Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 630547, 13 pages http://dx.doi.org/10.1155/2014/630547
Research Article Exact Penalization and Necessary Optimality Conditions for Multiobjective Optimization Problems with Equilibrium Constraints Shengkun Zhu1,2 and Shengjie Li2 1 2
Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
Correspondence should be addressed to Shengkun Zhu;
[email protected] Received 1 December 2013; Accepted 16 March 2014; Published 15 May 2014 Academic Editor: Geraldo Botelho Copyright Š 2014 S. Zhu and S. Li. 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. A calmness condition for a general multiobjective optimization problem with equilibrium constraints is proposed. Some exact penalization properties for two classes of multiobjective penalty problems are established and shown to be equivalent to the calmness condition. Subsequently, a Mordukhovich stationary necessary optimality condition based on the exact penalization results is obtained. Moreover, some applications to a multiobjective optimization problem with complementarity constraints and a multiobjective optimization problem with weak vector variational inequality constraints are given.
1. Introduction In this paper, we consider a general multiobjective optimization problem with equilibrium constraints as follows: (MOPEC) min
đ (đĽ)
s.t.
đ (đĽ) â âRđ+ ,
Ě â âRđ+ \ {0Rđ } (resp. â int Rđ+ ) , đ (đĽ) â đ (đĽ) (1)
â (đĽ) = 0Rđ , 0Rđ â đ (đĽ) + đ (đĽ) ,
đ is closed (i.e., gphđ is closed in Rđ Ă Rđ ), and the feasible set đ := {đĽ â Rđ | đ(đĽ) â âRđ+ , â(đĽ) = 0Rđ , 0Rđ â đ(đĽ) + đ(đĽ), đĽ â Î} of (MOPEC) is nonempty. Obviously, đ is a closed subset of Rđ . Recall that a point đĽĚ â đ is said to be an efficient (resp. weak efficient) solution for (MOPEC) if and only if
đĽ â Î,
where đ : Rđ â Rđ , đ(đĽ) = (đ1 (đĽ), đ2 (đĽ), . . . , đđ (đĽ)), đ : Rđ â Rđ , đ(đĽ) = (đ1 (đĽ), đ2 (đĽ), . . . , đđ (đĽ)) â : Rđ â Rđ , â(đĽ) = (â1 (đĽ), â2 (đĽ), . . . , âđ (đĽ)), đ : Rđ â Rđ , and đ(đĽ) = (đ1 (đĽ), đ2 (đĽ), . . . , đđ (đĽ)) are vector-valued maps, đ : Rđ ó´ó´ą Rđ is a set-valued map, and Î is a nonempty and closed subset of Rđ . As usual, we denote by int Î the interior of Î and by gphđ := {(đĽ, đŚ) â Rđ Ă Rđ | đŚ â đ(đĽ)} the graph of đ. Moreover, Rđ+ denotes the nonnegative quadrant in Rđ , and 0Rđ and 0Rđ denote, respectively, the origins of Rđ and Rđ . Throughout this paper, we assume that đ is locally Lipschitz, đ, â, and đ are continuously Fr´echet differentiable,
âđĽ â đ. (2)
A point đĽĚ â đ is said to be a local efficient (resp. local weak efficient) solution for (MOPEC) if and only if there exists a neighborhood đ of đĽĚ such that Ě â âRđ+ \ {0Rđ } (resp. â int Rđ+ ) , đ (đĽ) â đ (đĽ)
âđĽ â đ ⊠đ. (3)
During the past few decades, there have been a lot of papers devoted to study the scalar optimization problem (i.e., the case đ = 1) with equilibrium constraints, which plays an important role in engineering design, economic equilibria, operations research, and so on. It is well recognized that the scalar optimization problem with equilibrium constraints covers various classes of optimization-related problems and
2 models arisen in practical applications, such as mathematical programs with geometric constraints, mathematical programs with complementarity constraints, and mathematical programs with variational inequality constraints. For more details, we refer to [1â4]. It is worth noting that when đ is a general closed set-valued map, even if đ(đĽ) is a fixed closed subset of Rđ for all đĽ â Rđ , the general constraint system (1) fails to satisfy the standard linear independence constraint qualification and Mangasarian-Fromovitz constraint qualification at any feasible point [5]. Thus, it is a hard work to establish Karush-Kuhn-Tucker (in short, KKT) necessary optimality conditions for (MOPEC). Recently, by virtue of advanced tools of variational analysis and various coderivatives for set-valued maps developed in [6â8] and references therein, some necessary optimality conditions including the strong, Mordukhovich, Clarke, and Bouligand stationary conditions are obtained by using different reformulations under some generalized constraint qualifications. Simultaneously, Ye and Zhu [3] claimed that the Mordukhovich stationary (in short, M-stationary) condition is the strongest stationary condition except the strong stationary condition which is equivalent to the classical KKT condition, and proposed some new constraint qualifications for M-stationary conditions to hold. It is well known that the penalization method is a very important and effective tool for dealing with optimization theories and numerical algorithms of constrained extremum problems. In scalar optimization with equality and inequality constraints, the classical exact penalty function with order 1 was extensively used to investigate optimality conditions and convergence analysis; see [6, 9, 10] and references therein. Clarke [6] derived some Fritz-John necessary optimality conditions for a constrained mathematical programming problem on a Banach space by virtue of exact penalty functions with order 1. Moreover, Burke [9] showed that the existence of an exact penalization function is equivalent to a calmness condition involving with the objective function and the equality and inequality constrained system. Subsequently, Flegel and Kanzow [4] demonstrated that the corresponding relationships still held in a generalized bilevel programming problem and a mathematical programming problem with complementarity constraints, respectively. Simultaneously, they obtained some KKT necessary optimality conditions by using exact penalty formulations and nonsmooth analysis. Recently, the classical penalization theory has been widely generalized by various kinds of Lagrangian functions, especially the augmented Lagrangian function, introduced by Rockafellar and Wets [7], and the nonlinear Lagrangian function, proposed by Rubinov et al. [11]. It has also been proved that the exactness of these types of penalty functions is equivalent to some generalized calmness conditions; see more details in [11, 12]. However, to the best of our knowledge, there are only a few papers devoted to study the penalty method for constrained multiobjective optimization problems, especially, for (MOPEC). Huang and Yang [13] first introduced a vectorvalued nonlinear Lagrangian and penalty functions for multiobjective optimization problems with equality and inequality constraints and obtained some relationships between the exact penalization property and a generalized calmness-type
Abstract and Applied Analysis condition. Moreover, Mordukhovich [8] and Bao et al. [14] investigated some more general optimization problems with equilibrium constraints by methods of modern variational analysis. It is worth noting that the standard MangasarianFromovitz constraint qualification and error bound condition for a nonlinear programming problem with equality and inequality constraints implies the calmness condition; see [6, 15] for details. Taking into account this fact, it is necessary to further investigate the calmness condition and the penalty method for constrained multiobjective optimization problems. The main motivation of this work is that there has been no study on the penalization method and M-stationary condition for (MOPEC) by using an appropriate calmness condition associated with the objective function and the constraint system. Although there have been many papers dealing with constrained multiobjective optimization problems, for example, [3, 8] and references therein, the KKT necessary optimality conditions are obtained under some generalized qualification conditions only involved with the constraint system. Inspired by the ideas reported in [3, 4, 6, 8, 13], we introduce a so-called (MOPEC-) calmness condition with order đ > 0 at a local efficient (weak efficient) solution associated with the objective function and the constraint system for (MOPEC) and show that the (MOPEC-) calmness condition can be implied by an error bound condition of the constraint system. Moreover, we establish some equivalent relationships between the exact penalization property with order đ and the (MOPEC-) calmness condition. Simultaneously, we apply a nonlinear scalar technical to obtain a KKT necessary optimality condition for (MOPEC) by using Mordukhovich generalized differentiation and the (MOPEC-) calmness condition with order 1. The organization of this paper is as follows. In Section 2, we recall some basic concepts and tools generally used in variational analysis and set-valued analysis. In Section 3, we introduce a (MOPEC-) calmness condition for (MOPEC) and establish some relationships between the exact penalization property and the (MOPEC-) calmness condition. Moreover, we obtain a KKT necessary optimality condition under the (MOPEC-) calmness condition with order 1. In Section 4, we apply the obtained results to a multiobjective optimization problem with complementarity constraints and a multiobjective optimization problem with weak vector variational inequality constraints, respectively.
