 Research
 Open Access
 Published:
LevitinPolyak wellposedness for generalized semiinfinite multiobjective programming problems
Journal of Inequalities and Applications volume 2016, Article number: 12 (2016)
Abstract
In this paper, we introduce a notion of LevitinPolyak wellposedness for generalized semiinfinite multiobjective programming problems in terms of weakly efficient solutions. We obtain some metric characterizations of LevitinPolyak wellposedness for this problem. We derive the relations between the LevitinPolyak wellposedness and the upper semicontinuity of approximate solution maps for generalized semiinfinite multiobjective programming problems. Examples are given to illustrate our main results.
1 Introduction
In this paper, we consider the following generalized semiinfinite multiobjective programming problem:
where \(f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{p}\) is a vectorvalued function, \(C\subseteq\mathbb{R}^{p}\) is a closed, convex and cone, \(g, h_{l}:\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}\) are realvalued functions, and the index set \(L=\{1,2,\ldots,s\}\) with \(s<+\infty\).
If \(p=1\) and \(C=\mathbb{R}_{+}\), then (GSIMP) reduces to generalized semiinfinite 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 semiinfinite 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 semiinfinite 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 semiinfinite programming problems have been generalized to multiobjective case. Chuong et al. [10] derived necessary and sufficient conditions for lower and upper semicontinuity of Pareto solution maps for parametric semiinfinite multiobjective optimization problems. Chuong et al. [11] obtained the pseudoLipschitz property of Pareto solution maps for the parametric linear semiinfinite multiobjective optimization problem. Chuong and Yao [12] derived necessary and sufficient optimality conditions of strongly isolated solutions and positively properly efficient solutions for nonsmooth semiinfinite multiobjective optimization problems. Huy and Kim [13] established sufficient conditions for the Aubin Lipschitzlike property for nonconvex semiinfinite multiobjective optimization problems. Goberna et al. [14] derived some optimality conditions for linear semiinfinite vector optimization problems by using the constraint qualifications.
On the other hand, it is well known that the wellposedness 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 wellposedness was first introduced and studied by Tykhonov [15] for unconstrained optimization problems. One limitation in Tykhonov wellposedness is that every minimizing sequence needs to satisfy feasibility conditions. To overcome this drawback, Levitin and Polyak [16] introduced a notion of wellposedness which does not necessarily require the feasibility of the minimizing sequence. Konsulova and Revalski [17] investigated the LevitinPolyak wellposedness 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 LevitinPolyak wellposedness 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 LevitinPolyak wellposedness of quasiconvex vector optimization problems in terms of efficient solutions. Long et al. [22] introduced several types of LevitinPolyak wellposedness for equilibrium problems with functional constraints and obtained criteria and characterizations for these types of wellposedness. About the other wellposedness of optimization problems, we refer the readers to [23–29] and the references therein.
Very recently, Wang et al. [30] considered the generalized LevitinPolyak wellposedness for generalized semiinfinite programming problems. The criteria and characterizations of the generalized LevitinPolyak wellposedness for this problem are established.
We remark that, so far as we know, there are no papers dealing with the LevitinPolyak wellposedness for generalized semiinfinite 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 LevitinPolyak wellposedness for generalized semiinfinite multiobjective programming problems. We also give some criteria and characterizations for this kind of wellposedness. We discuss the relations between the LevitinPolyak wellposedness and the upper semicontinuity of approximate solution maps for generalized semiinfinite 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 \(yx\in C\). 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}}\ab\\), the distance from the point a to the set K.
Definition 2.1
A point \(x_{0}\in M\) is said to be a weakly efficient solution for problem (GSIMP) iff for any \(x\in M\),
Denote by S the set of weakly efficient solutions of problem (GSIMP).
