On strong KKT type sufficient optimality conditions for multiobjective semi-infinite programming problems with vanishing constraints
- Sy-Ming Guu^{1, 2}Email author,
- Yadvendra Singh^{3} and
- Shashi Kant Mishra^{3}
https://doi.org/10.1186/s13660-017-1558-x
© The Author(s) 2017
Received: 15 August 2017
Accepted: 26 October 2017
Published: 14 November 2017
Abstract
In this paper, we consider a nonsmooth multiobjective semi-infinite programming problem with vanishing constraints (MOSIPVC). We introduce stationary conditions for the MOSIPVCs and establish the strong Karush-Kuhn-Tucker type sufficient optimality conditions for the MOSIPVC under generalized convexity assumptions.
Keywords
MSC
1 Introduction
Multiobjective semi-infinite programming problems (MOSIPs) arise when more than one objective function is to be optimized over the feasible region described by an infinite number of constraints. If there is only one objective function in a MOSIP, then it is known as semi-infinite programming problem (SIP). SIPs have played an important role in several areas of modern research, such as transportation theory [1], engineering design [2], robot trajectory planning [3] and control of air pollution [4]. We refer to the books [5, 6] for more details as regards SIPs and their applications and to some recent papers [7–9] for details as regards MOSIPs.
Achtziger and Kanzow [10] introduced the mathematical programs with vanishing constraints (MPVCs) and showed that many problems from structural topology optimization can be reformulated as MPVCs. Hoheisel and Kanzow [11] defined stationary concepts for MPVCs and derived first order sufficient and second order necessary and sufficient optimality conditions for MPVCs. Hoheisel and Kanzow [12] established optimality conditions for weak constraint qualification. Mishra et al. [13] obtained various constraint qualifications and established Karush-Kuhn-Tucker (KKT) type necessary optimality conditions for multiobjective MPVCs. We refer to [14–16] and references therein for more details as regards MPVCs.
Recently, the idea of a strong KKT has been used to avoid the case where some of the Lagrange multipliers associated with the components of multiobjective functions vanish. Golestani and Nobakhtian [17] derived the strong KKT optimality conditions for nonsmooth multiobjective optimization. Kanzi [9] established strong KKT optimality conditions for MOSIPs. Pandey and Mishra [18] established the strong KKT type sufficient conditions for nonsmooth MOSIPs with equilibrium constraints.
Motivated by Achtziger and Kanzow [10], Golestani and Nobakhtian [17] and Pandey and Mishra [18], we extend the concept of the strong KKT optimality conditions for the MOSIPs with vanishing constraints (MOSIPVCs) that do not involve any constraint qualification. The paper is organized as follows. In Section 2, we present some known definitions and results which will be used in the sequel. In Section 3, we define stationary points and establish strong KKT type optimality for MOSIPVC. In Section 4, we conclude the results of the paper.
2 Definitions and preliminaries
Theorem 2.1
- 1.
\(f^{0}(\bar{x};v)= \max \{ \langle \xi, v\rangle: \xi \in \partial _{c} f(\bar{x}), \forall v \in \mathbb{R}^{n}\}\),
- 2.
\(\partial_{c}(\lambda f)(\bar{x})=\lambda \partial_{c}f(\bar{x})\), \(\forall \lambda \in \mathbb{R}\),
- 3.
\(\partial_{c}(f+g)(\bar{x}) \subseteq \partial_{c} f(\bar{x})+ \partial_{c} g(\bar{x})\).
The following definitions and lemma from Kanzi and Nobakhtian [8] will be used in the sequel.