2. Notations and Preliminaries Throughout this paper, all vectors are viewed as column vectors. Since all the norms on finite dimensional spaces are equivalent, we take specially the sum norm on Rđ and the product space Rđ Ă Rđ for simplicity; that is, for all đĽ = (đĽ1 , đĽ2 , . . . , đĽđ )đ â Rđ , âđĽâ = |đĽ1 | + |đĽ2 | + â
â
â
+ |đĽđ |, and, for all (đĽ, đŚ) â Rđ Ă Rđ , â(đĽ, đŚ)â = âđĽâ + âđŚâ. As usual, we denote by đĽđ the transposition of đĽ and by â¨đĽ, đŚâŠ := đĽđ đŚ the inner product of vectors đĽ and đŚ, respectively. For a given map đ : Rđ â Rđ and a vector đ â Rđ , the function â¨đ, đ⊠: Rđ â R is defined by â¨đ, đâŠ(đĽ) := â¨đ, đ(đĽ)⊠for
Abstract and Applied Analysis
3
all đĽ â Rđ . In general, we denote by BRđ the closed unit ball Ě đ) the open ball with center at đĽĚ and radius in Rđ and by B(đĽ, đ > 0 for any đĽĚ â Rđ . The main tools for our study in this paper are the Mordukhovich generalized differentiation notions which are generally used in variational analysis and set-valued analysis; see more details in [6â8, 16] and references therein. Recall that đ : Rđ â Rđ is said to be Fr´echet differentiable at đĽĚ if and only if there exists a matrix đ´ â RđĂđ such that
Ě Ě â(đĽ)))}, Ě â, (đĽ, or, equivalently, is defined in (đĽâ , â1) â đ(epi the analytical form by
óľŠóľŠ Ě â đ´ (đĽ â đĽ) Ě óľŠóľŠóľŠóľŠ óľŠđ (đĽ) â đ (đĽ) lim óľŠ = 0. đĽ â đĽĚ Ě âđĽ â đĽâ
Ě The Mordukhovich (or basic, limiting) subdifferential đâ(đĽ) Ě of â at đĽĚ are defined, and singular subdifferential đâ â(đĽ) Ě := {đĽâ â Rđ | (đĽâ , â1) â respectively, by đâ(đĽ) Ě := {đĽâ â Rđ | (đĽâ , 0) â Ě â(đĽ)))} Ě đ(epi â, (đĽ, and đâ â(đĽ) Ě â(đĽ)))}. Ě Ě â đâ(đĽ) Ě and đ(epi â, (đĽ, Clearly, we have Ěđâ(đĽ)
(4)
Ě As usual, đ´ is Obviously, đ´ is uniquely determined by đĽ. Ě called the Fr´echet derivative of đ at đĽĚ and denoted by âđ(đĽ). If đ is Fr´echet differentiable at every đĽĚ â Rđ , then đ is said to be Fr´echet differentiable on Rđ . đ is said to be continuously Fr´echet differentiable at đĽĚ if and only if the map âđ(â) : Ě Specially, we denote by Rđ â RđĂđ is continuous at đĽ. Ě that is, Ě â : Rđ â Rđ the adjoint operator of âđ(đĽ); (âđ(đĽ)) Ě Ě â (đŚ)⊠for all đĽ â Rđ and đŚ â Rđ . â¨âđ(đĽ)(đĽ), đŚâŠ = â¨đĽ, (âđ(đĽ)) Moreover, đ is said to be strictly differentiable at đĽĚ if and only if óľŠóľŠ Ě (đĽ â đ˘)óľŠóľŠóľŠóľŠ óľŠóľŠđ (đĽ) â đ (đ˘) â âđ (đĽ) = 0. (5) lim Ě â đĽ,đĽ Ě ≠ đ˘ đĽ â đĽ,đ˘ âđĽ â đ˘â Ě Obviously, if đ is continuously Fr´echet differentiable at đĽ, Ě then đ is strictly differentiable at đĽ. For a nonempty subset đ â Rđ , the indicator function đ(â, đ) : Rđ â R ⪠{+â} is defined by đ(đĽ, đ) := 0, âđĽ â đ and đ(đĽ, đ) := +â, âđĽ â đ, and the distance function đ(â, đ) : Rđ â R is defined by đ(đĽ, đ) := inf đŚâđ âđĽ â đŚâ for all đĽ â Rđ , respectively. Given a point đĽĚ â đ, recall that Ě đĽ) Ě of đ at đĽ, Ě which is a convex, the Fr´echet normal cone đ(đ, closed subset of Rđ and consisted of all the Fr´echet normals, has the form } { Ě â¨đĽâ , đĽ â đĽâŠ Ě (đ, đĽ) Ě := {đĽâ â Rđ | lim sup ⤠0} , đ Ě âđĽ â đĽâ đ đĽâ ół¨ đĽĚ } {
(6)
đ
Ě The Mordukhovich ół¨ đĽĚ means đĽ â đ and đĽ â đĽ. where đĽ â (or basic, limiting) normal cone of đ at đĽĚ is đ
with
Ě (đ, đĽđ ) , âđ â N} . âđ
(7)
Specially, if đ is convex, then we have Ě (đ, đĽ) Ě â¤ 0, âđĽ â đ} . Ě = đ (đ, đĽ) Ě = {đĽâ â Rđ | â¨đĽâ , đĽ â đĽâŠ đ (8) đ
Ě â â¨đĽâ , đĽ â đĽâŠ Ě â (đĽ) â â (đĽ) ⼠0} . Ě đĽ ≠ đĽĚ đĽ â đĽ, Ě âđĽ â đĽâ (9)
:= {đĽâ â Rđ | lim inf
â
Ě = {đĽâ â Rđ | âđĽđ ół¨â đĽ, Ě âđĽđâ ół¨â đĽâ đâ (đĽ) with
đĽđâ
(10)
â Ěđâ (đĽđ ) } ,
â
Ě Specially, where đĽđ â ół¨ đĽĚ means đĽđ â đĽĚ and â(đĽđ ) â â(đĽ). Ě đĽ) Ě đ) = đ(đ, Ě and đđ(đĽ, Ě đ) = for any đĽĚ â đ, it follows that Ěđđ(đĽ, Ě đ) = đ(đ, đĽ). Ě Furthermore, if â is a convex function, đâ đ(đĽ, then we have Ěđâ (đĽ) Ě = đâ (đĽ) Ě Ě â¤ â (đĽ) â â (đĽ) Ě , âđĽ â Rđ } , = {đĽâ â Rđ | â¨đĽâ , đĽ â đĽâŠ Ě â {đĽâ â Rđ | â¨đĽâ , đĽ â đĽâŠ Ě â¤ 0, âđĽ â dom â} đâ â (đĽ) Ě . = đ (dom â, đĽ) (11) Ěâ đš(đĽ, Ě đŚ) Ě and the MorRecall that the Fr´echet coderivative đˇ Ě đŚ) Ě of the dukhovich (or basic, limiting) coderivative đˇâ đš(đĽ, Ě đŚ) Ě â gphđš are the setset-valued map đš : Rđ ó´ó´ą Rđ at (đĽ, valued maps from Rđ to Rđ defined, respectively, by Ěâ đš (đĽ, Ě đŚ) Ě (đŚâ ) đˇ Ě (gphđš, (đĽ, Ě đŚ))} Ě , := {đĽâ â Rđ | (đĽâ , âđŚâ ) â đ âđŚâ â Rđ , Ě đŚ) Ě (đŚâ ) đˇâ đš (đĽ,
Ě âđĽđâ ół¨â đĽâ Ě : = {đĽâ â Rđ | âđĽđ ół¨â đĽ, đ (đ, đĽ) đĽđâ
Ěđâ (đĽ) Ě
Let â : R â R ⪠{+â} be an extended real-valued function and let đĽĚ â dom â, where dom â := {đĽ â Rđ | â(đĽ) < +â} Ě of denotes the domain of â. The Fr´echet subdifferential Ěđâ(đĽ) Ě := {đĽâ â Rđ | â at đĽĚ is defined in the geometric form by Ěđâ(đĽ)
(12)
Ě đŚ))} Ě , := {đĽâ â Rđ | (đĽâ , âđŚâ ) â đ (gphđš, (đĽ, âđŚâ â Rđ . Next, we collect some useful and important propositions and definitions for this paper. Proposition 1 (see [8]). For every nonempty subset Ί â Rđ and every đĽ â Ί, we have Ě Ě (i) Ěđđ(â, Ί)(đĽ) = BRđ ⊠đ(Ί, đĽ) and đ(Ί, đĽ) = Ě âđ>0 đđđ(â, Ί)(đĽ).
4
Abstract and Applied Analysis đ
In addition, if Ί is closed, then we get (ii) đđ(â, Ί)(đĽ) â BRđ ⊠đ(Ί, đĽ) and đ(Ί, đĽ) = âđ>0 đđđ(â, Ί)(đĽ). The following necessary optimality condition, called generalized Fermat rule, for a function to attain its local minimum is useful for our analysis. Proposition 2 (see [7, 8]). Let đ : Rđ â R ⪠{+â} be a proper lower semicontinuous function. If đ attains a local Ě and 0Rđ â đđ(đĽ). Ě minimum at đĽĚ â Rđ , then 0Rđ â Ěđđ(đĽ) We recall the following sum rule for the Mordukhovich subdifferential which is important in the sequel. Proposition 3 (see [8]). Let đ1 , đ2 : Rđ â R ⪠{+â} be proper lower semicontinuous functions and đĽ â dom đ1 ⊠dom đ2 . Suppose that the qualification condition đâ đ1 (đĽ) ⊠(âđâ đ2 (đĽ)) = {0Rđ } ,
(13)
đđ (đŚ) := inf {đź â R | đŚ â đźđ â Rđ+ } ,
âđŚ â Rđ ,
đ
(21)
đ
is convex, strictly int R+ -monotone, R+ -monotone, nonnegative homogeneous, globally Lipschitz with modulus đ đ(đ, bd R+ )â1 . Simultaneously, for every đź â R, it follows that đđ (đźđ) = đź, {đŚ â Rđ | đđ (đŚ) ⤠đź} = đźđ â Rđ+ , {đŚ â Rđ | đđ (đŚ) < đź} = đźđ â int Rđ+ .
(22)
Furthermore, for every đŚĚ â Rđ , đ
Ě = {đ â Rđ+ | âđ đ = 1, â¨đ, đŚâŠ Ě = đđ (đŚ)} Ě . đđđ (đŚ)
(23)
đ=1
Specially, one has đ
is fulfilled. Then one has đ (đ1 + đ2 ) (đĽ) â đđ1 (đĽ) + đđ2 (đĽ) .
(14)
Specially, if either đ1 or đ2 is locally Lipschitz around đĽ, then one always has đ (đ1 + đ2 ) (đĽ) â đđ1 (đĽ) + đđ2 (đĽ) .
(15)
The following propositions of the scalarization of Mordukhovich coderivatives and the chain rule of Mordukhovich subdifferentials are important for this paper. Proposition 4 (see [8, 16]). Let đ : Rđ â Rđ be continuous Ě Then around đĽ. Ě â đˇâ đ (đĽ) Ě (đŚâ ) , đ â¨đŚâ , đ⊠(đĽ)
âđŚâ â Rđ .
(16)
Ě then If in addition đ is locally Lipschitz around đĽ, Ě , Ě (đŚâ ) = đ â¨đŚâ , đ⊠(đĽ) đˇâ đ (đĽ)
âđŚâ â Rđ .
(17)
Proposition 5 (see [8, 16]). Let the vector-valued map đť : Rđ â Râ be locally Lipschitz and let â : Râ â R be lower semicontinuous. If đŚâ â đâ â (đť (đĽ)) ,
Lemma 6. Given đ = (1, 1, . . . , 1) â int R+ , the nonlinear scalar function đđ : Rđ â R, defined by
0Rđ â đˇâ đť (đĽ) (đŚâ ) đđđđđđđ đŚâ = 0Râ ,
(18)
(19)
Moreover, if đť is strictly differentiable and â is locally Lipschitz, then one always has đâ â đť (đĽ) â {(âđť (đĽ))â (đŚâ ) : đŚâ â đâ (đť (đĽ))} .
(24)
đ=1
3. Exact Penalization, Calmness Condition, and Necessary Optimality Condition for (MOPEC) In this section, we focus our attention on establishing some equivalent properties between a multiobjective exact penalization and a calmness condition, called (MOPEC-) calmness, for (MOPEC). Simultaneously, we show that a local error bound condition associated merely with the constraint system, equivalently, a calmness condition of the parametric constraint system, implies the (MOPEC-) calmness condition. Subsequently, we apply a nonlinear scalar method to obtain a M-stationary necessary optimality condition under the (MOPEC-) calmness condition. Consider the following parametric form of the feasible set đ with parameter (đ˘, V, đŚ, đ§) â Rđ+đ +đ+đ : đ (đĽ) + đ˘ â âRđ+ ,
â (đĽ) + V = 0Rđ ,
đ§ â đ (đĽ) + đ (đĽ + đŚ) ,
đĽ â Î.