Remark 2.1
From Definition 2.1, we have
To reformulate problem (GSIMP) as a finite nonlinear multiobjective programming problem, we define the value function of the lowerlevel problem by
Let \(X=\{x\in\mathbb{R}^{n}: Y(x)\neq\emptyset\}\). It is easy to see that problem (GSIMP) can be equivalently reformulated as the following multiobjective programming problem with a single nonsmooth constraint:
We will use the following definitions of continuity for a setvalued map.
Definition 2.2
[31]
Let \(G:K\rightrightarrows\mathbb{R}^{m}\) be a setvalued map. G is said to be upper semicontinuous 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 semicontinuous on K iff it is upper semicontinuous at all \(x\in{K}\).
Definition 2.3
[31]
Let \(G:K\rightrightarrows\mathbb {R}^{m}\) be a setvalued map. G is said to be lower semicontinuous 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 semicontinuous on K iff it is lower semicontinuous at each \(x\in K\).
Remark 2.2
[31]
G is lower semicontinuous 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]
Let \(G:K\rightrightarrows\mathbb {R}^{m}\) be a setvalued map. We say that G is Hausdorff upper continuous at \(x_{0}\in K\) iff for any neighborhood \(V(0)\) of 0, there exists a neighborhood \(W(x_{0})\) of \(x_{0}\) such that
We say that G is Hausdorff upper continuous iff G is Hausdorff upper continuous at every point of K.
Remark 2.3
If G is upper semicontinuous 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 setvalued mapping Y is lower semicontinuous on X. They also proved that if Y is lower semicontinuous on X and g is lower semicontinuous, then φ is lower semicontinuous on X.
Definition 2.5
[34]
Let A be a nonempty subset of \(\mathbb{R}^{n}\). The Kuratowski measure [34] of noncompactness μ of the set A is defined by
where \(\operatorname{diam}A_{i}\) is the diameter of \(A_{i}\) defined by \(\operatorname{diam}A_{i}=\sup\{d(x_{1},x_{2}):x_{1},x_{2}\in{A_{i}}\}\).
Definition 2.6
Let A and B be two nonempty subsets of \(\mathbb{R}^{n}\). The Hausdorff distance between A and B is defined by
where \(e(A,B)=\sup_{a\in{A}}d(a,B)\). Let \(\{A_{n}\}\) be a sequence of nonempty subsets of X. We say that \(A_{n}\) converges to A in the sense of Hausdorff distance if \(H(A_{n},A)\rightarrow0\). It is easy to see that \(e(A_{n},A)\rightarrow0\) if and only if \(d(a_{n},A)\rightarrow0\) for every \(a_{n}\in A_{n}\). For more details on this topic, we refer the reader to [34].
3 Metric characterizations of LevitinPolyak wellposedness
In this section, we introduce a notion of LevitinPolyak wellposedness for generalized semiinfinite multiobjective programming problems. We also obtain some metric characterizations of LevitinPolyak wellposedness by considering the noncompactness of approximate solution set.
We first introduce the notion of LevitinPolyak wellposedness for problem (GSIMP).
Definition 3.1
A sequence \(\{x_{n}\}\subseteq\mathbb{R}^{n}\) is said to be a LevitinPolyak minimizing sequence of problem (GSIMP) iff there exists a sequence \(\varepsilon_{n}>0\) with \(\varepsilon_{n}\rightarrow0\) such that
and
Definition 3.2
Problem (GSIMP) is said to be LevitinPolyak wellposed iff the solution set S is nonempty, and for every LevitinPolyak minimizing sequence has a subsequence which converges to an element of S.
Remark 3.1
We remark that:

(i)
The LevitinPolyak wellposedness implies that the set S of weakly efficient solutions of problem (GSIMP) is nonempty and compact.

(ii)
When f is a realvalued function and \(C=\mathbb{R}_{+} ^{1}\), the LevitinPolyak wellposedness reduces to generalized type II LevitinPolyak wellposedness for generalized semiinfinite 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 LevitinPolyak wellposedness for problem (GSIMP) is similar to the definition introduced by Huang and Yang [20].
Consider the following statement:
The proof of the following proposition is easy and so we omit it.
