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Journal of Applied Mathematics
Volume 2012 (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

1College of Science, Huazhong Agricultural University, Wuhan 430070, China
2School of Basic Science, East China Jiaotong University, Nanchang 330000, China

Received 4 June 2012; Accepted 17 July 2012

Academic Editor: Jian-Wen Peng

Copyright © 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.

Abstract

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 ( S I M 3 P, see [1, 2], etc. for related applications reference): m i n 𝑥 𝑓 ( 𝑥 ) s . t . 𝑔 ( 𝑥 ) = m a x 𝑦 ∈ 𝑌 m i n 1 ≤ 𝑖 ≤ 𝑞 𝑔 𝑖 ( 𝑥 , 𝑦 ) ≤ 0 . ( 1 . 1 ) With the local reduction technique, the S I M 3 P can be rewritten as a bilevel programming first, where the lower problem is the following parameterized min-max programming 𝑃 𝑥 (see [3–5] for related reference of local reduction strategy): m i n 𝑦 𝑔 ( 𝑥 , 𝑦 ) = m a x 1 ≤ 𝑖 ≤ 𝑞 − 𝑔 𝑖 ( 𝑥 , 𝑦 ) s . t . 𝑦 ∈ 𝑌 . ( 1 . 2 ) To make the bilevel strategy applicable to S I M 3 P, it is essential to discuss the second-order sufficient optimality of 𝑃 𝑥 and give sensitivity analysis of the parameterized minimum 𝑦 ( 𝑥 ) and corresponding marginal function 𝑔 ( 𝑥 , 𝑦 ( 𝑥 ) ) .

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 𝑃 𝑥 is given first. And then Lipschitzian continuousness of the parameterized isolated minimizer 𝑦 ( 𝑥 ) and the marginal function 𝑔 ( 𝑦 ( 𝑥 ) , 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 ∈ 𝑅 𝑚 ∶ ℎ 𝑖 ( 𝑦 ) ≤ 0 , 𝑖 = 1 , … , 𝑙 } , where ℎ 𝑖 ( ⋅ ) and 𝑖 = 1 , … , 𝑙 , are twice continuously differentiable functions on 𝑅 𝑚 , and 𝑔 𝑖 ( ⋅ , ⋅ ) in (1.2) are twice continuously differentiable functions on 𝑅 𝑛 × 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 𝑃 𝑥 .

2.1. Sufficient Optimality Conditions of 𝑃 𝑥

Definition 2.1 (see [10]). For a given parameter 𝑥 , a point 𝑦 ∗ ∈ 𝑌 is said to be an local minimum of problem 𝑃 𝑥 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 𝑃 𝑥 satisfying the following constraint qualification: 𝑑 ∈ 𝑅 𝑚 ∶ ∇ ℎ 𝑖 ( 𝑦 ) 𝑇 𝑑 < 0 , ∀ 𝑖 ∈ 𝐼 ℎ ( 𝑦 ) , 𝑦 ∈ 𝑌 ≠ ∅ , ( 2 . 2 ) where 𝐼 ℎ ( 𝑦 ) = { 𝑖 = { 1 , … , 𝑙 } ∶ ℎ 𝑖 ( 𝑦 ) = 0 } .
For a given parameter 𝑥 , denote the Lagrange function of 𝑃 𝑥 as ∑ 𝐿 ( 𝑥 , 𝑦 , 𝜆 ) = 𝑔 ( 𝑥 , 𝑦 ) + 𝑙 𝑖 = 1 𝜆 𝑖 ℎ 𝑖 ( 𝑦 ) , then the following holds.