(25)
Denote the corresponding feasible set by đ (đ˘, V, đŚ, đ§) : = {đĽ â Rđ | đ (đĽ) + đ˘ â âRđ+ , â (đĽ) + V = 0Rđ ,
then đâ â đť (đĽ) â {đˇâ đť (đĽ) (đŚâ ) : đŚâ â đâ (đť (đĽ))} .
đđđ (0Rđ ) = {đ â Rđ+ | âđ đ = 1} .
(20)
Finally in this section, we recall the following useful concept called nonlinear scalar function and some of its properties. For more details, we refer to [17â20].
đ§ â đ (đĽ) + đ (đĽ + đŚ) , đĽ â Î} . (26) Obviously, for the set-valued map đ : Rđ+đ +đ+đ ó´ó´ą Rđ , we have đ = đ(0Rđ+đ +đ+đ ). We are now in the position to introduce a (MOPEC-) calmness concept for (MOPEC). Definition 7. Given đ > 0 and đĽĚ â đ being a local efficient (resp. local weak efficient) solution for (MOPEC),
Abstract and Applied Analysis then (MOPEC) is said to be (MOPEC-) calm with order đ at đĽĚ if and only if there exist đż > 0 and đ > 0 such that, for all Ě đż), (đ˘, V, đŚ, đ§) â B(0Rđ+đ +đ+đ , đż) and all đĽ â đ(đ˘, V, đŚ, đ§) ⊠B(đĽ, one has óľŠđ óľŠ (27) Ě â int Rđ+ . đ (đĽ) + đóľŠóľŠóľŠ(đ˘, V, đŚ, đ§)óľŠóľŠóľŠ đ â đ (đĽ) Remark 8. Given đ > 0 and đĽĚ â đ being a local efficient (resp. local weak efficient) solution for (MOPEC), we can also characterize the (MOPEC-) calmness condition by means of sequences. It is easy to verify that (MOPEC) is (MOPEC-) calm with order đ at đĽĚ if and only if there exists đ > 0 such that, for every sequence {(đ˘đ , Vđ , đŚđ , đ§đ )} â Rđ+đ +đ+đ with (đ˘đ , Vđ , đŚđ , đ§đ ) â 0Rđ+đ +đ+đ and every sequence {đĽđ } â Î satisfying đ(đĽđ ) + đ˘đ â Rđ+ , â(đĽđ ) + Vđ = 0Rđ , đ§đ â đ(đĽđ ) + Ě it holds that đ(đĽđ + đŚđ ) and đĽđ â đĽ, óľŠ óľŠđ (28) Ě â int Rđ+ . đ (đĽđ ) + đóľŠóľŠóľŠ(đ˘đ , Vđ , đŚđ , đ§đ )óľŠóľŠóľŠ đ â đ (đĽ) Note that the (MOPEC-) calmness condition depends on not only the objective function but also the constraint system. In order to make up this deficiency, we propose the following local error bound notion for (MOPEC) associated merely with the constraint system. Definition 9. Given đ > 0 and đĽĚ â đ, then the constraint system of (MOPEC) is said to have a local error bound with order đ at đĽĚ if and only if there exist đż > 0 and đ > 0 such that, for all (đ˘, V, đŚ, đ§) â B(0Rđ+đ +đ+đ , đż) \ {0Rđ+đ +đ+đ } and all đĽ â Ě đż), one has đ(đ˘, V, đŚ, đ§) ⊠B(đĽ, óľŠđ óľŠ (29) đ (đĽ, đ) BRđ â đóľŠóľŠóľŠ(đ˘, V, đŚ, đ§)óľŠóľŠóľŠ đ â int Rđ+ . Next, we show that the local error bound implies the (MOPEC-) calmness.
5 Together with đĽĚ â đ being a local efficient (resp. local weak efficient) solution for (MOPEC), there exists some đ1 â N such that Ě â âRđ+ \ {0Rđ } (resp. â int Rđ+ ) , đ (đ (đĽđ , đ)) â đ (đĽ) âđ ⼠đ1 .
(33)
Moreover, since đ is locally Lipschitz, there exist đż > 0 and đ2 â N such that óľŠ óľŠ óľŠ óľŠóľŠ óľŠóľŠđ (đĽđ ) â đ (đ (đĽđ , đ))óľŠóľŠóľŠ ⤠đż óľŠóľŠóľŠđĽđ â đ (đĽđ , đ)óľŠóľŠóľŠ ,
âđ ⼠đ2 . (34)
By (31) and (33), we have for all đ ⼠đ1 đ (đ (đĽđ , đ)) â đ (đĽđ ) Ě = đ (đ (đĽđ , đ)) â đ (đĽ) đ Ě â đ (đĽđ )) â đóľŠóľŠóľŠóľŠ(đ˘đ , Vđ , đŚđ , đ§đ )óľŠóľŠóľŠóľŠ đ â int Rđ+ . + (đ (đĽ) (35)
Together with đ(đĽđ , đ) = âđĽđ â đ(đĽđ , đ)â and (34), we can conclude that đ (đĽđ , đ) BRđ â̸
đ óľŠóľŠ óľŠđ đ óľŠ(đ˘ , V , đŚ , đ§ )óľŠóľŠ đ â int R+ , đżóľŠ đ đ đ đ óľŠ
(36)
âđ ⼠max {đ1 , đ2 } . This is a contradiction to the assumption that (MOPEC) has a local error bound with order đ at đĽĚ since đ/đż â +â, (đ˘đ , Vđ , đŚđ , đ§đ ) ≠ 0Rđ+đ +đ+đ , (đ˘đ , Vđ , đŚđ , đ§đ ) â 0Rđ+đ +đ+đ , đĽđ â Ě đ(đ˘đ , Vđ , đŚđ , đ§đ ), and đĽđ â đĽ.
Theorem 10. Let đĽĚ â đ be a local efficient (resp. local weak efficient) solution for (MOPEC). If the constraint system of Ě then (MOPEC) has a local error bound with order đ at đĽ, Ě (MOPEC) is (MOPEC-) calm with order đ at đĽ.
Remark 11. Specially, if we consider the case đ = 1 for every given đ > 0 and đĽĚ â đ, then Definition 9 reduces to the fact that there exist đż > 0 and đ > 0 such that, for all (đ˘, V, đŚ, đ§) â Ě đż), B(0Rđ+đ +đ+đ , đż) \ {0Rđ+đ +đ+đ } and all đĽ â đ(đ˘, V, đŚ, đ§) ⊠B(đĽ, one has
Proof. Since đĽĚ â đ is a local efficient (resp. local weak efficient) solution for (MOPEC) and đ = đ(0Rđ+đ +đ+đ ), it immediately follows that óľŠđ óľŠ (30) Ě â int Rđ+ đ (đĽ) + đóľŠóľŠóľŠ(đ˘, V, đŚ, đ§)óľŠóľŠóľŠ đ â đ (đĽ)
óľŠđ óľŠ đ (đĽ, đ) < đóľŠóľŠóľŠ(đ˘, V, đŚ, đ§)óľŠóľŠóľŠ .
Ě đż) with (đ˘, V, đŚ, đ§) = holds for all đĽ â đ(đ˘, V, đŚ, đ§) ⊠B(đĽ, 0Rđ+đ +đ+đ and sufficiently small đż > 0. Thus, we only need to prove the case (đ˘, V, đŚ, đ§) ≠ 0Rđ+đ +đ+đ . Assume that (MOPEC) is Ě Then, for every đ â N, not (MOPEC-) calm with order đ at đĽ. there exist (đ˘đ , Vđ , đŚđ , đ§đ ) â B(0Rđ+đ +đ+đ , 1/đ) \ {0Rđ+đ +đ+đ } and Ě 1/đ) such that đĽđ â đ(đ˘đ , Vđ , đŚđ , đ§đ ) ⊠B(đĽ, óľŠóľŠ óľŠđ (31) Ě â int Rđ+ . đ (đĽđ ) + đóľŠóľŠ(đ˘đ , Vđ , đŚđ , đ§đ )óľŠóľŠóľŠ đ â đ (đĽ) Since đ is nonempty and closed, there exists a projection đ(đĽđ , đ) of đĽđ onto đ such that đ(đĽđ , đ) = âđĽđ â đ(đĽđ , đ)â for all đ â N. Note that đĽđ â đĽĚ and đĽĚ â đ. Then it follows that óľŠóľŠ óľŠ óľŠ óľŠ óľŠ óľŠ óľŠóľŠđ (đĽđ , đ) â đĽĚóľŠóľŠóľŠ ⤠óľŠóľŠóľŠđ (đĽđ , đ) â đĽđ óľŠóľŠóľŠ + óľŠóľŠóľŠđĽđ â đĽĚóľŠóľŠóľŠ (32) óľŠ óľŠ = đ (đĽđ , đ) + óľŠóľŠóľŠđĽđ â đĽĚóľŠóľŠóľŠ ół¨â 0.
(37)
It is worth noting that this condition is essentially sufficient and necessary for the situation đ > 1. Clearly, the necessity holds. In fact, since (1/đ)đ â BRđ , it follows that đ (đĽ, đ)
1 óľŠđ óľŠ đ â đóľŠóľŠóľŠ(đ˘, V, đŚ, đ§)óľŠóľŠóľŠ đ â int Rđ+ , đ
(38)
for all (đ˘, V, đŚ, đ§) â B(0Rđ+đ +đ+đ , đż) \ {0Rđ+đ +đ+đ } and all Ě đż). Moreover, đđ is nonnegative đĽ â đ(đ˘, V, đŚ, đ§) ⊠B(đĽ, homogeneous and đđ ((1/đ)đ) = 1/đ. By Lemma 6, we have đđ (đ(đĽ, đ)(1/đ)đ) < đâ(đ˘, V, đŚ, đ§)âđ , which implies đ(đĽ, đ) < đđâ(đ˘, V, đŚ, đ§)âđ . For the sufficiency, since đđ is continuous and BRđ is compact, there exists some đ â R such that đ = max đđ (đ¤) . đ¤âBRđ
(39)
Obviously, (1/đ)đ â BRđ and đđ ((1/đ)đ) = 1/đ > 0; then we have đ > 0. Thus, we get from the nonnegative homogeneity
6
Abstract and Applied Analysis Ě (i) (MOPEC) is (MOPEC-) calm with order đ > 0 at đĽ.
of đđ that, for all (đ˘, V, đŚ, đ§) â B(0Rđ+đ +đ+đ , đż) \ {0Rđ+đ +đ+đ }, all Ě đż) and all đ¤ â BRđ , đĽ â đ(đ˘, V, đŚ, đ§) ⊠B(đĽ, óľŠđ óľŠ đđ (đ (đĽ, đ) đ¤) = đ (đĽ, đ) đđ (đ¤) < đđóľŠóľŠóľŠ(đ˘, V, đŚ, đ§)óľŠóľŠóľŠ . (40)
Ě (ii) There exists some đĚ > 0 such that, for any đ ⼠đ, Ě 0Rđ+đ ) is a local efficient (resp. local weak efficient) (đĽ, solution for the following multiobjective penalty problem with order đ:
By Lemma 6, we have óľŠđ óľŠ đ (đĽ, đ) đ¤ â đđóľŠóľŠóľŠ(đ˘, V, đŚ, đ§)óľŠóľŠóľŠ đ â int Rđ+ ,
(đđđ)đź
(41)
which implies óľŠđ óľŠ đ (đĽ, đ) BRđ â đđóľŠóľŠóľŠ(đ˘, V, đŚ, đ§)óľŠóľŠóľŠ đ â int Rđ+ .
