Optimality conditions for strict minimizers of higher-order in semi-infinite multi-objective optimization
- Guolin Yu^{1}Email author
https://doi.org/10.1186/s13660-016-1209-7
© Yu 2016
Received: 27 August 2016
Accepted: 13 October 2016
Published: 24 October 2016
Abstract
This paper is devoted to the study of optimality conditions for strict minimizers of higher-order for a non-smooth semi-infinite multi-objective optimization problem. We propose a generalized Guignard constraint qualification and a generalized Abadie constraint qualification for this problem under which necessary optimality conditions are proved. Under the assumptions of generalized higher-order strong convexity for the functions appearing in the formulation of the non-smooth semi-infinite multi-objective optimization problem, three sufficient optimality conditions are derived.
Keywords
MSC
1 Introduction
In recent years, there has been considerable interest in the so-called semi-infinite multi-objective optimization problems (SIMOPs), which is the simultaneous minimization of finitely many scalar objective functions subject to an infinitely many constraints. SIMOPs have been investigated intensively by many researchers from several different perspectives. For example, the pseudo-Lipschitz property and the semicontinuity of the efficient solution map under some types of perturbation with respect to a parameter have been discussed in [1–5]. The density of the set of all stable convex semi-infinite vector optimization problems has been established in [6]. However, the work on optimality conditions for SIMOPs is limited. Here we should mention that the authors in [7] have examined the optimality conditions and duality relations in SIMOPs involving differentiable functions, whose constraints are required to depend continuously on an index t belonging to a compact set T. For non-smooth semi-infinite multi-objective optimization problems, work has been done to obtain necessary optimality conditions for weakly efficient solutions and sufficient optimality conditions for efficient solutions by presenting several kinds of constraint qualifications and imposing assumptions of generalized convexity (see [8]), and to establish necessary and sufficient conditions for (weakly) efficient solutions of SIMOPs by applying some advanced tools of variational analysis and generalized differentiation and proposing the concepts of (strictly) generalized convex functions defined by using the limiting subdifferential of locally Lipschitz functions (see [9]). It is worth noticing that all of the above mentioned literature studies only weakly efficient solutions or efficient solutions of SIMOPs.
On the other hand, a continuing interest in the theory of multi-objective optimization is to define and characterize its solutions. Besides the weak efficiency and efficiency mentioned above, a meaningful solution concept called a strict efficient solution of higher-order (also called a strict minimizer of higher-order) was recently extended by Jiménez in [10] from the strict minimizer of higher-order in scalar optimization given by Auslender in [11] and Ward in [12]. Recently, Bhatia [13] established necessary and sufficient optimality conditions for strict efficiency of higher-order in multi-objective optimization under the basic regularity condition and generalized higher-order strong convexity assumption, respectively.
In this paper, we introduce the notion of a semi-strict minimizer of higher-order for a semi-infinite multi-objective optimization problem, which includes arbitrary many (possibly infinite) inequality constraints. For the purpose of investigating this new solution concept, we found that the notion of convexity that appears to be most appropriate in the development of sufficient optimality conditions is the strong convexity of higher-order [14].
The rest of this paper is organized as follows. In Section 2, some basic notations and results of non-smooth and convex analysis are reviewed, and the concept of a semi-strict minimizer for a semi-infinite multi-objective optimization problem is presented. In Section 3, we introduce the generalized Guignard constraint qualification and Abadie constraint qualification for SIMOPs. Necessary optimality conditions of Karush-Kuhn-Tuchker type are derived under these two constraint qualifications. Finally, in Section 4, three sufficient optimality conditions for SIMOPs are obtained under the assumption of some generalized strong convexity of higher-order.
2 Notations and preliminaries
Throughout the paper, we let \(\mathbb{R}^{n}\) be the n-dimensional Euclidean space endowed with the Euclidean norm \(\|\cdot\|\), X be a convex subset of \(\mathbb{R}^{n}\), and \(m\geq1\) be a positive integer. Let W be a subset of \(\mathbb{R}^{n}\). We use clW, coW, and coneW to denote the closure of W, the convex hull of W, and the conic hull of W (i.e., the smallest convex cone containing W), respectively.
