x; d ≥ 0, this leads to a contradiction; 2 from Assumption 2.8 and Theorem 3.1 in 13, the conclusion is direct; 3 since gx, y is a locally Lipschitzian function with respect to x and y, then there exists δ > 0, δ > 0, and L2 > 0 such that for any x1 ∈ Ux, δ, y ∈ Uy∗ x, δ , it has g x1 , y∗ x − g x, y∗ x ≤ L2 x1 − x, g x, y − g x, y∗ x ≤ L2 y − y∗ x. 2.22 As to x1 ∈ Ux, δ, from the conclusion in 1.2, there exists a a L1 > 0 such that y∗ x1 − y∗ x ≤ L1 x1 − x. As a result, |υx1 − υx| g x1 , y∗ x1 − g x, y∗ x g x1 , y∗ x1 − g x1 , y∗ x g x1 , y∗ x − g x, y∗ x ≤ g x1 , y∗ x1 − g x1 , y∗ x g x1 , y∗ x − g x, y∗ x ≤ L2 y∗ x1 − y∗ x L2 x1 − x ≤ L2 1 L1 x1 − x. 2.23 Hence, the marginal function υx is a local Lipschitzian function with respect to x. We prove that Let Sx {−∇x gi x, yx, i ∈ Ix, yx}, then Sx co{ξ, ξ ∈ Sx}. k n k k k , zk → z, it Sx is closed first, that is, prove for any sequence {x } ⊂ R , x → x, z ∈ Sx has z ∈ Sx. k , there exist yk ∈ Y xk ; ik ∈ Ixk , yk such that zk −∇x gik xk , yk . From zk ∈ Sx Without loss of generality, suppose that {yk } converges to y; {ik } converges to i. From Proposition 3.3 in 14, it has y ∈ Y x, and i ∈ Ix, y and from ∇x gi x, y is a continuous As a result, Sx function, it has z limk → ∞ zk limk → ∞ ∇x gik xk , yk ∇x gi x, y ∈ Sx. is a closed set. 8 Journal of Applied Mathematics From Theorem 3.2.16 in 11, for any ξ ∈ ∂υx, there exists xk ∈ Rn , xk → x such that ∇υx exists and ξ limk → ∞ ∇υxk . In addition, for arbitrary d ∈ Rn , it has k ∇υ x k T d ≤ υ xk td − υ xk υ x ; d lim t↓0 t k g x td, y∗ xk td − g xk , y∗ xk lim t↓0 t k g x td, y∗ xk − g xk , y∗ xk lim t↓0 t T k max −∇x gi x , y d . i∈I xk ,y k T 2.24 T From the definition of Sxk , ∃zk ∈ Sxk such that zk d maxi∈Ixk ,y {−∇x gi xk , y d}. k T T Hence, it has ∇υx d ≤ zk d. T T ⊂ Sx, ∇υxk → ξ and ∇υxk d ≤ zk d, it has ξT d ≤ zT d, that From zk → z ∈ Sx is, for arbitrary d ∈ Rn and ξ ∈ ∂υx, there exists z ∈ Sx such that ξT d ≤ zT d. If ∂υx ⊂ Sx does not hold, then there exists a ξ ∈ ∂υx and ξ ∈ / Sx. From Sx is a compact convex set and separation theorem 15, there exists a d ∈ Rn such that ξT d < 0 and for arbitrary z ∈ Sx, zT d ≥ 0, which leads to a contradiction. As a result, ∂υx ⊂ Sx holds. From ∂υx ⊂ Sx and Sx co{∇x gi x, y∗ x, i ∈ Ix, y∗ x}, computation formula 2.17 is direct. 3. Discussion In this paper, suﬃcient optimality and sensitivity analysis of a parameterized min-max programming are given. A rule for computation the subdiﬀerential of υx is established. Though the assumptions in this paper are some restrictive compared to some existing work, the assumptions hold naturally for some applications. Moreover, the obtained computation formula is simple, it is beneficial for establishing a concise first-order necessary optimality system of 1.1, and then constructing eﬀective algorithms to solve the applications. Acknowledgments This research was supported by the National Natural Science Foundation of China no. 11001092 and the Fundamental Research Funds for the Central Universities no. 2011QC064. References 1 C. Kirjner-Neto and E. Polak, “On the conversion of optimization problems with max-min constraints to standard optimization problems,” SIAM Journal on Optimization, vol. 8, no. 4, pp. 887–915, 1998. 2 E. Polak and J. O. 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