Proposition 3.1
If problem (GSIMP) is LevitinPolyak wellposed, then (1) holds. Conversely, if (1) holds and S is nonempty compact, then problem (GSIMP) is LevitinPolyak wellposed.
For any \(\varepsilon>0\), we consider the following approximating solution set:
Theorem 3.1
Problem (GSIMP) is LevitinPolyak wellposed if and only if the solution set S is nonempty compact and
Proof
Suppose that problem (GSIMP) is LevitinPolyak wellposed. Then S is nonempty and compact. Now, we prove (2) holds. Suppose by contradiction that there exist \(\alpha>0\), \(\varepsilon_{n}>0\) with \(\varepsilon_{n}\rightarrow0\), and \(\{x_{n}\}\subset\Omega(\varepsilon _{n})\) such that
As \(\{x_{n}\}\subset\Omega(\varepsilon_{n})\), we know that \(\{x_{n}\}\) is a LevitinPolyak minimizing sequence for problem (GSIMP). By the LevitinPolyak wellposedness of problem (GSIMP), there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) converging to some point of S. This contradicts (3). It follows that (2) holds.
Conversely, suppose that S is nonempty compact and (2) holds. Let \(\{ x_{n}\}\) is a LevitinPolyak minimizing sequence for problem (GSIMP). Then there exists a sequence \(\varepsilon_{n}>0\) with \(\varepsilon_{n}\rightarrow0\) such that
It follows that \(\{x_{n}\}\subset\Omega(\varepsilon_{n})\). By (2), there exists a sequence \(\{z_{n}\}\subset S\) such that
Note that S is compact. Then there exists a subsequence \(\{z_{n_{k}}\}\) of \(\{z_{n}\}\) converging to \(x_{0}\in S\). Thus, the corresponding subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) converges to \(x_{0}\). Therefore, problem (GSIMP) is LevitinPolyak wellposed. The proof is complete. □
The following theorem shows that the LevitinPolyak wellposedness of problem (GSIMP) can be characterized by considering the noncompactness of approximate solution set.
Theorem 3.2
Assume that f is continuous, g is lower semicontinuous and the setvalued mapping Y is lower semicontinuous. Then, problem (GSIMP) is LevitinPolyak wellposed if and only if
Proof
Let problem (GSIMP) be LevitinPolyak wellposed. By Theorem 3.1, S is nonempty compact and
Clearly, \(\Omega(\varepsilon)\neq\emptyset\) for any \(\varepsilon>0\), since \(S\subset\Omega(\varepsilon)\). Observe that for \(\varepsilon>0\), we have
Since S is compact, \(\mu(S)=0\). It follows that
This fact together with (5) implies that (4) holds.
Conversely, assume that (4) holds. We first show that \(\Omega (\varepsilon)\) is a closed set for any \(\varepsilon>0\). Let \(x_{n}\in \Omega(\varepsilon)\) with \(x_{n}\rightarrow x_{0}\) such that
By (6), we have
Since f is continuous and \(R^{p}\backslash(\operatorname{int}C)\) is closed,
or equivalently,
On the other hand, for any \(y'\in Y(x_{0})\), since Y is lower semicontinuous, there exists a sequence \(\{y_{n}\}\) with \(y_{n}\in Y(x_{n})\) converging to \(y'\) such that
By the lower semicontinuity of g, we have
This fact together with (7) yields \(x_{0}\in\Omega(\varepsilon)\). It follows that \(\Omega(\varepsilon)\) is closed.
We next prove that
Obviously, \(S\subset\bigcap_{\varepsilon>0}\Omega(\varepsilon)\). Now suppose that \(\varepsilon_{n}>0\) with \(\varepsilon_{n}\rightarrow0\) and \(x_{0}\in\bigcap_{n=1}^{+\infty}\Omega(\varepsilon_{n})\). It follows that for any n,
Since \(R^{p}\backslash(\operatorname{int}C)\) is closed and \(\varepsilon _{n}\rightarrow0\),
This implies that \(x_{0}\in S\). Therefore, (8) holds.