(42)
Furthermore, recall that a set-valued map Ψ : RđĄ ó´ó´ą Rđ is said to be calm with order đ > 0 at (đĽ, đŚ) â gphΨ if and only if there exist neighborhoods đ of đĽ and đ of đŚ and a real number â > 0 such that
min
óľŠđ óľŠ óľŠ óľŠ đ (đĽ) + đ (óľŠóľŠóľŠđ+ (đĽ)óľŠóľŠóľŠ + ââ (đĽ)â + óľŠóľŠóľŠ(đŚ, đ§)óľŠóľŠóľŠ) đ,
đ .đĄ.
đ§ â đ (đĽ) + đ (đĽ + đŚ) ,
(45)
đĽ â Î, (đŚ, đ§) â Rđ+đ ,
(43)
where đ+ (đĽ) := (max{đ1 (đĽ), 0}, max{đ2 (đĽ), 0}, . . ., max{đđ (đĽ), 0}).
Then we can immediately obtain the following characterization of local error bounds for the constraint system of (MOPEC) based on the arguments in Remark 11.
Ě đĽĚ is a (iii) There exists some đĚ > 0 such that, for any đ ⼠đ, local efficient (resp. local weak efficient) solution for the following multiobjective penalty problem with order đ:
đ
Ψ (đĽ) ⊠đ â Ψ (đĽ) + ââđĽ â đĽâ BRđ ,
âđĽ â đ.
Proposition 12. Given đ > 0 and đĽĚ â đ, then the following assertions are equivalent.
(đđđ)đźđź min
(i) The constraint system of (MOPEC) has a local error Ě bound with order đ at đĽ. (ii) There exist đż > 0 and đ > 0 such that, for all (đ˘, V, đŚ, đ§) â B(0Rđ+đ +đ+đ , đż) \ {0Rđ+đ +đ+đ } and all đĽ â Ě đż), đ(đ˘, V, đŚ, đ§) ⊠B(đĽ, óľŠđ óľŠ (44) đ (đĽ, đ (0Rđ+đ +đ+đ )) < đóľŠóľŠóľŠ(đ˘, V, đŚ, đ§)óľŠóľŠóľŠ . (iii) The set-valued map đ : Rđ+đ +đ+đ ó´ó´ą Rđ , defined by Ě (26), is calm with order đ at (0Rđ+đ +đ+đ , đĽ). Proof. As discussed in Remark 11, (i) is equivalent to (ii). We only need to prove the equivalence of (ii) and (iii). In fact, it follows from the definition of calmness for a set-valued map that (ii) is obviously equivalent to the calmness with order đ Ě of the set-valued map đ. at (0Rđ+đ +đ+đ , đĽ) As we know, there have been many papers devoted to investigate the calmness of a set-valued map Ψ (which is equivalent to the metric subregularity of its converse Ψâ1 ). For more details, we refer to [21â24] and references therein. It has been shown in Remark 11 and Proposition 12 that there have been no differences between the scalar (đ = 1) and the multiobjective (đ > 1) settings when we only consider the calmness or the local error bound for the constraint system of (MOPEC). However, if we pay attention to the weaker (MOPEC-) calmness, we cannot negative the differences between them. We now give the following equivalent characterizations of two classes of multiobjective penalty problems and the (MOPEC-) calmness condition. Theorem 13. Let đĽĚ â đ be a local efficient (resp. local weak efficient) solution for (MOPEC). Then the following assertions are equivalent.
đ (đĽ) đ óľŠ óľŠ + đ[óľŠóľŠóľŠđ+ (đĽ)óľŠóľŠóľŠ + ââ (đĽ)â + đ ((đĽ, âđ (đĽ)) , gphđ)] đ,
đ .đĄ.
đĽ â Î. (46)
Proof. We only prove the case for đĽĚ being a local weak efficient solution since the proof of the case for đĽĚ being a local efficient solution is similar. (i)â(ii). Suppose to the contrary that, for every đ â N, Ě 0Rđ+đ ), 1/đ) with đĽđ â Î and there exists (đĽđ , đŚđ , đ§đ ) â B((đĽ, đ§đ â đ(đĽđ ) + đ(đĽđ + đŚđ ) such that óľŠ óľŠ óľŠ óľŠ đ (đĽđ ) + đ (óľŠóľŠóľŠđ+ (đĽđ )óľŠóľŠóľŠ + óľŠóľŠóľŠâ (đĽđ )óľŠóľŠóľŠ óľŠ óľŠđ Ě â int Rđ+ . + óľŠóľŠóľŠ(đŚđ , đ§đ )óľŠóľŠóľŠ) đ â đ (đĽ)
(47)
Take đ˘đ = âđ+ (đĽđ ) and Vđ = ââ(đĽđ ). Then it follows that đ(đĽđ ) + đ˘đ â âRđ+ and â(đĽđ ) + Vđ = 0Rđ . Together with đ§đ â đ(đĽđ ) + đ(đĽđ + đŚđ ) and đĽđ â Î, we get đĽđ â đ(đ˘đ , Vđ , đŚđ , đ§đ ) for all đ â N. Moreover, by (47), we have óľŠ óľŠđ Ě â int Rđ+ . đ (đĽđ ) + đóľŠóľŠóľŠ(đ˘đ , Vđ , đŚđ , đ§đ )óľŠóľŠóľŠ đ â đ (đĽ)
(48)
Ě đ(đĽ) Ě â âRđ+ , â(đĽ) Ě = 0Rđ , and đ and â Note that đĽđ â đĽ, are continuously Fr´echet differentiable. Then it follows that đ˘đ = đ+ (đĽđ ) â 0Rđ and Vđ = â(đĽđ ) â 0Rđ . Together with (đŚđ , đ§đ ) â 0Rđ+đ and (48), this is a contradiction to the Ě (MOPEC-) calmness with order đ of (MOPEC) at đĽ. (ii)â(i). Suppose that (MOPEC) is not (MOPEC-) calm Ě Then, for every đ â N, there exist with order đ > 0 at đĽ. (đ˘đ , Vđ , đŚđ , đ§đ ) â B(0Rđ+đ +đ+đ , 1/đ) and đĽđ â đ(đ˘đ , Vđ , đŚđ , đ§đ ) âŠ Ě 1/đ) such that (31) holds. Since đĽđ â đ(đ˘đ , Vđ , đŚđ , đ§đ ), it B(đĽ, follows that đĽđ â Î, đ(đĽđ ) + đ˘đ â âRđ+ , â(đĽđ ) + Vđ = 0Rđ and
Abstract and Applied Analysis
7
đ§đ â đ(đĽđ ) + đ(đĽđ + đŚđ ); that is, (đĽđ + đŚđ , đ§đ â đ(đĽđ )) â gphđ for all đ â N. Thus, we have óľŠ óľŠ óľŠ óľŠóľŠ óľŠóľŠđ+ (đĽđ )óľŠóľŠóľŠ + óľŠóľŠóľŠâ (đĽđ )óľŠóľŠóľŠ óľŠ óľŠ óľŠ óľŠ â¤ óľŠóľŠóľŠđ (đĽđ ) â (đ (đĽđ ) + đ˘đ )óľŠóľŠóľŠ + óľŠóľŠóľŠâ (đĽđ ) â (â (đĽđ ) + Vđ )óľŠóľŠóľŠ óľŠ óľŠ óľŠ óľŠ = óľŠóľŠóľŠđ˘đ óľŠóľŠóľŠ + óľŠóľŠóľŠVđ óľŠóľŠóľŠ , âđ â N, (49) which implies (âđ+ (đĽđ )â + ââ(đĽđ )â + â(đŚđ , đ§đ )â)đ â(đ˘đ , Vđ , đŚđ , đ§đ )âđ , âđ â N. Together with (31), we get
đ
Connecting đ â int R+ , (51), and (54), we have for any đ â N
đ (đĽđ ) +
đđ+1 óľŠóľŠ óľŠđ Ě óľŠ(đ˘ , V , đŚ , đ§ )óľŠóľŠ đ â đ (đĽ) (đ + 1)đ óľŠ đ đ đ đ óľŠ
óľŠ óľŠ óľŠ óľŠ = đ (đĽđ ) + đ [óľŠóľŠóľŠđ+ (đĽđ )óľŠóľŠóľŠ + óľŠóľŠóľŠâ (đĽđ )óľŠóľŠóľŠ đ
Ě +đ ((đĽđ , âđ (đĽđ )) , gphđ)] đ â đ (đĽ)
⤠+
óľŠ óľŠ óľŠ óľŠ óľŠ óľŠđ đ (đĽđ ) + đ(óľŠóľŠóľŠđ+ (đĽđ )óľŠóľŠóľŠ + óľŠóľŠóľŠâ (đĽđ )óľŠóľŠóľŠ + óľŠóľŠóľŠ(đŚđ , đ§đ )óľŠóľŠóľŠ) đ
đđ+1 1 đ óľŠóľŠ óľŠóľŠ óľŠóľŠ óľŠóľŠ óľŠóľŠ óľŠđ đ˘đ óľŠóľŠ + óľŠóľŠVđ óľŠóľŠ + óľŠóľŠ(đŚđ , đ§đ )óľŠóľŠóľŠ) â (1 + ) đ {(óľŠ óľŠ đ (đ + 1) óľŠ óľŠ óľŠ óľŠ Ă [óľŠóľŠóľŠđ+ (đĽđ )óľŠóľŠóľŠ + óľŠóľŠóľŠâ (đĽđ )óľŠóľŠóľŠ
óľŠ óľŠđ = đ (đĽđ ) + đóľŠóľŠóľŠ(đ˘đ , Vđ , đŚđ , đ§đ )óľŠóľŠóľŠ đ óľŠ óľŠ óľŠ óľŠ óľŠ óľŠđ + đ [(óľŠóľŠóľŠđ+ (đĽđ )óľŠóľŠóľŠ + óľŠóľŠóľŠâ (đĽđ )óľŠóľŠóľŠ + óľŠóľŠóľŠ(đŚđ , đ§đ )óľŠóľŠóľŠ) óľŠ óľŠđ Ě â óľŠóľŠóľŠ(đ˘đ , Vđ , đŚđ , đ§đ )óľŠóľŠóľŠ ] đ â đ (đĽ) â
int Rđ+
â
â â int Rđ+ â int Rđ+ = â int Rđ+ .
int Rđ+
Ě â int Rđ+ , = đ (đĽ)
đ
+ đ ((đĽđ , âđ (đĽđ )) , gphđ)] } đ
(50)
(55)
âđ â N.