Theorem 2.3. For a given parameter 𝑥 , if 𝑦 ∗ is a minimum of 𝑃 𝑥 , Assumption 2.2 holds, then there exists a 𝜆 ∗ ∈ 𝑅 𝑙 + such that 0 ∈ 𝜕 𝑦 𝐿 ( 𝑥 , 𝑦 ∗ , 𝜆 ) , where 𝜕 𝑦 𝐿 ( 𝑥 , 𝑦 ∗ , 𝜆 ∗ ) denotes the Clarke's subdifferential of 𝐿 ( 𝑥 , 𝑦 ∗ , 𝜆 ∗ ) . Specifically, the following system holds: 0 ∈ 𝜕 𝑦 𝑔 𝑥 , 𝑦 ∗ + 𝑙 𝑖 = 1 𝜆 𝑖 ∇ ℎ 𝑖 𝑦 ∗ , ( 2 . 3 ) where 𝜕 𝑦 𝑔 ( 𝑥 , 𝑦 ) denotes Clarke's subdifferential of 𝑔 ( 𝑥 , 𝑦 ) with respect to 𝑦 , it can be computed as c o { ∇ 𝑦 𝑔 𝑖 ( 𝑥 , 𝑦 ∗ ) ∶ 𝑖 ∈ 𝐼 ( 𝑥 , 𝑦 ∗ ) } , c o { ⋅ } is an operation of making convex hull of the elements, 𝐼 ( 𝑥 , 𝑦 ∗ ) = { 𝑖 ∈ { 1 , … , 𝑞 } ∶ 𝑔 ( 𝑥 , 𝑦 ∗ ) = 𝑔 ( 𝑥 , 𝑦 ∗ ) } .

Proof. The conclusion is direct from Theorem 3 . 2 . 6 and Corollary 5 . 1 . 8 in [11].

Since 𝑔 ( 𝑥 , 𝑦 ) = m a x 1 ≤ 𝑖 ≤ 𝑝 { 𝑔 𝑖 ( 𝑥 , 𝑦 ) } is a directional differentiable function (Theorem 3 . 2 . 1 3 in [11]), the directional derivative of 𝑔 ( 𝑥 , 𝑦 ) with respect to 𝑦 in direction 𝑑 can be computed as follows: 𝑔 𝑦 𝜉 ( 𝑥 , 𝑦 ; 𝑑 ) = m a x 𝑇 𝑑 ∶ ∀ 𝜉 ∈ 𝜕 𝑦 𝑔 ( 𝑥 , 𝑦 ) , ∀ 𝑑 ∈ 𝑅 𝑚 . ( 2 . 4 )

