Levitin-Polyak well-posedness for generalized semi-infinite multiobjective programming problems
- Xian-Jun Long^{1}Email author,
- Zai-Yun Peng^{2} and
- Xiang-Kai Sun^{1}
https://doi.org/10.1186/s13660-015-0958-z
© Long et al. 2016
Received: 24 September 2015
Accepted: 27 December 2015
Published: 7 January 2016
Abstract
In this paper, we introduce a notion of Levitin-Polyak well-posedness for generalized semi-infinite multiobjective programming problems in terms of weakly efficient solutions. We obtain some metric characterizations of Levitin-Polyak well-posedness for this problem. We derive the relations between the Levitin-Polyak well-posedness and the upper semi-continuity of approximate solution maps for generalized semi-infinite multiobjective programming problems. Examples are given to illustrate our main results.
Keywords
MSC
1 Introduction
If \(p=1\) and \(C=\mathbb{R}_{+}\), then (GSIMP) reduces to generalized semi-infinite programming problems (for short, GSIP). If the index set does not depend on the decision variable x, i.e., \(Y(x)=Y\) where Y is some nonempty set, then (GSIP) reduces to a standard semi-infinite programming problem and if the index set is finite, say \(Y(x)=\{y_{1},y_{2},\ldots,y_{t}\}\) for all \(x\in\mathbb{R}^{n}\), then (GSIP) reduces to a finite programming problem.
In recent years, generalized semi-infinite programming problems became an active research topic in mathematical programming due to its extensive applications in many fields such as reverse Chebyshev approximate, robust optimization, minimax problems, design centering and disjunctive programming: see [1–3]. A large number of results have appeared in the literature: see, e.g., [4–9] and the references therein. Recently, standard semi-infinite programming problems have been generalized to multiobjective case. Chuong et al. [10] derived necessary and sufficient conditions for lower and upper semi-continuity of Pareto solution maps for parametric semi-infinite multiobjective optimization problems. Chuong et al. [11] obtained the pseudo-Lipschitz property of Pareto solution maps for the parametric linear semi-infinite multiobjective optimization problem. Chuong and Yao [12] derived necessary and sufficient optimality conditions of strongly isolated solutions and positively properly efficient solutions for nonsmooth semi-infinite multiobjective optimization problems. Huy and Kim [13] established sufficient conditions for the Aubin Lipschitz-like property for nonconvex semi-infinite multiobjective optimization problems. Goberna et al. [14] derived some optimality conditions for linear semi-infinite vector optimization problems by using the constraint qualifications.
On the other hand, it is well known that the well-posedness is very important for both optimization theory and numerical methods of optimization problems, which guarantees that, for approximating solution sequences, there is a subsequence which converges to a solution. The notion of well-posedness was first introduced and studied by Tykhonov [15] for unconstrained optimization problems. One limitation in Tykhonov well-posedness is that every minimizing sequence needs to satisfy feasibility conditions. To overcome this drawback, Levitin and Polyak [16] introduced a notion of well-posedness which does not necessarily require the feasibility of the minimizing sequence. Konsulova and Revalski [17] investigated the Levitin-Polyak well-posedness for convex optimization problems with functional constraints. Huang and Yang [18] extended the results of Konsulova and Revalski [17] to nonconvex case. Huang and Yang [19, 20] studied the Levitin-Polyak well-posedness for vector optimization problems with functional constraints. They also derived characterizations for the nonemptiness and compactness of weakly efficient solutions for a convex vector optimization problem with functional constraints in finite dimensional spaces. Lalitha and Chatterjee [21] gave some characterizations for the Levitin-Polyak well-posedness of quasiconvex vector optimization problems in terms of efficient solutions. Long et al. [22] introduced several types of Levitin-Polyak well-posedness for equilibrium problems with functional constraints and obtained criteria and characterizations for these types of well-posedness. About the other well-posedness of optimization problems, we refer the readers to [23–29] and the references therein.
Very recently, Wang et al. [30] considered the generalized Levitin-Polyak well-posedness for generalized semi-infinite programming problems. The criteria and characterizations of the generalized Levitin-Polyak well-posedness for this problem are established.
We remark that, so far as we know, there are no papers dealing with the Levitin-Polyak well-posedness for generalized semi-infinite multiobjective programming problems. This paper is the effort in this direction.
The rest of this article is organized as follows. In Section 2, we recall some basic definitions required in the sequel. In Section 3, we introduced a notion of Levitin-Polyak well-posedness for generalized semi-infinite multiobjective programming problems. We also give some criteria and characterizations for this kind of well-posedness. We discuss the relations between the Levitin-Polyak well-posedness and the upper semi-continuity of approximate solution maps for generalized semi-infinite multiobjective programming problems in Section 4.