This shows that the multiobjective penalty problem (MPP)I with order đ does not admit a local exact penalization at Ě 0Rđ+đ ) since đĽđ â Î, đ§đ â đ(đĽđ ) + đ(đĽđ + đŚđ ), and (đĽ, Ě 0Rđ+đ ). (đĽđ , đŚđ , đ§đ ) â (đĽ, (i)â(iii). Assume that, for every đ â N, there exists đĽđ â Ě 1/đ) such that Π⊠B(đĽ, đ (đĽđ )
Moreover, it follows from [25, Lemma 3.21] and (51) that for đ đ any đ â R+ with âđ=1 đ đ = 1 we get đ óľŠ óľŠ óľŠ óľŠ [óľŠóľŠóľŠđ+ (đĽđ )óľŠóľŠóľŠ + óľŠóľŠóľŠâ (đĽđ )óľŠóľŠóľŠ + đ ((đĽđ , âđ (đĽđ )) , gphđ)]
đ óľŠ óľŠ óľŠ óľŠ + đ[óľŠóľŠóľŠđ+ (đĽđ )óľŠóľŠóľŠ + óľŠóľŠóľŠâ (đĽđ )óľŠóľŠóľŠ + đ ((đĽđ , âđ (đĽđ )) , gphđ)] đ
Ě â â int Rđ+ . â đ (đĽ)
(51)
Note that óľŠ óľŠ đ ((đĽđ , âđ (đĽđ )) , gphđ) = inf óľŠóľŠóľŠ(đĽđ , âđ (đĽđ )) â (đź, đ˝)óľŠóľŠóľŠ . đ˝âđ(đź) (52) Thus, for every đ â N, there exists (đźđ , đ˝đ ) â Rđ+đ with đ˝đ â đ(đźđ ) such that óľŠóľŠ óľŠ óľŠóľŠ(đĽđ , âđ (đĽđ )) â (đźđ , đ˝đ )óľŠóľŠóľŠ (53) 1 ⤠(1 + ) đ ((đĽđ , âđ (đĽđ )) , gphđ) . đ Take đ˘đ = âđ+ (đĽđ ), Vđ = ââ(đĽđ ), đŚđ = đźđ â đĽđ , and đ§đ = đ(đĽđ ) + đ˝đ . Then it follows that đ(đĽđ ) + đ˘đ â âRđ+ , â(đĽđ ) + Vđ = 0Rđ , and đ§đ â đ(đĽđ ) + đ(đĽđ + đŚđ ), which implies đĽđ â đ(đ˘đ , Vđ , đŚđ , đ§đ ) since đĽđ â Î, and óľŠđ óľŠ óľŠ óľŠ óľŠ óľŠ (óľŠóľŠóľŠđ˘đ óľŠóľŠóľŠ + óľŠóľŠóľŠVđ óľŠóľŠóľŠ + óľŠóľŠóľŠ(đŚđ , đ§đ )óľŠóľŠóľŠ) đ
1 óľŠ óľŠ óľŠ óľŠ â¤ (1 + ) [óľŠóľŠóľŠđ+ (đĽđ )óľŠóľŠóľŠ + óľŠóľŠóľŠâ (đĽđ )óľŠóľŠóľŠ đ
(54) đ
+đ ((đĽđ , âđ (đĽđ )) , gphđ)] .
(56)
đ
â¤
1 Ě â đđ (đĽđ )) , âđ (đ (đĽ) đ đ=1 đ đ
âđ â N.
Ě Then it follows Note that đ is locally Lipschitz and đĽđ â đĽ. that [âđ+ (đĽđ )â + ââ(đĽđ )â + đ((đĽđ , âđ(đĽđ )), gphđ)]đ â 0, which implies (đ˘đ , Vđ , đŚđ , đ§đ ) â 0Rđ+đ +đ+đ from (54). Together Ě with đđ+1 /(đ + 1)đ â +â, đĽđ â đ(đ˘đ , Vđ , đŚđ , đ§đ ), đĽđ â đĽ, and (55), this is a contradiction to the (MOPEC-) calmness Ě with order đ of (MOPEC) at đĽ. (iii)â(i). Assume that (MOPEC) is not (MOPEC-) calm Ě Then, by the same argument to the with order đ > 0 at đĽ. proof of (ii)â(i), it follows that, for every đ â N, there Ě 1/đ) exist (đ˘đ , Vđ , đŚđ , đ§đ ) â B(0Rđ+đ +đ+đ , 1/đ) and đĽđ â B(đĽ, with đĽđ â Î, đ(đĽđ ) + đ˘đ â âRđ+ , â(đĽđ ) + Vđ = 0Rđ and đ§đ â đ(đĽđ ) + đ(đĽđ + đŚđ ); that is, (đĽđ + đŚđ , đ§đ â đ(đĽđ )) â gphđ such that (31) holds. Thus, we have đ óľŠ óľŠ óľŠ óľŠ [óľŠóľŠóľŠđ+ (đĽđ )óľŠóľŠóľŠ + óľŠóľŠóľŠâ (đĽđ )óľŠóľŠóľŠ + đ ((đĽđ , âđ (đĽđ )) , gphđ)]
óľŠ óľŠ â¤ [óľŠóľŠóľŠđ (đĽđ ) â (đ (đĽđ ) + đ˘đ )óľŠóľŠóľŠ óľŠ óľŠ + óľŠóľŠóľŠâ (đĽđ ) â (â (đĽđ ) + Vđ )óľŠóľŠóľŠ óľŠ óľŠđ + óľŠóľŠóľŠ(đĽđ , âđ (đĽđ )) â (đĽđ + đŚđ , đ§đ â đ (đĽđ ))óľŠóľŠóľŠ] óľŠ óľŠđ = óľŠóľŠóľŠ(đ˘đ , Vđ , đŚđ , đ§đ )óľŠóľŠóľŠ ,
âđ â N.
(57)
8
Abstract and Applied Analysis đ
đ
Together with (31) and đ â int R+ , we get óľŠ óľŠ óľŠ óľŠ đ (đĽđ ) + đ [óľŠóľŠóľŠđ+ (đĽđ )óľŠóľŠóľŠ + óľŠóľŠóľŠâ (đĽđ )óľŠóľŠóľŠ
Ě (đ) + đ (Ί, đĽ) Ě . 0Rđ â đˇâ đ (đĽ)
đ
+đ ((đĽđ , âđ (đĽđ )) , gphđ)] đ đ óľŠ óľŠ óľŠ óľŠ + đ ([óľŠóľŠóľŠđ+ (đĽđ )óľŠóľŠóľŠ + óľŠóľŠóľŠâ (đĽđ )óľŠóľŠóľŠ + đ ((đĽđ , âđ (đĽđ )) , gphđ)] óľŠ óľŠđ Ě â int Rđ+ â int Rđ+ â óľŠóľŠóľŠ(đ˘đ , Vđ , đŚđ , đ§đ )óľŠóľŠóľŠ ) đ â đ (đĽ)
âđ â N, (58)
which implies that the multiobjective penalty problem (MPP)II with order đ does not admit a local exact penalizaĚ tion at đĽĚ since the sequence {đĽđ } â Î and đĽđ â đĽ. It is well known that a calmness condition with order 1 for standard nonlinear programming can lead to a KKT condition. In fact, we can also obtain a M-stationary condition for (MOPEC) under the (MOPEC-) calmness condition with order 1. To this end, we need the following generalized Fermat rule for a multiobjective optimization problem with an abstract constraint, which is established by applying the nonlinear scalar function in Lemma 6. Lemma 14. Let đ : Rđ â Rđ be a locally Lipschitz vectorvalued map, and let Ί â Rđ be a nonempty and closed subset. If đĽĚ â Ί is a local weak efficient solution for the multiobjective optimization problem min
đ (đĽ)
đ .đĄ.
đĽ â Ί, đ
(59)
đ
then there exists some đ â R+ with âđ=1 đ đ = 1 such that đ
Ě + đ (Ί, đĽ) Ě . 0Rđ â âđ â¨đ, đ⊠(đĽ)
(60)
đ=1
Proof. Define the function ÎŚ : Rđ â R by Ě + đ (Ί, đĽ) , ÎŚ (đĽ) := đđ (đ (đĽ) â đ (đĽ))
âđĽ â Rđ .
(61)
Since đĽĚ â Ί is a local weak efficient solution, ÎŚ attains a local Ě Otherwise, there exists a sequence {đĽđ } â Rđ minimum at đĽ. Ě = 0. Then we converging to đĽĚ such that ÎŚ(đĽđ ) < 0 since ÎŚ(đĽ) Ě < 0. Together have {đĽđ } â Ί and ÎŚ(đĽđ ) = đđ (đ(đĽđ ) â đ(đĽ)) with Lemma 6, we get Ě â â int Rđ+ , đ (đĽđ ) â đ (đĽ)
âđ â N.