Definition 2.4 (see [10]). Let 𝑓 ( 𝑥 ) is a locally Lipschitzian function on 𝑅 𝑛 , 𝑢 be a nonzero vector in 𝑅 𝑛 . Suppose that 𝑑 ∈ 𝜕 𝑢 𝑓 ( 𝑥 ) = 𝜐 ∈ 𝑅 𝑛 𝑥 ∶ ∃ 𝑘 , 𝜐 𝑘 , s . t . 𝑥 𝑘 𝑥 ⟶ , 𝜐 𝑘 ⟶ 𝜐 , 𝜐 𝑘 𝑥 ∈ 𝜕 𝑓 𝑘 f o r e a c h 𝑘 ( 2 . 5 ) define Chaney's lower second-order directional derivative as follows: 𝑓 − 𝑓 𝑥 ( 𝑥 , 𝜐 , 𝑢 ) = l i m i n f 𝑘 − 𝑓 ( 𝑥 ) − 𝜐 𝑇 𝑥 𝑘 − 𝑥 𝑡 2 𝑘 , ( 2 . 6 ) taking over all triples of sequences { 𝑥 𝑘 } , { 𝜐 𝑘 } , and { 𝑡 𝑘 } for which (a) 𝑡 𝑘 > 0 for each 𝑘 and { 𝑥 𝑘 } → 𝑥 ; (b) 𝑡 𝑘 → 0 and ( 𝑥 𝑘 − 𝑥 ∗ ) / 𝑡 𝑘 converges to 𝑢 ; (c) { 𝜐 𝑘 } → 𝜐 with 𝜐 𝑘 ∈ 𝜕 𝑓 ( 𝑥 𝑘 ) for each 𝑘 .
Similarly, Chaney's upper second-order directional derivative can be defined as 𝑓 + 𝑓 𝑥 = l i m s u p 𝑘 − 𝑓 ( 𝑥 ) − 𝜐 𝑇 𝑥 𝑘 − 𝑥 𝑡 2 𝑘 , ( 2 . 7 ) taking over all triples of sequences { 𝑥 𝑘 } , { 𝜐 𝑘 } , and { 𝑡 𝑘 } for which (a), (b), and (c) above hold.
For parameterized max-type function 𝑔 ( 𝑥 , 𝑦 ) = m a x 1 ≤ 𝑖 ≤ 𝑝 { − 𝑔 𝑖 ( 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 𝑔 ( 𝑥 , 𝑦 ) with respect to 𝑦 exist; moreover, for any given 0 ≠ 𝑢 ∈ 𝑅 𝑞 , 𝜐 ∈ 𝜕 𝑢 𝑔 ( 𝑥 , 𝑦 ) , it has 𝑔 − 1 ( 𝑥 , 𝑦 ; 𝑑 ) = m i n 2 𝑞 𝑖 = 1 𝑎 𝑗 𝑢 𝑇 ∇ 2 𝑦 𝑔 𝑖 ( 𝑥 , 𝑦 ) 𝑢 ∶ 𝑎 ∈ 𝑇 𝑢 , 𝑔 ( 𝑔 , 𝑦 , 𝜐 ) + 1 ( 𝑥 , 𝑦 ; 𝑑 ) = m a x 2 𝑞 𝑖 = 1 𝑎 𝑗 𝑢 𝑇 ∇ 2 𝑦 𝑔 𝑖 ( 𝑥 , 𝑦 ) 𝑢 ∶ 𝑎 ∈ 𝑇 𝑢 , ( 𝑔 , 𝑦 , 𝜐 ) ( 2 . 8 ) where 𝑇 𝑢 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ∃ 𝑦 ( 𝑔 , 𝑦 , 𝜐 ) = ( 𝑘 ) , 𝑎 ( 𝑘 ) , 𝜐 ( 𝑘 ) ( , s u c h t h a t 1 ) 𝑦 ( 𝑘 ) ⟶ 𝑦 i n d i r e c t i o n 𝑢 , 𝑎 ∈ 𝑅 𝑞 + ∶ ( 2 ) 𝜐 ( 𝑘 ) ⟶ 𝜐 , a n d 𝜐 ( 𝑘 ) ∈ 𝜕 𝑦 𝑔 𝑥 , 𝑦 ( 𝑘 ) , 𝑘 = 1 , 2 , … , ( 3 ) 𝑎 ( 𝑘 ) ⟶ 𝑎 , 𝑎 ( 𝑘 ) ∈ 𝐸 𝑞 , 𝜐 ( 𝑘 ) = 𝑝 ∑ 𝑖 = 1 𝑎 𝑖 ( 𝑘 ) ∇ 𝑦 𝑔 𝑖 𝑥 , 𝑦 ( 𝑘 ) , ( 4 ) 𝑎 𝑗 ( 𝑘 ) = 0 , f o r 𝑗 ∉ 𝐾 𝑔 𝑦 ( 𝑘 ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ , ( 2 . 9 ) where 𝐾 𝑔 ( 𝑦 ( 𝑘 ) ) = { 𝑖 ∈ 𝑄 ∶ 𝑔 𝑖 ( 𝑥 , 𝑦 ( 𝑛 ) ) = 𝑔 ( 𝑥 , 𝑦 ( 𝑛 ) ) , ∃ 𝑦 ( 𝑛 ) ∈ 𝐵 ( 𝑦 , 1 / 𝑛 ) , ∀ 𝑛 ∈ 𝑁 } , 𝐸 𝑞 = { 𝑎 ∈ 𝑅 𝑞 + ∶ ∑ 𝑝 𝑖 = 1 𝑎 𝑖 = 1 } , 𝑄 = { 1 , … , 𝑞 } , and 𝐵 ( 𝑦 , 1 / 𝑛 ) denotes the ball centered in 𝑦 with radius 1 / 𝑛 .