Definition 2.1
Definition 2.2
see [18]
Lemma 2.1
see [19]
- (a)
\(\varphi^{0}(\bar{x},v)=\max \{\langle\xi,v\rangle: \xi\in\partial \varphi(\bar{x}) \}\), for all \(v\in\mathbb{R}^{n}\).
- (b)
\(\partial (\lambda\varphi(\bar{x}) )=\lambda\partial\varphi (\bar{x})\), for all \(\lambda\in\mathbb{R}\).
- (c)
\(\partial (\varphi_{1}+\varphi_{2} )(\bar{x})\subset\partial \varphi_{1}(\bar{x})+\partial\varphi_{2}(\bar{x})\).
Now, we recall the definition of the strong convexity of order m for a Lipschitz function.
Definition 2.3
- (a)φ is said to be strongly convex of order m at x̄ if there exists a constant \(c>0\) such that for each \(x\in X\) and \(\xi\in\partial\varphi(\bar{x})\)$$ \varphi(x)-\varphi(\bar{x})\geq\langle\xi,x-\bar{x}\rangle+c\|x-\bar {x} \|^{m} . $$
- (b)φ is said to be strongly quasiconvex of order m at x̄ if there exists a constant \(c>0\) such that, for each \(x\in X\) and \(\xi\in\partial\varphi(\bar{x})\),$$ \varphi(x)\leq\varphi(\bar{x})\quad \Rightarrow \quad \langle\xi,x-\bar{x} \rangle +c\|x-\bar{x}\|^{m}\leq0. $$
Based upon the above definition of a strongly convex function of order m, we define the following generalized strong convexities of order m for a Lipschitz function.
Definition 2.4
- (a)φ is strictly strong convex of order m at x̄ if there exists a constant \(c>0\) such that, for each \(x\in X\) with \(x\neq\bar{x}\) and \(\xi\in\partial\varphi(\bar{x})\),$$ \varphi(x)-\varphi(\bar{x})> \langle\xi,x-\bar{x}\rangle+c\|x-\bar{x}\| ^{m}. $$
- (b)φ is strictly strong quasiconvex of order m at x̄ if there exists a constant \(c>0\) such that, for each \(x\in X\) with \(x\neq\bar{x}\) and any \(\xi\in\partial\varphi(\bar{x})\),$$ \varphi(x)\leq\varphi(\bar{x}) \quad \Rightarrow\quad \langle\xi,x-\bar{x} \rangle +c\|x-\bar{x}\|^{m}< 0. $$
The next Lemma gives a basic property of generalized higher-order strong convexities, which will be used in Section 4.
Proposition 2.1
Let \(\varphi_{i}:X\rightarrow\mathbb{R}\) be Lipschitz at \(\bar{x}\in X\), \(i=0,1,2,\ldots, s\). Suppose that \(\varphi_{0}\) is a strictly strong convex function of order m and \(\varphi_{1},\varphi_{2},\ldots, \varphi_{s}\) are strongly convex functions of order m at x̄. If \(\lambda _{0}>0\) and \(\lambda_{i}\geq0\) for \(i=1,2,\ldots,s\), then \(\sum_{i=0}^{s}\lambda_{i}\varphi_{i}\) is strictly strong convex of order m at x̄.
Proof
- (i)
\(f(x)< f(y) \Leftrightarrow f_{i}(x)< f_{i}(y)\) for every \(i\in P\);
- (ii)
\(f(x)\nless f(y)\) is the negation of \(f(x)< f(y)\);
- (iii)
\(f(x)\leqslant f(y) \Leftrightarrow f_{i}(x)\leq f_{i}(y)\) for every \(i\in P\), but there is at least one \(i_{0}\in P\) such that \(f_{i_{0}}(x)< f_{i_{0}}(y)\);
- (iv)
\(f(x)\nleq f(y)\) is the negation of \(f(x)\leqslant f(y)\).