(62)
This is a contradiction to đĽĚ â Ί being a local weak efficient Ě Note that đ is locally solution since {đĽđ } â Ί and đĽđ â đĽ. Lipschitz and Ί is closed. It follows from Propositions 2, 3, and 5 and Lemma 6 that Ě â đđđ (đ (â) â đ (đĽ)) Ě (đĽ) Ě + đđ (Ί, â) (đĽ) Ě 0Rđ â đÎŚ (đĽ) â
Ě Ě (đ) : đ â đđđ (0Rđ )} + đ (Ί, đĽ) â {đˇ đ (đĽ) đ
Ě . Ě (đ) : đ â Rđ+ , âđ đ = 1} + đ (Ί, đĽ) = {đˇâ đ (đĽ) đ=1
(64)
By Proposition 4, it follows that
óľŠ óľŠđ = đ (đĽđ ) + đóľŠóľŠóľŠ(đ˘đ , Vđ , đŚđ , đ§đ )óľŠóľŠóľŠ đ
Ě â int Rđ+ , = đ (đĽ)
đ
Therefore, there exists some đ â R+ with âđ=1 đ đ = 1 such that
(63)
Ě + đ (Ί, đĽ) Ě . 0Rđ â đ â¨đ, đ⊠(đĽ)
(65)
This completes the proof. Next, we show that the (MOPEC-) calmness condition with order 1 is sufficient to establish a M-stationary condition for (MOPEC). Theorem 15. Suppose that đĽĚ â đ is a local weak efficient solution for (MOPEC) and (MOPEC) is (MOPEC-) calm with Ě Then đĽĚ is a M-stationary point for (MOPEC); that order 1 at đĽ. đ đ is, there exist đ â R+ with âđ=1 đ đ = 1, đ˝ â Rđ+ , đž â Rđ , đ > 0, â â đ+đ â â Ě âđ(đĽ))(đŚ Ě and (đĽ , đŚ ) â R with đĽ â đˇâ đ(đĽ, ) such that đ
đ
đ=1
đ=1
Ě + âđ˝đ âđđ (đĽ) Ě + âđžđ ââđ (đĽ) Ě 0Rđ â đ â¨đ, đ⊠(đĽ) â
Ě (đŚâ )) + đ (Î, đĽ) Ě , + đ (đĽâ + (âđ (đĽ)) Ě = 0, đ˝đ đđ (đĽ)
(66)
âđ = 1, 2, . . . , đ.
Proof. Since đĽĚ â đ is a local weak efficient solution for (MOPEC) and (MOPEC) is (MOPEC-) calm with order 1 Ě it follows from Theorem 13 (i)â(iii) that there exists at đĽ, some đĚ > 0 such that đĽĚ is a local weak efficient solution for the multiobjective penalty problem (MPP)II with order 1. For simplicity, let the real-valued function T : Rđ â R defined by óľŠ óľŠ T (đĽ) = óľŠóľŠóľŠđ+ (đĽ)óľŠóľŠóľŠ + ââ (đĽ)â + đ ((đĽ, âđ (đĽ)) , gphđ) , (67) âđĽ â Rđ . Note that đ is locally Lipschitz, đ, â, and đ are continuously Fr´echet differentiable, and đ is closed. Then T is locally Ě Lipschitz and the penalty function đ(â)+ đT(â)đ : Rđ â R đ is also locally Lipschitz. Together with the closedness of Î đ and Lemma 14, it follows that there exists some đ â R+ with đ âđ=1 đ đ = 1 such that Ě (â) đ⊠(đĽ) Ě + đ (Î, đĽ) Ě . 0Rđ â đ â¨đ, đ (â) + đT Moreover, by using â¨đ, đ⊠= we have
đ âđ=1
(68)
đ đ = 1 and Proposition 3,
Ě (â) đ⊠(đĽ) Ě â đ â¨đ, đ⊠(đĽ) Ě + đđT Ě Ě , (69) đ â¨đ, đ (â) + đT (đĽ) Ě + đ ââ (â)â (đĽ) Ě Ě â đ óľŠóľŠóľŠóľŠđ+ (â)óľŠóľŠóľŠóľŠ (đĽ) đT (đĽ) Ě . + đđ ((â, âđ (â)) , gphđ) (đĽ)
(70)
Note that đ, â, and đ are continuously Fr´echet differentiable and đ is closed. Then it follows from Propositions 1 (ii), 4, and 5 that, for all đ â {1, 2, . . . , đ}, {0 đ } , Ě ={ R đ max {0, đđ (â)} (đĽ) Ě , [0, 1] âđđ (đĽ)
Ě < 0, if đđ (đĽ) (71) Ě = 0, if đđ (đĽ)
Abstract and Applied Analysis
9
for all đ â {1, 2, . . . , đ }, óľ¨ Ě óľ¨ Ě , = [â1, 1] ââđ (đĽ) đ óľ¨óľ¨óľ¨âđ (â)óľ¨óľ¨óľ¨ (đĽ) Ě đđ ((â, âđ (â)) , gphđ) (đĽ) â
Ě (đĽâ , âđŚâ ) : (đĽâ , âđŚâ ) â {(IRđ , ââđ (đĽ))
(72)
Ě đ (đĽ))) Ě }, â đ (gphđ, (âđĽ, where IRđ denotes the identity map from Rđ to itself. Let Ě = 0} be the set of active Ě := {đ â {1, 2, . . . , đ} | đđ (đĽ) J(đĽ) Ě Then we can conclude from (70)â(72) constraints of đ at đĽ. that đ
Ě + â [â1, 1] ââđ (đĽ) Ě Ě â â [0, 1] âđđ (đĽ) đT (đĽ) đ=1
Ě đâJ(đĽ)
â
Ě (đŚâ ) : (đĽâ , âđŚâ ) + {đĽâ + (âđ (đĽ))
(73)
Ě đ (đĽ))) Ě }. â đ (gphđ, (âđĽ, Ě Together with (68) and (69), there exist đ˝đ ⼠0 with đ â J(đĽ), Ě đ(đĽ))); Ě đž â Rđ , and (đĽâ , âđŚâ ) â đ(gphđ, (âđĽ, that is, đĽâ â â Ě đ(đĽ))(đŚ Ě ), such that đˇâ đ(âđĽ, Ě 0Rđ â đ â¨đ, đ⊠(đĽ) đ
Ě + âđžđ ââđ (đĽ) Ě + đĚ ( â đ˝đ âđđ (đĽ)
Corollary 16. Let đĽĚ â đ be a local efficient solution for (MOPEC). Suppose that the constraint system of (MOPEC) has Ě or, equivalently, the seta local error bound with order 1 at đĽ, valued map đ : Rđ+đ +đ+đ ó´ó´ą Rđ , defined by (26), is calm Ě Then đĽĚ is a M-stationary point with order 1 at (0Rđ+đ +đ+đ , đĽ). for (MOPEC). Remark 17. Recently, Kanzow and Schwartz [26] discussed the enhanced Fritz-John conditions for a smooth scalar optimization problem with equilibrium constraints and proposed some new constraint qualifications for the enhanced Mstationary condition. In particular, they obtained some sufficient conditions for the existence of a local error bound for the constraint system and the exactness of penalty functions with order 1 by using an appropriate condition. Subsequently, Ye and Zhang [27] extended Kanzow and Schwartzâs results to the nonsmooth case. It is worth noting that the exactness of the penalty function with order 1 in [26, 27] was established by using various qualification conditions, which were actually sufficient for the local error bound property of the constraint system; see [28, 29] for more details. However, just as shown in Theorem 13, the exactness for the two types of multiobjective penalty functions with order đ is obtained by means of the equivalent (MOPEC-) calmness condition, which is associated with not only the objective function but also the constraint system. Simultaneously, it follows from Theorem 10 and Proposition 12 that the (MOPEC-) calmness condition is weaker than the local error bound property of the constraint system.
đ=1
Ě đâJ(đĽ)
â
â
4. Applications
â
Ě (đŚ ))) + đ (Î, đĽ) Ě + (đĽ + (âđ (đĽ)) Ě + â đđ˝ Ě đ âđđ (đĽ) Ě = đ â¨đ, đ⊠(đĽ) Ě đâJ(đĽ)
đ
â
Ě . Ě + đĚ (đĽâ + (âđ (đĽ)) Ě (đŚâ )) + đ (Î, đĽ) + âđĚ đžđ ââđ (đĽ) đ=1
The main purpose of this section is to apply the obtained results for (MOPEC) to a multiobjective optimization problem with complementarity constraints (in short, (MOPCC)) and a multiobjective optimization problem with weak vector variational inequality constraints (in short, (MOPWVVI)) and establish corresponding calmness conditions and Mstationary conditions.
(74) Ě đ , đ â J(đĽ), Ě and đ˝đ = 0, Take đ˝ â Rđ+ with đ˝đ = đđ˝ Ě Ě đž â Rđ with đž = đĚ đž and đ = đ. đ â {1, 2, . . . , đ} \ J(đĽ), Then we have
4.1. Applications to (MOPCC). In this subsection, we consider the following multiobjective optimization problem with complementarity constraints: (MOPCC)
đ
Ě + âđ˝đ âđđ (đĽ) Ě 0Rđ â đ â¨đ, đ⊠(đĽ) đ=1
đ
â
â
â
Ě , Ě + đ (đĽ + (âđ (đĽ)) Ě (đŚ )) + đ (Î, đĽ) + âđžđ ââđ (đĽ) đ=1
Ě = 0, đ˝đ đđ (đĽ)
đ = 1, 2, . . . , đ. (75)
min
đ (đĽ)
s.t.
đ (đĽ) â âRđ+ , â (đĽ) = 0Rđ ,
(76)
đş (đĽ) â Rđ+ , đť (đĽ) â Rđ+ , đş(đĽ)đ đť (đĽ) = 0, đĽ â Î,
This completes the proof. Combining Proposition 12 and Theorem 15, we immediately have the following corollary.
where đ : Rđ â Rđ is locally Lipschitz, đ : Rđ â Rđ , â : Rđ â Rđ , đş, đť : Rđ â Rđ are continuously Fr´echet
10
Abstract and Applied Analysis
differentiable, and Î is a nonempty and closed subset of Rđ . As usual, we denote Ě = 0, đťđ (đĽ) Ě > 0} , đź0+ := {đ | đşđ (đĽ) Ě = 0, đťđ (đĽ) Ě = 0} , đź00 := {đ | đşđ (đĽ)
(77)
Ě > 0, đťđ (đĽ) Ě = 0} . đź+0 := {đ | đşđ (đĽ) Obviously, the feasible set đĚ := {đĽ â Rđ | đ(đĽ) â âRđ+ , â(đĽ) = 0Rđ , đş(đĽ) â Rđ+ , đť(đĽ) â Rđ+ , đş(đĽ)đ đť(đĽ) = 0, đĽ â Î} is a closed subset of Rđ . It is easy to verify that (MOPCC) can be reformulated as a special case of (MOPEC) if we let đ = 2đ, đş1 (đĽ) đť1 (đĽ) đ (đĽ) := ( ... ) ,
đ (đĽ) := đśđ , âđĽ â Rđ ,
(78)
đşđ (đĽ) đťđ (đĽ)
Definition 20 (see [4]). A point đĽĚ â đĚ is called a M-stationary point of (MOPCC) if and only if there exists a Lagrange đ multiplier đâ = (đđ , đđ , đâ , đđş, đđť) â Rđ+đ+đ +2đ with đđ â R+ đ đ and âđ=1 đ đ = 1 such that đ
đ
đ
Ě + âđ đ âđđ (đĽ) Ě + âđâđ ââđ (đĽ) Ě 0Rđ â đ â¨đđ , đ⊠(đĽ) đ=1
where đś := {(đ, đ) â R2 | 0 ⤠âđ ⼠âđ ⼠0}. Note that đ is constant and equals to đśđ . Then the parametric form Ě V, đŚ, đ§) of đĚ with parameter (đ˘, V, đŚ, đ§) â Rđ+đ +2đ is đ(đ˘, đĚ (đ˘, V, đŚ, đ§) = {đĽ â Î | đ (đĽ) + đ˘ â âRđ+ , â (đĽ) + V = 0Rđ , đş (đĽ) + đŚ â
only if the set-valued map đĚ : Rđ+đ +2đ ó´ó´ą Rđ is calm with Ě Specially, if we take đ = 1 and đ = 1, order đ at (0Rđ+đ +2đ , đĽ). then Definitions 18 and 19 reduce to Definitions 3.3 and 3.6 in [4], respectively. Simultaneously, the corresponding results to Propositions 3.4 and 3.7 in [4] also hold. As mentioned in the introduction, there have been various stationary concepts proposed for (MOPCC). Here we only recall the notion of the M-stationary point.