Theorem 2.6 (sufficiency theorem). For a given parameter 𝑥 ∈ 𝑅 𝑛 , Assumption 2.2 holds, then there exists 𝑦 ∗ ∈ 𝑅 𝑚 such that (2.3) holds. Moreover, for any feasible direction 𝑑 ∈ 𝑅 𝑚 of 𝑌 , that is, m a x { ∇ ℎ 𝑖 ( 𝑦 ∗ ) 𝑇 𝑑 ∶ 1 ≤ 𝑖 ≤ 𝑙 } ≤ 0 , if 𝑑 satisfying one of the following conditions: (1) 𝑔 𝑦 ( 𝑥 , 𝑦 ∗ ; 𝑑 ) ≠ 0 ; (2) 𝑔 𝑦 ( 𝑥 , 𝑦 ∗ ; 𝑑 ) = 0 , ∑ 𝑙 𝑖 = 1 𝜆 𝑖 ∇ ℎ 𝑖 ( 𝑦 ) 𝑇 𝑑 = 0 , that is, 𝐿 𝑦 ( 𝑥 , 𝑦 ; 𝑑 ) = 0 , and 1 m i n 2 𝑞 𝑖 = 1 𝑎 𝑖 𝑑 𝑇 ∇ 2 𝑦 𝑔 𝑖 𝑥 , 𝑦 ∗ 𝑑 ∶ 𝑎 ∈ 𝐸 𝑞 + 𝑙 𝑖 = 1 𝜆 𝑖 𝑑 𝑇 ∇ 2 ℎ 𝑖 𝑦 ∗ 𝑑 > 0 , ( 2 . 1 0 ) then 𝑦 ∗ is a local minimum of 𝑃 𝑥 .

Proof. (1) If not, then there exists sequences 𝑡 𝑘 ↓ 0 , 𝑑 𝑘 → 𝑑 , 𝑦 𝑘 = 𝑦 ∗ + 𝑡 𝑘 𝑑 𝑘 ∈ 𝑌 such that 𝑔 𝑥 , 𝑦 𝑘 < 𝑔 𝑥 , 𝑦 ∗ . ( 2 . 1 1 ) As a result, 𝑔 𝑦 ( 𝑥 , 𝑦 ∗ ; 𝑑 ) = l i m 𝑡 ↓ 0 ( 𝑔 ( 𝑥 , 𝑦 ∗ + 𝑡 𝑑 ) − 𝑔 ( 𝑥 , 𝑦 ∗ ) ) / 𝑡 = l i m 𝑘 → + ∞ ( 𝑔 ( 𝑥 , 𝑦 ∗ + 𝑡 𝑘 𝑑 𝑘 ) − 𝑔 ( 𝑥 , 𝑦 ∗ ) ) / 𝑡 𝑘 ≤ 0 . If 𝑔 𝑦 ( 𝑥 , 𝑦 ∗ ; 𝑑 ) ≠ 0 , then 𝑔 𝑦 ( 𝑥 , 𝑦 ∗ ; 𝑑 ) < 0 . From (2.4), we know that 𝜉 𝑇 𝑑 < 0 f o r a l l 𝜉 ∈ 𝜕 𝑦 𝑔 ( 𝑥 , 𝑦 ∗ ) . Hence, for the direction 𝑑 ∈ 𝑅 𝑚 , we have 𝜉 𝑇 𝑑 + 𝑙 𝑖 = 1 ∇ ℎ 𝑖 𝑦 ∗ 𝑇 𝑑 < 0 , 𝜉 ∈ 𝜕 𝑦 𝑔 𝑥 , 𝑦 ∗ . ( 2 . 1 2 ) On the other hand, from 𝑦 ∗ satisfying (2.3), we know that there exists a 𝜉 ∈ 𝜕 𝑦 𝑔 ( 𝑥 , 𝑦 ∗ ) such that 𝜉 𝑇 𝑑 + 𝑙 𝑖 = 1 ∇ ℎ 𝑖 𝑦 ∗ 𝑇 𝑑 = 0 , ( 2 . 1 3 ) 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 𝑃 𝑥

In the following, we make sensitivity analysis of parameterized min-max programming 𝑃 𝑥 , 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 𝑃 𝑥 , 𝜐 ( 𝑥 ) = m i n { 𝑔 ( 𝑥 , 𝑦 ) ∶ 𝑦 ∈ 𝑌 } as the corresponding marginal function value and make the following assumptions first.

Assumption 2.7. For given 𝑥 ∈ 𝑅 𝑛 , the parametric problem 𝑃 𝑥 is a convex problem, specifically, 𝑔 𝑖 ( 𝑥 , 𝑦 ) and 𝑖 = 1 , … , 𝑞 are concave functions with respect to that variables 𝑦 and ℎ 𝑗 ( 𝑦 ) , 𝑗 = 1 , … , 𝑙 are convex functions.