Definition 2.1
- 1.f is said to be generalized convex at x̄ if, for each \(x \in \mathbb{R}^{n}\) and any \(\xi \in \partial_{c} f(\bar{x})\),$$f(x) - f(\bar{x}) \geq \langle \xi, x-\bar{x}\rangle, $$
- 2.f is said to be strictly generalized convex at x̄ if, for each \(x \in \mathbb{R}^{n}\), \(x\neq \bar{x}\) and any \(\xi \in \partial_{c} f( \bar{x})\),$$f(x) - f(\bar{x}) > \langle \xi, x-\bar{x}\rangle, $$
- 3.f is said to be generalized quasiconvex at x̄ if, for each \(x \in \mathbb{R}^{n}\) and any \(\xi \in \partial_{c} f(\bar{x})\),$$f(x) \leq f(\bar{x})\quad \Rightarrow\quad \langle \xi, x-\bar{x}\rangle \leq 0, $$
- 4.f is said to be strictly generalized quasiconvex at x̄ if, for each \(x \in \mathbb{R}^{n}\) and any \(\xi \in \partial_{c} f(\bar{x})\),$$f(x) \leq f(\bar{x}) \quad \Rightarrow\quad \langle \xi, x-\bar{x}\rangle < 0. $$
Lemma 2.1
Let \(f_{0}\) be strictly generalized convex and \(f_{1},f_{2},\dots,f _{s}\) be generalized convex function at x. If \(\lambda_{0} > 0\) and \(\lambda_{l} \geq 0\) for \(l=1,\dots,s\), then \(\sum_{l=1}^{s} \lambda_{l}f_{l}\) is strictly generalized convex at x.
3 Strong KKT type sufficient optimality conditions
We extend Definitions 2.1 and 2.2 of Hoheisel and Kanzow [11] to the case of the MOSIPVC.
Definition 3.1
(MOSIPVC S-stationary point)
Definition 3.2
(MOSIPVC M-stationary point)
Remark 3.1
The difference between MOSIPVC M-stationary points and MOSIPVC S-stationary points occurs only for the index set \(I_{00}\). For MOSIPVC M-stationary points, \(\eta_{i}^{G} \geq 0\) and \(\eta_{i}^{H}\cdot\eta_{i} ^{G}=0\) for \(i\in I_{00}\), whereas for MOSIPVC S-stationary points, \(\eta_{i}^{H} \geq 0\) and \(\eta_{i}^{G}= 0\) for \(i\in I_{00}\).
In the following theorem, we establish the strong KKT type sufficient optimality result for the MOSIPVC under generalized convexity assumptions.
Theorem 3.1
Let x̄ be a MOSIPVC M-stationary point. Suppose that \(f_{i}\), \(i=1,\ldots,m\), \(g_{t}\), \(t \in T(\bar{x})\), \(-H_{i}\), \(G_{i}\), \(i=1, \dots, l\), are generalized convex at x̄ on M and at least one of them is strictly generalized convex at x̄ on M. Then x̄ is a weakly efficient solution for the MOSIPVC.
Proof
The following result is a direct consequence of Theorem 3.1, where the MOSIPVC M-stationary point is replaced by a MOSIPVC S-stationary point.
Corollary 3.1
Let x̄ be a MOSIPVC S-stationary point. Suppose that \(f_{i}\), \(i=1,\ldots,m\), \(g_{t}\), \(t \in T(\bar{x})\), \(-H_{i}\), \(G_{i}\), \(i=1, \dots, l\), are generalized convex at x̄ on M and at least one of them is strictly generalized convex at x̄ on M. Then x̄ is a weakly efficient solution for the MOSIPVC.
The strong KKT type sufficient condition for the MOSIPVC given in Theorem 3.1 can be obtained under further relaxations on generalized convexity requirements.
Theorem 3.2
Let x̄ be a MOSIPVC M-stationary point. Suppose that \(f_{i}\), \(i=1,\ldots,m\), \(g_{t}\), \(t \in T(\bar{x})\), \(-H_{i}\), \(G_{i}\), \(i=1, \dots, l\), are generalized quasiconvex at x̄ on M and at least one of them is strictly generalized quasiconvex at x̄ on M. Then x̄ is a weakly efficient solution for the MOSIPVC.
The following example satisfies the assumptions of Theorem 3.1.
Example 3.1
It is easy to see that \(\bar{x}= ( 0,0 ) \) is a feasible point of the problem, \(T(\bar{x})=\mathbb{N}\) and \(I_{00}(\bar{x})=\{1\}\). The feasible point x̄ is a MOSIPVC M-stationary point with \(\lambda_{1}>0\), \(\lambda_{2}=1\), \(\mu_{1}=1\), \(\mu_{2}=\frac{1}{2}\), \(\mu_{3}= \mu_{4}=\cdots=0\), \(\eta^{H}=-1\), \(\eta^{G}=0\), \(\xi^{f_{1}}=(0,0) \in \partial_{c} f_{1}(\bar{x})=\{(0,0)\}\), \(\xi^{f_{2}}=(1,0)\in \partial_{c} f_{2}(\bar{x})=[-1,1]\times [-1,1]\), \(\xi_{1}^{g_{t}}=(-t,0)\in \partial _{c} g_{t}(\bar{x})=\{(-t,0)\}\), \(\xi^{H}=(1,0) \in \partial_{c} H( \bar{x})=\{(1,0)\}\) and \(\xi^{G}=(0,1) \in \partial_{c} G(\bar{x})=[-1,1] \times \{1\}\).