Suppose that (4) holds. Note that \(\Omega(\varepsilon)\) is closed and \(\Omega(\varepsilon_{1})\subset\Omega(\varepsilon_{2})\) whenever \(\varepsilon_{1}<\varepsilon_{2}\). By the Kuratowski theorem ([34], p.412),
and S is nonempty and compact.
Let \(\{x_{n}\}\) be a LevitinPolyak minimizing sequence for problem (GSIMP). Then there exists a sequence \(\varepsilon_{n}>0\) with \(\varepsilon_{n}\rightarrow0\) such that
Thus, \(\{x_{n}\}\subset\Omega(\varepsilon_{n})\). This fact together with (9) yields that \(d(x_{n},S)\rightarrow0\). By Proposition 3.1, problem (GSIMP) is LevitinPolyak wellposed. This completes the proof. □
We now give an example to illustrate Theorem 3.2.
Example 3.1
Let \(C=\mathbb{R}_{+}^{2}\) and \(e=(1,1)\). We consider the following generalized semiinfinite multiobjective programming problem:
By simple calculations, \(Y(x)=(\infty,x^{2}]\) and \(M=(\infty,1]\). It is easy to verify that f and g are continuous and \(Y(x)\) is lower semicontinuous. It is clear that \(S=[0,1]\) and
It follows that \(\lim_{\varepsilon\rightarrow0}\mu(\Omega(\varepsilon ))=0\). By Theorem 3.2, problem (GSIMP) is LevitinPolyak wellposed.
The following example illustrates that the continuity of f in Theorem 3.2 is essential.
Example 3.2
Let C, e, g, and Y be considered in Example 3.1. Let \(f:\mathbb{R}\rightarrow\mathbb{R}^{2}\) be defined by
Then \(Y(x)=(\infty,x^{2}]\), \(M=(\infty,1]\), and \(S=[0,1]\). It is easy to see that g are continuous and \(Y(x)\) is lower semicontinuous. Moreover,
Obviously, f is not continuous. By Theorem 3.2, problem (GSIMP) is not LevitinPolyak wellposed. In fact, for sequence \(\{x_{n}\}=\{11/n \}\) is a LevitinPolyak minimizing sequence for problem (GSIMP), but any subsequence of \(\{x_{n}\}\) converges to \(1\notin S\).
Theorem 3.3
Assume that f is continuous, g is lower semicontinuous and the setvalued mapping Y is lower semicontinuous. If there exists some \(\varepsilon>0\) such that \(\Omega(\varepsilon)\) is nonempty bounded, then problem (GSIMP) is LevitinPolyak wellposed.
Proof
Let \(\{x_{n}\}\) be a LevitinPolyak minimizing sequence for problem (GSIMP). Then there exists a sequence \(\varepsilon_{n}>0\) with \(\varepsilon_{n}\rightarrow0\) such that
Let \(\varepsilon>0\) be such that \(\Omega(\varepsilon)\) is nonempty bounded. Then there exists \(n_{0}\) such that \(\{x_{n}\}\subset\Omega (\varepsilon)\) for all \(n> n_{0}\). This implies that \(\{x_{n}\}\) is bounded. It follows that there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{ x_{n}\}\) such that \(x_{n_{k}}\rightarrow x_{0}\). From (10), we have
Since \(\mathbb{R}^{p}\backslash(\operatorname{int}C)\) is closed, f is continuous and \(\varepsilon_{n_{k}}\rightarrow0\),
which implies that
Note that (11) also holds for \(x_{n_{k}}\) and \(\varepsilon_{n_{k}}\). For any \(y'\in Y(x_{0})\), by the lower semicontinuity of Y, there exists a sequence \(\{y_{n_{k}}\}\) with \(y_{n_{k}}\in Y(x_{n_{k}})\) converging to \(y'\) such that
Since g is lower semicontinuous and \(\varepsilon_{n_{k}}\rightarrow0\),
Thus, \(x_{0}\in M\). This fact together with (12) yields \(x_{0}\in S\). Therefore, problem (GSIMP) is LevitinPolyak wellposed. This completes the proof. □
Remark 3.2
Theorem 3.3 illustrates that under suitable conditions, LevitinPolyak wellposedness 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
Let \(C=\mathbb{R}_{+}^{2}\) and \(e=(1,1)\). We consider the following generalized semiinfinite multiobjective programming problem:
Then, it is easy to check that \(Y(x)=[x,+\infty)\) and \(M=[1,+\infty)\). Clearly, f and g are continuous and \(Y(x)\) is lower semicontinuous. By simple calculations, \(S=[0,+\infty)\) and for any \(\varepsilon>0\),
It follows that \(\Omega(\varepsilon)\) is not bounded. By Theorem 3.3, problem (GSIMP) is not LevitinPolyak wellposed. In fact, for sequence \(\{x_{n}\}=\{n\}\) is a LevitinPolyak minimizing sequence for problem (GSIMP), but it does not have any subsequence which converges to an element of S.