Assumption 2.8. Let 𝐼 ℎ ( 𝑦 ) = { 𝑖 ∈ 𝐿 ∶ ℎ 𝑖 ( 𝑦 ) = 0 } , { ∇ ℎ 𝑖 ( 𝑦 ) ∶ 𝑖 ∈ 𝐼 ℎ ( 𝑦 ) } are linearly independent.

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 𝑃 𝑥 if there exists a real 𝑚 > 0 and a neighborhood 𝑉 of 𝑦 such that 𝑔 𝑥 , 𝑦 > 𝑔 𝑥 , 𝑦 + 1 2 𝑚 ‖ ‖ 𝑦 − 𝑦 ‖ ‖ 𝑖 , ∀ 𝑦 ∈ 𝑉 ∩ 𝑌 , 𝑦 ≠ 𝑦 . ( 2 . 1 4 )

Theorem 2.10. For a given 𝑥 ∈ 𝑅 𝑛 , 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 𝑃 𝑥 ;(2)for any minimum 𝑦 ∗ ( 𝑥 ) , it is a locally Lipschitzian function with respect to 𝑥 , that is, there exists a 𝐿 1 > 0 , 𝛿 > 0 such that ‖ ‖ 𝑦 ∗ 𝑥 𝑘 − 𝑦 ∗ ‖ ‖ ( 𝑥 ) ≤ 𝐿 1 ‖ ‖ 𝑥 𝑘 ‖ ‖ − 𝑥 , ∀ 𝑥 𝑘 ∈ 𝑈 ( 𝑥 , 𝛿 ) , 𝑦 ∗ 𝑥 𝑘 𝑥 ∈ 𝑌 𝑘 , ( 2 . 1 5 ) where 𝑌 ( 𝑥 𝑘 ) denotes minima set of 𝑃 𝑥 𝑘 ;(3)for any minimum 𝑦 ∗ ( 𝑥 ) , marginal function 𝜐 ( 𝑥 ) = 𝑔 ( 𝑥 , 𝑦 ∗ ( 𝑥 ) ) is also a locally Lipschitz function with respect to 𝑥 , and 𝜕 𝜐 ( 𝑥 ) ⊆ 𝑆 ( 𝑥 ) , where 𝑆 ∇ ( 𝑥 ) = c o 𝑥 𝑔 𝑖 𝑥 , 𝑦 ∗ ( 𝑥 ) , 𝑖 ∈ 𝐼 𝑥 , 𝑦 ∗ ( 𝑥 ) , ( 2 . 1 6 ) and 𝐼 ( 𝑥 , 𝑦 ∗ ( 𝑥 ) ) = { 𝑖 ∈ { 1 , … , 𝑞 } ∶ 𝑔 𝑖 ( 𝑥 , 𝑦 ∗ ( 𝑥 ) ) = 𝑔 ( 𝑥 , 𝑦 ∗ ( 𝑥 ) ) } . As a result, 𝜕 𝜐 ( 𝑥 ) = 𝑖 ∈ 𝐼 ( 𝑥 , 𝑦 ∗ ( 𝑥 ) ) 𝜆 𝑖 ∇ 𝑥 𝑔 𝑖 𝑥 , 𝑦 ∗ ( 𝑥 ) ∶ 𝜆 𝑖 ≥ 0 , 𝑖 ∈ 𝐼 ( 𝑥 , 𝑦 ∗ ( 𝑥 ) ) 𝜆 𝑖 . = 1 ( 2 . 1 7 )