Definition 2.5
see [10]
Definition 2.6
Remark 2.1
It is obvious that if \(\bar{x}\in\Omega\) is a semi-strict minimizer of order m for (P), then \(\bar{x}\in\Omega\) is a strict minimizer of order m for (P).
Example 2.1
Motivated by the notion of a linearizing cone at a point to the feasible set of a differentiable multi-objective optimization problem, which was introduced by Maeda in [15], we give the definition of a linearizing cone for the semi-infinite multi-objective optimization problem (P). We first need to define a set.
Definition 2.7
3 Necessary conditions
In this section, we shall examine necessary optimality conditions for a semi-strict (strict) minimizer of order m for (P). we begin with presenting two constraint qualifications, which are the non-smooth, semi-infinite version of the generalized Guignard constraint qualification and generalized Abadie constraint qualification presented in [15] and [16].
Definition 3.1
The problem (P) satisfies the generalized Guignard constraint qualification at a given point \(\bar{x}\in\Omega\) which is a semi-strict minimizer of order m for (P) if the following holds: \(C(\bar{x}) \subseteq\bigcap_{i=1}^{p} \operatorname{cl} [\operatorname{co} (T(Q^{i};\bar{x}) ) ]\), where \(Q^{i}:= Q^{i}(\bar{x})\).
Definition 3.2
The problem (P) satisfies the generalized Abadie constraint qualification at a given point \(\bar{x}\in\Omega\) which is a semi-strict minimizer of order m for (P) if the following holds: \(C(\bar{x}) \subseteq\bigcap_{i=1}^{p} T(Q^{i};\bar{x})\).
Next, we recall the generalized Motzkin theorem discussed in [20].
Lemma 3.1
see [20]
The following lemma is a generalization of the classical Tucker theorem of the alternative. We use this lemma in the proof of our necessary efficiency result. The proof of the lemma is similar to that of Lemma 3.6 in [17], hence it is omitted.
Lemma 3.2
Lemma 3.3
Proof
Replacing \(t_{mln}\) by \(t_{n}\), \(x^{mln}\) by \(x^{n}\), and \(z^{ml}\) by z in (3.6), we arrive at \(f_{1}^{0}(\bar{x}; z)\ge0\). By Proposition 2.1.2 in [18], we know that there is a \(\xi\in \partial f_{1}(\bar{x})\) such that \(\langle\xi, z \rangle= f_{1}^{0}(\bar{x}; z) \ge0\), contradicting the assumption that z is a solution of the system (3.1). Therefore, (3.1) has no solution \(z\in\mathbb{R}^{n}\). □
Now we are ready to prove the following necessary optimality condition for (P).
Theorem 3.1
Proof
Remark 3.1
4 Sufficient conditions
In this section we discuss sufficient optimality results under various generalized higher-order strong convexity (introduced in Section 2) hypotheses imposed on the involved functions.
Theorem 4.1
Sufficient optimality conditions I
Proof
Theorem 4.2
Sufficient optimality conditions II
Proof
Theorem 4.3
Sufficient optimality conditions III
5 Conclusions
We have defined a strict minimizer of higher-order and a semi-strict minimizer of higher-order for a semi-infinite multi-objective optimization problem in this paper. We have presented a non-smooth semi-infinite version of the generealized Guignard and Abadie constraint qualifications. Under those constraint qualifications, utilizing the method in [15, 16] we have proved necessary optimality conditions for a semi-strict minimizer of higher-order and a strict minimizer of higher-order. Three sufficient optimality conditions have been proved under the assumption of strong convexities.
Declarations
Acknowledgements
This research was supported by Natural Science Foundation of China under Grant No. 11361001; Natural Science Foundation of Ningxia under Grant No. NZ14101.
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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