Remark 3.3
It is worth mentioning that Huang and Yang [20] established the equivalence between the generalized type I LevitinPolyak wellposedness 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 semicontinuity of approximate solution maps
In this section, we investigate the relationship between the LevitinPolyak wellposedness of problem (GSIMP) and the upper semicontinuity of approximate solution maps. We first have the following result concerning the necessary condition for problem (GSIMP) to be LevitinPolyak wellposed.
Theorem 4.1
If problem (GSIMP) is LevitinPolyak wellposed, then the setvalued map \(\Omega:\mathbb{R}_{+}\rightrightarrows\mathbb {R}^{n}\) is upper semicontinuous at \(\varepsilon=0\).
Proof
Let problem (GSIMP) be LevitinPolyak wellposed. Suppose by contradiction that Ω is not upper semicontinuous at \(\varepsilon=0\). Then there exists an open set U with \(\Omega (0)\subset U\), and for any \(\varepsilon_{n}>0\) with \(\varepsilon _{n}\rightarrow0\), there exists \(x_{n}\in\Omega(\varepsilon_{n})\) such that \(x_{n}\notin U\). Since \(x_{n}\in\Omega(\varepsilon_{n})\), we have
It follows that \(\{x_{n}\}\) is a LevitinPolyak minimizing sequence for problem (GSIMP). Note that problem (GSIMP) is LevitinPolyak wellposed. Then there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) which converges to some point \(x_{0}\in S\). It is easy to see that \(S=\Omega(0)\). This implies \(x_{0}\in\Omega(0)\). It follows that
As \(x_{n}\notin U\), we have \(x_{n}\in\mathbb{R}^{n}\backslash U\). By the closedness of \(\mathbb{R}^{n}\backslash U\) and \(x_{n_{k}}\rightarrow x_{0}\), we get \(x_{0}\in\mathbb{R}^{n}\backslash U\). This gives a contradiction. Therefore, Ω is upper semicontinuous at \(\varepsilon=0\). This completes the proof. □
By Theorem 4.1 and Remark 2.3, we have the following corollary.
Corollary 4.1
If problem (GSIMP) is LevitinPolyak wellposed, then for every LevitinPolyak 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 LevitinPolyak wellposed.
Theorem 4.2
If S is nonempty compact and Ω is upper semicontinuous at \(\varepsilon=0\), then problem (GSIMP) is LevitinPolyak wellposed.