Proof. (1) From Assumption 2.7, it is direct that 𝑦 ∗ ( 𝑥 ) is a global minimizer of 𝑃 𝑥 . We only prove 𝑦 ∗ ( 𝑥 ) is a first-order isolated minimizer.
If the conclusion does not hold, then there exists a sequence { 𝑦 𝑘 } ∈ 𝑌 ( 𝑥 ) converging to 𝑦 ∗ ( 𝑥 ) , 𝑦 𝑘 ≠ 𝑦 ∗ ( 𝑥 ) , and a sequence 𝑚 𝑘 , 𝑚 𝑘 > 0 , and 𝑚 𝑘 converges to 0 such that 𝑔 𝑥 , 𝑦 𝑘 ≤ 𝑔 𝑥 , 𝑦 ∗ + 1 ( 𝑥 ) 2 𝑚 𝑘 ‖ ‖ 𝑦 𝑘 − 𝑦 ∗ ‖ ‖ , 𝑦 𝑘 ∈ 𝑌 . ( 2 . 1 8 ) Take 𝑑 𝑘 = ( 𝑦 𝑘 − 𝑦 ∗ ( 𝑥 ) ) / ‖ 𝑦 𝑘 − 𝑦 ∗ ( 𝑥 ) ‖ , for simplicity, we suppose 𝑑 𝑘 → 𝑑 , with ‖ 𝑑 ‖ = 1 . Let 𝑡 𝑘 = ‖ 𝑦 𝑘 − 𝑦 ∗ ( 𝑥 ) ‖ , then from 𝑦 𝑘 ∈ 𝑌 , 𝑑 𝑘 → 𝑑 and 𝑌 is compact, we have 𝑦 ∗ ( 𝑥 ) + 𝑡 𝑘 𝑑 ∈ 𝑌 , 𝑡 𝑘 ⟶ 0 , ( 2 . 1 9 ) that is, ∇ ℎ 𝑖 𝑦 ∗ ( 𝑥 ) 𝑇 𝑑 ≤ 0 , ∀ 𝑖 ∈ 𝐼 𝑥 , 𝑦 ∗ . ( 𝑥 ) ( 2 . 2 0 ) From Assumption 2.8, we know that ∑ 𝑖 ∈ 𝐼 ( 𝑥 , 𝑦 ∗ ( 𝑥 ) ) ∇ ℎ 𝑖 ( 𝑦 ∗ ( 𝑥 ) ) 𝑇 𝑑 ≠ 0 . As a result, we have ∑ 𝑖 ∈ 𝐼 ( 𝑥 , 𝑦 ∗ ( 𝑥 ) ) ∇ ℎ 𝑖 ( 𝑦 ∗ ( 𝑥 ) ) 𝑇 𝑑 < 0 .
From the first equation of (2.3), we know that there exists a 𝑧 ∈ 𝜕 𝑦 𝑔 ( 𝑥 , 𝑦 ∗ ( 𝑥 ) ) such that for any feasible direction 𝑑 , 𝑧 𝑇 ∑ 𝑑 = − 𝑖 ∈ 𝐼 ( 𝑥 , 𝑦 ∗ ( 𝑥 ) ) 𝜆 𝑖 ∇ ℎ 𝑖 ( 𝑦 ∗ ( 𝑥 ) ) 𝑇 𝑑 > 0 . Hence, 𝑔 𝑦 𝑥 , 𝑦 ∗ 𝜉 ( 𝑥 ) ; 𝑑 = m a x 𝑇 𝑑 ∶ 𝜉 ∈ 𝜕 𝑦 𝑔 𝑥 , 𝑦 ∗ ( 𝑥 ) ≥ 𝑧 𝑇 𝑑 > 0 . ( 2 . 2 1 ) On the other hand, from 𝑦 ∗ ( 𝑥 ) is a minimizer, we know that 𝑔 𝑦 ( 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 𝑔 ( 𝑥 , 𝑦 ) is a locally Lipschitzian function with respect to 𝑥 and 𝑦 , then there exists 𝛿 > 0 , 𝛿 ′ > 0 , and 𝐿 2 > 0 such that for any 𝑥 1 ∈ 𝑈 ( 𝑥 , 𝛿 ) , 𝑦 ∈ 𝑈 ( 𝑦 ∗ ( 𝑥 ) , 𝛿 ) , it has | | 𝑔 𝑥 1 , 𝑦 ∗ ( 𝑥 ) − 𝑔 𝑥 , 𝑦 ∗ | | ( 𝑥 ) ≤ 𝐿 2 ‖ ‖ 𝑥 1 ‖ ‖ , | | − 𝑥 𝑔 ( 𝑥 , 𝑦 ) − 𝑔 𝑥 , 𝑦 ∗ ( | | 𝑥 ) ≤ 𝐿 2 ‖ 𝑦 − 𝑦 ∗ ( 𝑥 ) ‖ . ( 2 . 