Rđ+ , đť (đĽ)
+đ§â
Rđ+ ,
đ
đ
đ
Ě + đđť Ě + đ (Î, đĽ) Ě , â â [đ đ âđşđ (đĽ) đ âđťđ (đĽ)] đ=1
đđşđ
= 0,
âđ â đź+0 ,
đđşđ â R,
âđ â đź0+ ;
đđť đ = 0,
âđ â đź0+ ,
đđť đ â R,
âđ â đź+0 ,
either đđşđ > 0, đđ â Rđ+ ,
(đş (đĽ) + đŚ) (đť (đĽ) + đ§) = 0} .
đ=1
đş đť đđť đ > 0 or đ đ đ đ = 0, đ
Ě = 0, đ đ đđ (đĽ)
(82)
âđ â đź00 ,
âđ = 1, 2, . . . , đ.
(79) Clearly, for the set-valued map đĚ : Rđ+đ +2đ ó´ó´ą Rđ , one has Ě Rđ+đ +2đ ) = đ. Ě đ(0 Inspired by Definitions 7 and 9, we give the following concepts, called (MOPCC-) calm and local error bound, for (MOPCC). Definition 18. Given đ > 0 and đĽĚ â đĚ being a local efficient (resp. local weak efficient) solution for (MOPCC), then (MOPCC) is said to be (MOPCC-) calm with order đ at đĽĚ if and only if there exist đż > 0 and đ > 0 such that, for Ě V, đŚ, đ§) ⊠B(đĽ, Ě đż), all (đ˘, V, đŚ, đ§) â B(0Rđ+đ +2đ , đż) and all đĽ â đ(đ˘, one has óľŠđ óľŠ Ě â int Rđ+ . đ (đĽ) + đóľŠóľŠóľŠ(đ˘, V, đŚ, đ§)óľŠóľŠóľŠ đ â đ (đĽ)
(80)
Definition 19. Given đ > 0 and đĽĚ â đ, then the constraint system of (MOPCC) is said to have a local error bound with order đ at đĽĚ if and only if there exist đż > 0 and đ > 0 such that, for all (đ˘, V, đŚ, đ§) â B(0Rđ+đ +2đ , đż) \ {0Rđ+đ +2đ } and all đĽ â Ě V, đŚ, đ§) ⊠B(đĽ, Ě đż), one has đ(đ˘, Ě BRđ â đóľŠóľŠóľŠ(đ˘, V, đŚ, đ§)óľŠóľŠóľŠđ đ â int Rđ . đ (đĽ, đ) + óľŠ óľŠ
(81)
Similarly, it follows from Theorem 10 that if the constraint system of (MOPCC) has a local error bound with order Ě then (MOPCC) is (MOPCC-) calm with order đ đ at đĽ, Ě Moreover, by Proposition 12, the constraint system of at đĽ. (MOPCC) has a local error bound with order đ at đĽĚ if and
The following formula for the Mordukhovich normal cone of the set đś is useful in the sequel. Lemma 21 (see [4]). For every (đ, đ) â đś, we have
(đ1 , đ2 ) | đ1 â R, đ2 = 0 { { { { { đ1 = 0, đ2 â R { { đ (đś, (đ, đ)) = { either đ1 > 0, { { { đ2 > 0 { { { or đ1 đ2 = 0 {
đđ đ = 0 > đ đđ đ < 0 = đ
đđ đ = 0 = đ. (83)
We now apply Theorem 15 to establish a M-stationary condition for (MOPCC) by virtue of the (MOPCC-) calmness condition. Theorem 22. Suppose that đĽĚ â đĚ is a local weak efficient solution for (MOPCC) and (MOPCC) is (MOPCC-) calm with Ě Then đĽĚ is a M-stationary point of (MOPCC). order 1 at đĽ. Proof. As stated above, (MOPCC) is equivalent to (MOPCC) with đ = 2đ and đ, đ given by (78). Note that the (MOPCC-) calmness of (MOPCC) implies the (MOPEC-) calmness of (MOPCC). Then it follows from Theorem 15 that there exist
Abstract and Applied Analysis đ
11
đ
đ â R+ with âđ=1 đ đ = 1, đ˝ â Rđ+ , đž â Rđ , đ > 0 and â Ě âđ(đĽ))(đŚ Ě ) such that (đĽâ , đŚâ ) â Rđ+2đ with đĽâ â đˇâ đ(đĽ, đ
đ
Ě + âđžđ ââđ (đĽ) Ě Ě + âđ˝đ âđđ (đĽ) 0Rđ â đ â¨đ, đ⊠(đĽ) đ=1
where đ : Rđ â Rđ is locally Lipschitz, đ : Rđ â Rđ and Ě is â : Rđ â Rđ are continuously Fr´echet differentiable. Î the solution set of the weak vector variational inequality (in short, (WVVI)): find a vector đĽĚ â Î such that Ě , đ¤ â đĽâŠ Ě , â¨đš2 (đĽ) Ě , đ¤ â đĽâŠ Ě ,..., (â¨đš1 (đĽ)
đ=1
â
Ě (đŚâ )) + đ (Î, đĽ) Ě , + đ (đĽâ + (âđ (đĽ)) Ě = 0, đ˝đ đđ (đĽ)
(84)
âđ = 1, 2, . . . , đ.
Note that đ(đĽ) = đśđ for all đĽ â Rđ . Then it follows that gphđ = Rđ Ă đśđ and Ě âđ (đĽ))) Ě đ (gphđ, (đĽ, Ě Ă đ (đśđ , âđ (đĽ)) Ě = đ (Rđ , đĽ) Ě , âđť1 (đĽ))) Ě = {0Rđ } Ă đ (đś, (âđş1 (đĽ)
(85)
Ě , đ¤ â đĽâŠ, Ě â¨đš2 (đĽ) Ě , đ¤ â đĽâŠ, Ě ..., inf đđ0 ( (â¨đš1 (đĽ)
đ¤âÎ
Ě đ¤ â đĽâŠ) Ě ) = 0. â¨đšđ (đĽ), (90) Moreover, since Î is nonempty, closed, and convex, and đđ0 is a convex function, we can conclude from Theorem 8.15 in [7] and Proposition 5 that đĽĚ is a solution of (WVVI) if and only if
đĽâ = 0Rđ , (86)
Ě , â â đĽâŠ Ě , â¨đš2 (đĽ) Ě , â â đĽâŠ Ě ,..., 0Rđ â đđđ0 ((â¨đš1 (đĽ) đ
if đ â đź00 .
Ě , â â đĽâŠ) Ě ) (đĽ) Ě + đ (Î, đĽ) Ě â¨đšđ (đĽ) Ě . Ě , đš2 (đĽ) Ě , . . . , đšđ (đĽ)) Ě đđđ0 (0Rđ ) + đ (Î, đĽ) â (đš1 (đĽ) (91)
Moreover, we get Ěđ âđş1 (đĽ) Ěđ âđť1 (đĽ) .. ). Ě =( âđ (đĽ) .
(89)
đ
Together with Lemma 21, we have for every (đĽâ , đŚâ ) â Rđ+2đ â Ě âđ(đĽ))(đŚ Ě with đĽâ â đˇâ đ(đĽ, ),
if đ â đź0+ if đ â đź+0
âđ¤ â Î,
where đšđ : Rđ â Rđ , đ = 1, 2, . . . , đ are continuously Fr´echet differentiable and Î is a nonempty, closed, and convex subset of Rđ . In the sequel, we denote đĚ := {đĽ â Rđ | đ(đĽ) â Ě by the feasible set of (MOPWVVI). âRđ+ , â(đĽ) = 0Rđ , đĽ â Î} Ě Then it is clear that đ is closed. Take đ0 := (1, 1, . . . , 1) â Rđ . Then it follows from Lemma 6 that đĽĚ â Rđ is a solution of (WVVI) if and only if đĽĚ â Î,
Ě , âđťđ (đĽ))) Ě . Ă â
â
â
Ă đ (đś, (âđşđ (đĽ)
đ1 | đđ â R, đđ = 0 { { { đ { 1 đđ = 0, đđ â R { { âđŚâ â { ( ... ) either đđ > 0, { { đđ > 0 { đđ { { or đđ đđ = 0 đđ {
đ
Ě , đ¤ â đĽâŠ) Ě â â int Rđ â¨đšđ (đĽ) +,
(87)
This shows that there exists some đ = (đ1 , đ2 , . . . , đđ ) â đ Rđ + with âđ=1 đđ = 1 such that đ
đ
Ě âđşđ (đĽ) Ě đ) ( âđťđ (đĽ)
Ě + đ (Î, đĽ) Ě . 0Rđ â âđđ đšđ (đĽ)
(92)
đ=1
Taking đâ = (đđ , đđ , đâ , đđş, đđť) â Rđ+đ+đ +2đ with đđ = đ, đđ = đ˝, đâ = đž, đđşđ = đđđ , and đđť đ = đđđ , for all đ â {1, 2, . . . , đ}, and substituting (86) and (87) into (84), then we can conclude that đĽĚ â đĚ is a M-stationary point of (MOPCC) with respect to the Lagrange multiplier đâ .