Proof
Let B be an open unit ball in \(\mathbb{R}^{n}\). For any \(\rho>0\), \(\Omega(0)+\rho B\) is a neighborhood of \(\Omega(0)\). Since Ω is upper semicontinuous at \(\varepsilon=0\), there exists a neighborhood of V of 0 such that
Let \(\{x_{n}\}\) be a LevitinPolyak minimizing sequence for problem (GSIMP). Thus, there exists \(\varepsilon'\in V\) and \(n_{0}\in\mathbb{N}\) such that \(\{x_{n}\}\subset\Omega(\varepsilon')\) when \(n>n_{0}\). It follows that
Let \(s_{n}\in S\) and \(b_{n}\in\rho B\) be such that
Since S is nonempty compact, there exists a subsequence \(\{s_{n_{k}}\}\) of \(\{s_{n}\}\) which converges to some point \(s_{0}\in S\), and for the above \(\rho>0\), there exists \(N\in\mathbb{N}\) such that \(\s_{n_{k}}s_{0}\ <\rho\) for all \(k>N\). It follows that
By the arbitrariness of ρ, we get \(x_{n_{k}}\rightarrow s_{0}\in S\). Hence, problem (GSIMP) is LevitinPolyak wellposed. This completes the 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 semicontinuous at \(\varepsilon=0\). But the problem is not LevitinPolyak wellposed.
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 LevitinPolyak wellposed.
From Theorems 4.1 and 4.2, we obtain the equivalent relation between the LevitinPolyak wellposed of problem (GSIMP) and the upper semicontinuity of approximate solution maps.
Corollary 4.3
If S is nonempty compact, then problem (GSIMP) is LevitinPolyak wellposed if and only if Ω is upper semicontinuous at \(\varepsilon=0\).
5 Conclusion
The purpose of this paper is to study the LevitinPolyak wellposedness for generalized semiinfinite multiobjective programming problems, where the objective function is vectorvalued and the generalized semiinfinite constraint functions are realvalued. Metric characterizations for this kind of LevitinPolyak wellposedness are obtained. The relations between the LevitinPolyak wellposedness and the upper semicontinuity of approximate solution maps for generalized semiinfinite multiobjective programming problems are established. It would be interesting to consider the LevitinPolyak wellposedness for semiinfinite vector optimization problems, where the objective function and the semiinfinite constraint functions are also vectorvalued. This may be the topic of some of our forthcoming papers.
References
Goberna, MA, López, MA: Linear SemiInfinite Optimization. Wiley, Chichester (2001)
Reemtsen, R, Ruckmann, JJ (eds.): SemiInfinite Programming. Kluwer Academic, Boston (1998)
Stein, O: How to solve a semiinfinite optimization problem. Eur. J. Oper. Res. 223, 312320 (2012)
Stein, O: BiLevel Strategies in SemiInfinite Programming. Kluwer Academic, Boston (2003)
Jongen, HTH, Shikhman, V: Generalized semiinfinite programming: the nonsmooth symmetric reduction ansatz. SIAM J. Optim. 21, 193211 (2011)
Kanzi, N, Nobakhtian, S: Necessary optimality conditions for nonsmooth generalized semiinfinite programming problems. Eur. J. Oper. Res. 205, 253261 (2010)
Ruckmann, JJ, Shapiro, A: Firstorder optimality conditions in generalized semiinfinite programming. J. Optim. Theory Appl. 101, 677691 (1999)
Vaquez, FG, Rukmann, JJ, Stein, O, Still, G: Generalized semiinfinite programming: a tutorial. J. Comput. Appl. Math. 217, 394419 (2008)
Ye, JJ, Wu, SY: First order optimality conditions for generalized semiinfinite programming problems. J. Optim. Theory Appl. 137, 419434 (2008)
Chuong, TD, Huy, NQ, Yao, JC: Stability of semiinfinite vector optimization problems under functional perturbations. J. Glob. Optim. 45, 583595 (2009)
Chuong, TD, Huy, NQ, Yao, JC: PseudoLipschitz property of linear semiinfinite vector optimization problems. Eur. J. Oper. Res. 200, 639644 (2010)