2 Preliminaries
Let \(C\subseteq\mathbb{R}^{p}\) be a closed convex and cone with nonempty interior intC, which induces an order in \(\mathbb{R}^{p}\), i.e., for any \(x,y\in\mathbb{R}^{p}\), \(x\leq_{C} y\) if and only if \(y-x\in C\). The corresponding ordered vector space is denoted by \((\mathbb{R}^{p}, C)\). Arbitrarily fix an \(e\in\operatorname{int}C\). Let \((\mathbb {R}^{n},d)\) be a metric space and \(K\subset{\mathbb{R}^{n}}\). We denote by \(d(a,K):=\inf_{b\in{K}}\|a-b\|\), the distance from the point a to the set K.
Definition 2.1
Remark 2.1
We will use the following definitions of continuity for a set-valued map.
Definition 2.2
[31]
Let \(G:K\rightrightarrows\mathbb{R}^{m}\) be a set-valued map. G is said to be upper semi-continuous at \(x_{0}\in {K}\) iff for any open set V containing \(G(x_{0})\), there exists an open set U containing \(x_{0}\) such that, for all \(t\in{U}\cap K\), \(G(t)\subset{V}\). G is said to be upper semi-continuous on K iff it is upper semi-continuous at all \(x\in{K}\).
Definition 2.3
[31]
Let \(G:K\rightrightarrows\mathbb {R}^{m}\) be a set-valued map. G is said to be lower semi-continuous at \(x_{0}\in K\) iff for any \(y_{0}\in G(x_{0})\) and any neighborhood \(V(y_{0})\) of \(y_{0}\), there exists a neighbourhood \(U(x_{0})\) of \(x_{0}\) such that \(G(x)\cap V(y_{0})\neq\emptyset\), \(\forall x\in U(x_{0})\cap K\). G is said to be lower semi-continuous on K iff it is lower semi-continuous at each \(x\in K\).
Remark 2.2
[31]
G is lower semi-continuous at \(x_{0}\in K\) if and only if for any \(x_{n}\rightarrow x_{0}\) and any \(y\in G(x_{0})\), there exists \(y_{n}\in G(x_{n})\) such that \(y_{n}\rightarrow y\).
Definition 2.4
[32]
Remark 2.3
If G is upper semi-continuous at \(x_{0}\in{K}\), then G is Hausdorff upper continuous at \(x_{0}\in{K}\); the converse implication is true when \(G(x_{0})\) is compact (see [33]).
Remark 2.4
For the index set \(Y(x)\) in problem (GSIMP), Wang et al. [30] gave a condition ensuring that the set-valued mapping Y is lower semi-continuous on X. They also proved that if Y is lower semi-continuous on X and g is lower semi-continuous, then φ is lower semi-continuous on X.
Definition 2.5
[34]
Definition 2.6
3 Metric characterizations of Levitin-Polyak well-posedness
In this section, we introduce a notion of Levitin-Polyak well-posedness for generalized semi-infinite multiobjective programming problems. We also obtain some metric characterizations of Levitin-Polyak well-posedness by considering the non-compactness of approximate solution set.
We first introduce the notion of Levitin-Polyak well-posedness for problem (GSIMP).
Definition 3.1
Definition 3.2
Problem (GSIMP) is said to be Levitin-Polyak well-posed iff the solution set S is nonempty, and for every Levitin-Polyak minimizing sequence has a subsequence which converges to an element of S.
Remark 3.1
- (i)
The Levitin-Polyak well-posedness implies that the set S of weakly efficient solutions of problem (GSIMP) is nonempty and compact.
- (ii)
When f is a real-valued function and \(C=\mathbb{R}_{+} ^{1}\), the Levitin-Polyak well-posedness reduces to generalized type II Levitin-Polyak well-posedness for generalized semi-infinite programming problems considered by Wang et al. [30].
- (iii)
When the index set is finite, e.g., \(Y(x)=\{ y_{1},y_{2},\ldots,y_{t}\}\) for all \(x\in\mathbb{R}^{n}\), the concept of the Levitin-Polyak well-posedness for problem (GSIMP) is similar to the definition introduced by Huang and Yang [20].
The proof of the following proposition is easy and so we omit it.
Proposition 3.1
If problem (GSIMP) is Levitin-Polyak well-posed, then (1) holds. Conversely, if (1) holds and S is nonempty compact, then problem (GSIMP) is Levitin-Polyak well-posed.
Theorem 3.1
Proof
The following theorem shows that the Levitin-Polyak well-posedness of problem (GSIMP) can be characterized by considering the non-compactness of approximate solution set.