Given đĽĚ â đĚ being a local efficient (resp. local weak efficient) solution for (MOPWVVI), then đĽĚ is a solution of (MOPWVVI). Next, we define the concept of (MOPWVVI-) calmness with order đ > 0 at đĽĚ for (MOPWVVI) with respect đ to the corresponding đ = (đ1 , đ2 , . . . , đđ ) â Rđ + with âđ=1 đđ = 1 satisfying (92).
4.2. Applications to (MOPWVVI). Consider the following multiobjective optimization problem with weak vector variational inequality constraints: (MOPWVVI)
Definition 23. Given đ > 0, đĽĚ â đĚ being a local efficient (resp. local weak efficient) solution for (MOPWVVI) and đ đ = (đ1 , đ2 , . . . , đđ ) â Rđ + with âđ=1 đđ = 1 satisfying (92), then (MOPWVVI) is said to be (MOPWVVI-) calm with order đ at đĽĚ if and only if there exists đ > 0 such that, for every sequence {(đ˘đ , Vđ , đŚđ , đ§đ )} â Rđ+đ +đ+đ with (đ˘đ , Vđ , đŚđ , đ§đ ) â 0Rđ+đ +đ+đ and every sequence {đĽđ } â Î satisfying đ(đĽđ ) + đ˘đ â Rđ+ , â(đĽđ ) + Vđ = 0Rđ , đ§đ â âđ đ=1 đđ đšđ (đĽđ ) + đ(Î, đĽđ + đŚđ ), and Ě it holds that đĽđ â đĽ,
min
đ (đĽ)
s.t.
đ (đĽ) â âRđ+ , â (đĽ) = 0Rđ , Ě đĽ â Î,
(88)
óľŠ óľŠđ Ě â int Rđ+ . đ (đĽđ ) + đóľŠóľŠóľŠ(đ˘đ , Vđ , đŚđ , đ§đ )óľŠóľŠóľŠ đ â đ (đĽ)
(93)
12
Abstract and Applied Analysis
Obviously, we can define a similar local error bound condition at a local weak efficient solution đĽĚ â đĚ for (MOPWVVI) with respect to đ = (đ1 , đ2 , . . . , đđ ) â Rđ + with âđ đ=1 đđ = 1 satisfying (92). Moreover, we can obtain a corresponding relationship between the (MOPWVVI-) calmness condition and the local error bound condition. However, we omit the details here for simplicity. Next, we establish a M-stationary condition for (MOPWVVI) under the (MOPWVVI-) calmness with order 1 assumption. Theorem 24. Suppose that đĽĚ â đĚ is a local weak efficient solution for (MOPWVVI) and đ = (đ1 , đ2 , . . . , đđ ) â Rđ + with âđ đ=1 đđ = 1 satisfy (92). If in addition (MOPWVVI) is đ Ě then there exist đ â R+ (MOPWVVI-) calm with order 1 at đĽ, đ đ đ â â 2đ with âđ=1 đ đ = 1, đ˝ â R+ , đž â R , đ > 0, and (đĽ , đŚ ) â R â Ě â âđ Ě with đĽâ â đˇâ đÎ (đĽ, ) such that đ=1 đđ đšđ (đĽ))(đŚ
[4]
[5]
[6] [7]
đ=1
[8]
â
đ
[3]
đ
đ
Ě + âđžđ ââđ (đĽ) Ě Ě + âđ˝đ âđđ (đĽ) 0Rđ â đ â¨đ, đ⊠(đĽ) đ=1
[2]
Ě , Ě (đŚâ )] + đ (Î, đĽ) + đ [đĽâ + (âđđ âđšđ (đĽ))
(94)
đ=1
Ě = 0, đ˝đ đđ (đĽ)
âđ = 1, 2, . . . , đ,
where the set-valued map đÎ : Rđ ó´ó´ą Rđ is defined by đÎ (đĽ) = đ(Î, đĽ) for all đĽ â Rđ . Proof. Consider the problem (MOPEC) with đ(đĽ) = đ âđ đ=1 đđ đšđ (đĽ) and đ(đĽ) = đ(Î, đĽ) for all đĽ â R . Obviously, đĽĚ is a feasible point of (MOPEC) and the feasible set of Ě By assumption, đĽĚ is a local weak (MOPEC) is contained in đ. efficient solution for (MOPEC). Moreover, it is easy to verify that the (MOPWVVI-) calmness of (MOPWVVI) with order 1 at đĽĚ implies the (MOPEC-) calmness of (MOPEC) with Ě Thus, together with đšđ , đ = 1, 2, . . . , đ being conorder 1 at đĽ. Ě Ě = âđ tinuously Fr´echet differentiable and âđ(đĽ) đ=1 đđ âđšđ (đĽ), we immediately complete the proof by Theorem 15.
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
[9] [10]
[11]
[12]
[13]
[14]
[15]
[16]
Acknowledgments The authors are grateful to the anonymous referee for his/her valuable comments and suggestions, which helped to improve the paper. This research was supported by the National Natural Science Foundation of China (Grant no. 11171362) and the Fundamental Research Funds for the Central Universities (Grant no. CDJXS12100021).
[17]
[18]
[19]
References [1] Z. Luo, J. Pang, D. Ralph, and S. Wu, âExact penalization and stationarity conditions of mathematical programs with
[20]
equilibrium constraints,â Mathematical Programming B, vol. 75, no. 1, pp. 19â76, 1996. J. J. Ye and X. Y. Ye, âNecessary optimality conditions for optimization problems with variational inequality constraints,â Mathematics of Operations Research, vol. 22, no. 4, pp. 977â997, 1997. J. J. Ye and Q. J. Zhu, âMultiobjective optimization problem with variational inequality constraints,â Mathematical Programming B, vol. 96, no. 1, pp. 139â160, 2003. M. L. Flegel and C. Kanzow, âOn M-stationary points for mathematical programs with equilibrium constraints,â Journal of Mathematical Analysis and Applications, vol. 310, no. 1, pp. 286â302, 2005. Y. Chen and M. Florian, âThe nonlinear bilevel programming problem: formulations, regularity and optimality conditions,â Optimization, vol. 32, pp. 120â145, 1995. F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, NY, USA, 1983. R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer, Berlin, Germany, 1998. B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Volume I: Basic Theory, Volume II: Applications, Springer, Berlin, Germany, 2006. J. V. Burke, âCalmness and exact penalization,â SIAM Journal on Control and Optimization, vol. 29, no. 2, pp. 493â497, 1991. A. Fiacco and G. McCormic, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley & Sons, New York, NY, USA, 1987. A. M. Rubinov, B. M. Glover, and X. Q. Yang, âDecreasing functions with applications to penalization,â SIAM Journal on Optimization, vol. 10, no. 1, pp. 289â313, 1999. A. M. Rubinov and X. Q. Yang, Lagrange-Type Functions in Constrained Non-Convex Optimzation, Kluwer Academic Publishers, New York, NY, USA, 2003. X. X. Huang and X. Q. Yang, âNonlinear Lagrangian for multiobjective optimization and applications to duality and exact penalization,â SIAM Journal on Optimization, vol. 13, no. 3, pp. 675â692, 2002. T. Q. Bao, P. Gupta, and B. S. Mordukhovich, âNecessary conditions in multiobjective optimization with equilibrium constraints,â Journal of Optimization Theory and Applications, vol. 135, no. 2, pp. 179â203, 2007. S. M. Robinson, âStability theory for systems of inequalities, part II: differentiable nonlinear systems,â SIAM Journal on Numerical Analysis, vol. 13, no. 4, pp. 497â513, 1976. J. S. Treiman, âThe linear nonconvex generalized gradient and Lagrange multipliers,â SIAM Journal on Optimization, vol. 5, pp. 670â680, 1995. A. Gopfert, H. Riahi, C. Tammer, and C. Zalinescu, Variational Methods in Partially Ordered Spaces, Springer, Berlin, Germany, 2003. G. Y. Chen and X. Q. Yang, âCharacterizations of variable domination structures via nonlinear scalarization,â Journal of Optimization Theory and Applications, vol. 112, no. 1, pp. 97â110, 2002. M. Durea and C. Tammer, âFuzzy necessary optimality conditions for vector optimization problems,â Optimization, vol. 58, no. 4, pp. 449â467, 2009. C. R. Chen, S. J. Li, and Z. M. Fang, âOn the solution semicontinuity to a parametric generalized vector quasivariational
Abstract and Applied Analysis
[21] [22] [23]
[24]
[25] [26]
[27]
[28]
[29]
inequality,â Computers and Mathematics with Applications, vol. 60, no. 8, pp. 2417â2425, 2010. A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings, Springer, Dordrecht, The Netherlands, 2009. A. D. Ioffe, âMetric regularity and subdifferential calculus,â Russian Mathematical Surveys, vol. 55, no. 3, pp. 501â558, 2000. R. Henrion, A. Jourani, and J. Outrata, âOn the calmness of a class of multifunction,â SIAM Journal on Optimization, vol. 13, no. 2, pp. 603â618, 2002. X. Y. Zheng and K. F. Ng, âMetric subregularity and constraint qualifications for convex generalized equations in banach spaces,â SIAM Journal on Optimization, vol. 18, no. 2, pp. 437â 460, 2007. J. Jahn, Vecor Optimization, Theory, Applications and Extension, Springer, Berlin, Germany, 2004. C. Kanzow and A. Schwartz, âMathematical programs with equilibrium constraints: enhanced Fritz John-conditions, new constraint qualifications, and improved exact penalty results,â SIAM Journal on Optimization, vol. 20, no. 5, pp. 2730â2753, 2010. J. J. Ye and J. Zhang, âEnhanced Karush-Kuhn-Tucker conditions for mathematical programs with equilibrium constraints,â Journal of Optimization Theory and Applications, 2013. A. Schwartz, Mathematical programs with complementarity constraints: theory, methods, and applications [Ph.D. dissertation], Institute of Applied Mathematics and Statistics, University of Wurzburg, 2011. L. Guo, J. J. Ye, and J. Zhang, âMathematical programs with geometric constraints in Banach spaces: enhanced optimality, exact penalty, and sensitivity,â SIAM Journal on Optimization, vol. 23, pp. 2295â2319, 2013.
13
Advances in
Operations Research Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Applied Mathematics
Algebra
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Probability and Statistics Volume 2014
The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at http://www.hindawi.com International Journal of
Advances in
Combinatorics Hindawi Publishing Corporation http://www.hindawi.com
Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of Mathematics and Mathematical Sciences
Mathematical Problems in Engineering
Journal of
Mathematics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Discrete Dynamics in Nature and Society
Journal of
Function Spaces Hindawi Publishing Corporation http://www.hindawi.com
Abstract and Applied Analysis
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014