The strong KKT type sufficient optimality condition for the MOSIPVC can also be obtained in the following way.
Theorem 3.3
Let x̄ be a MOSIPVC M-stationary point. Suppose that each \(f_{i}\), \(i=1,\ldots,m\), is generalized convex at x̄ on M and \(\sum_{t\in T(\bar{x})} \mu_{t}g_{t}({x})-\sum_{i=1} ^{l} \eta^{H}_{i} H_{i}({x})+ \sum_{i=1}^{l} \eta^{G}_{i} G _{i}({x})\) is generalized convex at x̄ on M. Then x̄ is a weakly efficient solution for the MOSIPVC.
The following result is a direct consequence of Theorem 3.3, where the MOSIPVC M-stationary point is replaced by a MOSIPVC S-stationary point.
Corollary 3.2
Let x̄ be a MOSIPVC S-stationary point. Suppose that each \(f_{i}\), \(i=1,\ldots,m\) is generalized convex and \(\sum_{t\in T(\bar{x})} \mu_{t}g_{t}({x})-\sum_{i=1}^{l} \eta ^{H}_{i} H_{i}({x})+ \sum_{i=1}^{l} \eta^{G}_{i} G_{i}({x})\) is generalized convex at x̄ on M. Then x̄ is a weakly efficient solution for the MOSIPVC.
The following example satisfies the assumptions of Theorem 3.3.
Example 3.2
It is easy to see that \(\bar{x}= ( 0,0 ) \) is a feasible point of the problem, \(T(\bar{x})=\mathbb{N}\) and \(I_{00}(\bar{x})=\{1\}\). The feasible point x̄ is a MOSIPVC M-stationary point with \(\lambda_{1} > 0\), \(\lambda_{2}=1\), \(\mu_{1}=1\), \(\mu_{2}=\mu_{3}=\cdots=0\), \(\eta ^{H}_{1}=-1\), \(\eta^{G}_{1}=0\), \(\xi^{f_{1}}=(0,0) \in \partial_{c} f _{1}(\bar{x})=[-1,1]\times \{0\}\), \(\xi^{f_{2}}=(0,-1)\in \partial_{c} f _{2}(\bar{x})=\{0\}\times [-1,1]\), \(\xi_{1}^{g_{t}}=(0,0)\in \partial_{c} g_{t}(\bar{x})=\{(0,0)\}\), \(\xi^{H}=(0,1) \in \partial_{c} H(\bar{x})= \{(0,1)\}\) and \(\xi^{G}=(1,0) \in \partial_{c} G(\bar{x})=[-1,1] \times \{0\}\). Also, \(\mu_{1}g_{1}({x})+\mu_{2}g_{2}({x})+\cdots- \eta ^{H}_{1} H({x})+ \eta^{G}_{1} G({x})=-x_{1}^{3}+x_{1}^{3}+x_{2}-0\). \(\vert x _{1}\vert = x_{2}\) is generalized convex at x̄ on M.
4 Results and discussion
In this paper, we consider a MOSIPVC. We introduce stationary conditions for the MOSIPVC and establish the strong KKT type sufficient optimality conditions for the MOSIPVC under generalized convexity assumptions. We extend the concept of the strong KKT optimality conditions for the MOSIPVC that do not involve any constraint qualification. Furthermore, the results of this paper may be extended to strong KKT type necessary optimality conditions for the MOSIPVC involving constraint qualification.
Declarations
Acknowledgements
The research of S-MG was partially supported by MOST 106-2221-E-182-038-MY2 of the Ministry of Science and Technology, Taiwan and BMRPD017 of Chang Gung Memorial Hospital LinKou, Taiwan. The research of YS was supported by the Council of Scientific and Industrial Research (CSIR), New Delhi, India, through grant no. 09/013(0474)/2012-EMR-1.
Authors’ contributions
YS conceived of the study and drafted the manuscript initially. S-MG participated in its design and coordination and finalized the manuscript. SKM outlined the scope and design of the study. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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