2 2 ) As to 𝑥 1 ∈ 𝑈 ( 𝑥 , 𝛿 ) , from the conclusion in (1.2), there exists a a 𝐿 1 > 0 such that ‖ 𝑦 ∗ ( 𝑥 1 ) − 𝑦 ∗ ( 𝑥 ) ‖ ≤ 𝐿 1 ‖ 𝑥 1 − 𝑥 ‖ . As a result, | | 𝜐 𝑥 1 | | = | | 𝑔 𝑥 − 𝜐 ( 𝑥 ) 1 , 𝑦 ∗ 𝑥 1 − 𝑔 𝑥 , 𝑦 ∗ ( | | = | | 𝑔 𝑥 𝑥 ) 1 , 𝑦 ∗ 𝑥 1 𝑥 − 𝑔 1 , 𝑦 ∗ 𝑥 ( 𝑥 ) + 𝑔 1 , 𝑦 ∗ ( 𝑥 ) − 𝑔 𝑥 , 𝑦 ∗ | | ≤ | | 𝑔 𝑥 ( 𝑥 ) 1 , 𝑦 ∗ 𝑥 1 𝑥 − 𝑔 1 , 𝑦 ∗ ( | | + | | 𝑔 𝑥 𝑥 ) 1 , 𝑦 ∗ ( 𝑥 ) − 𝑔 𝑥 , 𝑦 ∗ ( | | 𝑥 ) ≤ 𝐿 2 ‖ ‖ 𝑦 ∗ 𝑥 1 − 𝑦 ∗ ‖ ‖ ( 𝑥 ) + 𝐿 2 ‖ ‖ 𝑥 1 ‖ ‖ − 𝑥 ≤ 𝐿 2 1 + 𝐿 1 ‖ ‖ 𝑥 1 ‖ ‖ . − 𝑥 ( 2 . 2 3 ) Hence, the marginal function 𝜐 ( 𝑥 ) is a local Lipschitzian function with respect to 𝑥 .
Let 𝑆 ( 𝑥 ) = { − ∇ 𝑥 𝑔 𝑖 ( 𝑥 , 𝑦 ( 𝑥 ) ) , 𝑖 ∈ 𝐼 ( 𝑥 , 𝑦 ( 𝑥 ) ) } , then 𝑆 ( 𝑥 ) = c o { 𝜉 , 𝜉 ∈ 𝑆 ( 𝑥 ) } . We prove that 𝑆 ( 𝑥 ) is closed first, that is, prove for any sequence { 𝑥 𝑘 } ⊂ 𝑅 𝑛 , 𝑥 𝑘 → 𝑥 , 𝑧 𝑘 ∈ 𝑆 ( 𝑥 𝑘 ) , 𝑧 𝑘 → 𝑧 , it has 𝑧 ∈ 𝑆 ( 𝑥 ) .
From 𝑧 𝑘 ∈ 𝑆 ( 𝑥 𝑘 ) , there exist 𝑦 𝑘 ∈ 𝑌 ( 𝑥 𝑘 ) ; 𝑖 𝑘 ∈ 𝐼 ( 𝑥 𝑘 , 𝑦 𝑘 ) such that 𝑧 𝑘 = − ∇ 𝑥 𝑔 𝑖 𝑘 ( 𝑥 𝑘 , 𝑦 𝑘 ) . Without loss of generality, suppose that { 𝑦 𝑘 } converges to 𝑦 ; { 𝑖 𝑘 } converges to 𝑖 . From Proposition 3.3 in [14], it has 𝑦 ∈ 𝑌 ( 𝑥 ) , and 𝑖 ∈ 𝐼 ( 𝑥 , 𝑦 ) and from ∇ 𝑥 𝑔 𝑖 ( 𝑥 , 𝑦 ) is a continuous function, it has 𝑧 = l i m 𝑘 → + ∞ 𝑧 𝑘 = l i m 𝑘 → + ∞ ∇ 𝑥 𝑔 𝑖 𝑘 ( 𝑥 𝑘 , 𝑦 𝑘 ) = ∇ 𝑥 𝑔 𝑖 ( 𝑥 , 𝑦 ) ∈ 𝑆 ( 𝑥 ) . As a result, 𝑆 ( 𝑥 ) is a closed set.