Chuong, TD, Yao, JC: Isolated and proper efficiencies in semiinfinite vector optimization problems. J. Optim. Theory Appl. 162, 447462 (2014)
Huy, NQ, Kim, DS: Lipschitz behavior of solutions to nonconvex semiinfinite vector optimization problems. J. Glob. Optim. 56, 431448 (2013)
Goberna, MA, GuerraVazquez, F, Todorov, MI: Constraint qualifications in linear vector semiinfinite optimization. Eur. J. Oper. Res. 227, 1221 (2013)
Tykhonov, AN: On the stability of the functional optimization problems. USSR Comput. Math. Math. Phys. 6, 2833 (1966)
Levitin, ES, Polyak, BT: Convergence of minimizing sequences in conditional extremum problem. Sov. Math. Dokl. 7, 764767 (1966)
Konsulova, AS, Revalski, JP: Constrained convex optimization problemswellposedness and stability. Numer. Funct. Anal. Optim. 15, 889907 (1994)
Huang, XX, Yang, XQ: Generalized LevitinPolyak wellposedness in constrained optimization. SIAM J. Optim. 17, 243258 (2006)
Huang, XX, Yang, XQ: LevitinPolyak wellposedness of constrained vector optimization problems. J. Glob. Optim. 37, 287304 (2007)
Huang, XX, Yang, XQ: Further study on the LevitinPolyak wellposedness of constrained vector optimization problems. Nonlinear Anal. 75, 13411347 (2012)
Lalitha, CS, Chatterjee, P: LevitinPolyak wellposedness for constrained quasiconvex vector optimization problems. J. Glob. Optim. 59, 191205 (2014)
Long, XJ, Huang, NJ, Teo, KL: LevitinPolyak wellposedness for equilibrium problems with functional constraints. J. Inequal. Appl. 2008, Article ID 657329 (2008)
Bednarczuck, EM: An approach to wellposedness in vector optimization: consequences to stability and parametric optimization. Control Cybern. 23, 107122 (1994)
Dontchev, AL, Zolezzi, T: WellPosed Optimization Problems. Lecture Notes in Mathematics, vol. 1543. Springer, Berlin (1993)
Long, XJ, Huang, NJ: Metric characterizations of αwellposedness for symmetric quasiequilibrium problems. J. Glob. Optim. 45, 459471 (2009)
Long, XJ, Peng, JW: Generalized Bwellposedness for set optimization problems. J. Optim. Theory Appl. 157, 612623 (2013)
Long, XJ, Peng, JW, Peng, ZY: Scalarization and pointwise wellposedness for set optimization problems. J. Glob. Optim. 62, 763773 (2015)
Loridan, P: Wellposedness in vector optimization. In: Lucchetti, R, Recalski, J (eds.) Recent Developments in Variational WellPosedness Problems. Mathematics and Its Applications, vol. 331, pp. 171192. Kluwer Academic, Dordrecht (1995)
Miglierina, E, Molho, E, Rocca, M: Wellposedness and scalarization in vector optimization. J. Optim. Theory Appl. 126, 391409 (2005)
Wang, G, Yang, XQ, Cheng, TCE: Generalized LevitinPolyak wellposedness for generalized semiinfinite programs. Numer. Funct. Anal. Optim. 34, 695711 (2013)
Aubin, JP, Ekeland, I: Applied Nonlinear Analysis. Wiley, New York (1984)
Nikodem, K: Continuity of Kconvex setvalued function. Bull. Pol. Acad. Sci., Math. 34, 393400 (1986)
Göpfert, A, Riahi, H, Tammer, C, Zălinescu, C: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)
Kuratowski, K: Topology, vols. 1 and 2. Academic Press, New York (1968)
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
Rights and permissions
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.
About this article
Cite this article
Long, XJ., Peng, ZY. & Sun, XK. LevitinPolyak wellposedness for generalized semiinfinite multiobjective programming problems. J Inequal Appl 2016, 12 (2016). https://doi.org/10.1186/s136600150958z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s136600150958z
MSC
 90C29
 90C34
 49K40
Keywords
 generalized semiinfinite multiobjective programming problem
 LevitinPolyak wellposedness
 approximate solution map
 metric characterization
 upper semicontinuity