Theorem 3.2
Proof
We now give an example to illustrate Theorem 3.2.
Example 3.1
The following example illustrates that the continuity of f in Theorem 3.2 is essential.
Example 3.2
Theorem 3.3
Assume that f is continuous, g is lower semi-continuous and the set-valued mapping Y is lower semi-continuous. If there exists some \(\varepsilon>0\) such that \(\Omega(\varepsilon)\) is nonempty bounded, then problem (GSIMP) is Levitin-Polyak well-posed.
Proof
Remark 3.2
Theorem 3.3 illustrates that under suitable conditions, Levitin-Polyak well-posedness of problem (GSIMP) is equivalent to the existence of solutions.
The following example illustrates that the boundedness condition in Theorem 3.3 is essential.
Example 3.3
Remark 3.3
It is worth mentioning that Huang and Yang [20] established the equivalence between the generalized type I Levitin-Polyak well-posedness and the nonemptiness and compactness of weakly efficient solution set for convex vector optimization problems with a cone constraint by the linear scalarization method (see Theorem 3.1 in [20]). However, based on different problems and different approaches, their result and ours cannot include each other; for more details, see [20].
4 Links with upper semi-continuity of approximate solution maps
In this section, we investigate the relationship between the Levitin-Polyak well-posedness of problem (GSIMP) and the upper semi-continuity of approximate solution maps. We first have the following result concerning the necessary condition for problem (GSIMP) to be Levitin-Polyak well-posed.
Theorem 4.1
If problem (GSIMP) is Levitin-Polyak well-posed, then the set-valued map \(\Omega:\mathbb{R}_{+}\rightrightarrows\mathbb {R}^{n}\) is upper semi-continuous at \(\varepsilon=0\).
Proof
By Theorem 4.1 and Remark 2.3, we have the following corollary.
Corollary 4.1
If problem (GSIMP) is Levitin-Polyak well-posed, then for every Levitin-Polyak minimizing sequence \(\{x_{n}\} \subset\mathbb{R}^{n}\) and for every neighborhood W of 0, there exists \(n_{0}\in\mathbb{N}\) such that \(x_{n}\in S+W\) for all \(n>n_{0}\).
The next theorem gives a sufficient condition for problem (GSIMP) to be Levitin-Polyak well-posed.
Theorem 4.2
If S is nonempty compact and Ω is upper semi-continuous at \(\varepsilon=0\), then problem (GSIMP) is Levitin-Polyak well-posed.
Proof
Remark 4.1
It is worth mentioning that the compactness assumption of S cannot be dropped in the above theorem. Let us consider Example 3.2. Clearly, \(S=\Omega(0)=[0,+\infty)\) is not compact and for any \(\rho>0\), \(V=(-\rho,+\infty)\) is an open set with \(\Omega (0)\subset V\). It is easy to see that Ω is upper semi-continuous at \(\varepsilon=0\). But the problem is not Levitin-Polyak well-posed.
As a consequence of Theorem 4.2 and Remark 2.3, we have the following corollary.
Corollary 4.2
If S is nonempty compact and Ω is Hausdorff upper continuous at \(\varepsilon=0\), then problem (GSIMP) is Levitin-Polyak well-posed.
From Theorems 4.1 and 4.2, we obtain the equivalent relation between the Levitin-Polyak well-posed of problem (GSIMP) and the upper semi-continuity of approximate solution maps.
Corollary 4.3
If S is nonempty compact, then problem (GSIMP) is Levitin-Polyak well-posed if and only if Ω is upper semi-continuous at \(\varepsilon=0\).
5 Conclusion
The purpose of this paper is to study the Levitin-Polyak well-posedness for generalized semi-infinite multiobjective programming problems, where the objective function is vector-valued and the generalized semi-infinite constraint functions are real-valued. Metric characterizations for this kind of Levitin-Polyak well-posedness are obtained. The relations between the Levitin-Polyak well-posedness and the upper semi-continuity of approximate solution maps for generalized semi-infinite multiobjective programming problems are established. It would be interesting to consider the Levitin-Polyak well-posedness for semi-infinite vector optimization problems, where the objective function and the semi-infinite constraint functions are also vector-valued. This may be the topic of some of our forthcoming papers.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (11471059, 11301571, 11301570), the Chongqing Research Program of Basic Research and Frontier Technology (cstc2014jcyjA00037, cstc2015jcyjB00001, cstc2015jcyjA00025, cstc2015jcyjA00002), the Education Committee Project Research Foundation of Chongqing (KJ1400618, KJ1500626), the Postdoctoral Science Foundation of China (2015M580774) and the Program for Core Young Teacher of the Municipal Higher Education of Chongqing ([2014]47).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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