From Theorem 3 . 2 . 1 6 in [11], for any 𝜉 ∈ 𝜕 𝜐 ( 𝑥 ) , there exists 𝑥 𝑘 ∈ 𝑅 𝑛 , 𝑥 𝑘 → 𝑥 such that ∇ 𝜐 ( 𝑥 𝑘 ) exists and 𝜉 = l i m 𝑘 → + ∞ ∇ 𝜐 ( 𝑥 𝑘 ) . In addition, for arbitrary 𝑑 ∈ 𝑅 𝑛 , it has 𝑥 ∇ 𝜐 𝑘 𝑇 𝑑 = 𝜐 𝑥 𝑘 ; 𝑑 = l i m 𝑡 ↓ 0 𝜐 𝑥 𝑘 𝑥 + 𝑡 𝑑 − 𝜐 𝑘 𝑡 = l i m 𝑡 ↓ 0 𝑔 𝑥 𝑘 + 𝑡 𝑑 , 𝑦 ∗ 𝑥 𝑘 𝑥 + 𝑡 𝑑 − 𝑔 𝑘 , 𝑦 ∗ 𝑥 𝑘 𝑡 ≤ l i m 𝑡 ↓ 0 𝑔 𝑥 𝑘 + 𝑡 𝑑 , 𝑦 ∗ 𝑥 𝑘 𝑥 − 𝑔 𝑘 , 𝑦 ∗ 𝑥 𝑘 𝑡 = m a x 𝑥 𝑖 ∈ 𝐼 𝑘 , 𝑦 − ∇ 𝑥 𝑔 𝑖 𝑥 𝑘 , 𝑦 𝑇 𝑑 . ( 2 . 2 4 ) From the definition of 𝑆 ( 𝑥 𝑘 ) , ∃ 𝑧 𝑘 ∈ 𝑆 ( 𝑥 𝑘 ) such that 𝑧 𝑘 𝑇 𝑑 = m a x 𝑖 ∈ 𝐼 ( 𝑥 𝑘 , 𝑦 ) { − ∇ 𝑥 𝑔 𝑖 ( 𝑥 𝑘 , 𝑦 ) 𝑇 𝑑 } . Hence, it has ∇ 𝜐 ( 𝑥 𝑘 ) 𝑇 𝑑 ≤ 𝑧 𝑘 𝑇 𝑑 .
From 𝑧 𝑘 → 𝑧 ∈ 𝑆 ( 𝑥 ) ⊂ 𝑆 ( 𝑥 ) , ∇ 𝜐 ( 𝑥 𝑘 ) → 𝜉 and ∇ 𝜐 ( 𝑥 𝑘 ) 𝑇 𝑑 ≤ 𝑧 𝑘 𝑇 𝑑 , it has 𝜉 𝑇 𝑑 ≤ 𝑧 𝑇 𝑑 , that is, for arbitrary 𝑑 ∈ 𝑅 𝑛 and 𝜉 ∈ 𝜕 𝜐 ( 𝑥 ) , there exists 𝑧 ∈ 𝑆 ( 𝑥 ) such that 𝜉 𝑇 𝑑 ≤ 𝑧 𝑇 𝑑 .
If 𝜕 𝜐 ( 𝑥 ) ⊂ 𝑆 ( 𝑥 ) does not hold, then there exists a 𝜉 ∈ 𝜕 𝜐 ( 𝑥 ) and 𝜉 ∉ 𝑆 ( 𝑥 ) . From 𝑆 ( 𝑥 ) is a compact convex set and separation theorem ([15]), there exists a 𝑑 ∈ 𝑅 𝑛 such that 𝜉 𝑇 𝑑 < 0 and for arbitrary 𝑧 ∈ 𝑆 ( 𝑥 ) , 𝑧 𝑇 𝑑 ≥ 0 , which leads to a contradiction. As a result, 𝜕 𝜐 ( 𝑥 ) ⊂ 𝑆 ( 𝑥 ) holds. From 𝜕 𝜐 ( 𝑥 ) ⊂ 𝑆 ( 𝑥 ) and 𝑆 ( 𝑥 ) = c o { ∇ 𝑥 𝑔 𝑖 ( 𝑥 , 𝑦 ∗ ( 𝑥 ) ) , 𝑖 ∈ 𝐼 ( 𝑥 , 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 𝜐 ( 𝑥 ) 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.

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