Abstract
In this paper, we consider a nonsmooth multiobjective semiinfinite programming problem with vanishing constraints (MOSIPVC). We introduce stationary conditions for the MOSIPVCs and establish the strong KarushKuhnTucker type sufficient optimality conditions for the MOSIPVC under generalized convexity assumptions.
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1 Introduction
Multiobjective semiinfinite 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 semiinfinite 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 KarushKuhnTucker (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
In this paper, we consider the following MOSIPVC:
where \(f_{i}: \mathbb{R}^{n} \rightarrow \mathbb{R}\), \(g_{t}: \mathbb{R}^{n} \rightarrow \mathbb{R}\cup \{+\infty \}\), \(G_{i}: \mathbb{R}^{n} \rightarrow \mathbb{R}\), \(H_{i}:\mathbb{R}^{n} \rightarrow \mathbb{R}\) are given locally Lipschitz functions and the index set T is arbitrary (possibly infinite). Let \(M:=\{ x \in \mathbb{R}^{n}: g _{t}(x) \leq 0, t \in T, H_{i}(x) \geq 0, G_{i}(x)H_{i}(x) \leq 0, i=1,\dots,l \}\), denote the feasible set of the MOSIPVC. A point \(\bar{x} \in M\) is said to be a weakly efficient solution for the MOSIPVC if there exists no \(x \in M\) such that
Let \(\bar{x} \in M\). The following index sets will be used in the sequel.
Furthermore, the index set \(I_{+}(\bar{x})\) can be divided as follows:
Similarly, the index set \(I_{0}(\bar{x})\) can be partitioned as follows:
The Clarke directional derivative of a locally Lipschitz function \(f:\mathbb{R}^{n}\rightarrow \mathbb{R}\) around x̄ in the direction \(v \in \mathbb{R}^{n}\) and the Clarke subdifferential of f at x̄ are, respectively, given by
We recall the following results from [19].
Theorem 2.1
Let f and g be locally Lipschitz from \(\mathbb{R}^{n}\) to \(\mathbb{R}\) around x̄. Then the following properties hold:

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
Let \(f:\mathbb{R}^{n}\rightarrow \mathbb{R}\) be a locally Lipschitz function around x̄. Then

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 Sstationary point)
A feasible point x̄ of the MOSIPVC is called a MOSIPVC strong (S)stationary point if there exist Lagrange multipliers \(\lambda_{i} > 0\), \(i=1,\dots,m\), and \(\mu_{t} \geq 0\), \(t\in T(\bar{x})\), with \(\mu_{t} \neq 0\) for at most finitely many indices and \(\eta_{i}^{H}, \eta_{i}^{G} \in \mathbb{R}\), \(i=1, \dots, l\) such that the following conditions hold:
Definition 3.2
(MOSIPVC Mstationary point)
A feasible point x̄ of the MOSIPVC is called a MOSIPVC Mordukhovich (M)stationary point if there exist Lagrange multipliers \(\lambda_{i} > 0\), \(i=1,\dots,m\), and \(\mu_{t} \geq 0\), \(t\in T(\bar{x})\), with \(\mu_{t} \neq 0\) for at most finitely many indices and \(\eta_{i}^{H}\), \(\eta_{i}^{G} \in \mathbb{R}\), \(i=1, \dots, l\), such that the following conditions hold:
Remark 3.1
The difference between MOSIPVC Mstationary points and MOSIPVC Sstationary points occurs only for the index set \(I_{00}\). For MOSIPVC Mstationary points, \(\eta_{i}^{G} \geq 0\) and \(\eta_{i}^{H}\cdot\eta_{i} ^{G}=0\) for \(i\in I_{00}\), whereas for MOSIPVC Sstationary 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 Mstationary 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
Since x̄ is a MOSIPVC Mstationary point, there exist \(\bar{\xi }^{f}_{i} \in \partial_{c}f_{i}(\bar{x})\), \(i=1, \dots,m\), \(\bar{\xi }^{g}_{t} \in \partial_{c}g_{t}(\bar{x})\), \(t \in T(\bar{x})\), and \(\bar{\xi }^{H}_{i} \in \partial_{c}H_{i}( \bar{x})\), \(\bar{\xi }^{G}_{i} \in \partial_{c}G_{i}(\bar{x})\), \(i=1,\ldots,l\), such that
Suppose on the contrary that x̄ is not a weakly efficient solution for the MOSIPVC, that is, there exists \(\tilde{x} \in M\), such that
From the MOSIPVC Mstationary point, we have \(\lambda_{i}> 0\) for \(i=1,\dots, m\). Thus, we get
Since x̄ is a MOSIPVC Mstationary point and x̃ is a feasible point of the MOSIPVC, we have
which implies that
It follows from Lemma 2.1 that \(\sum_{i=1}^{m} \lambda _{i} f_{i}({x})+ \sum_{t\in T(\bar{x})} \mu_{t}g_{t}({x}) \sum_{i=1}^{l} \eta^{H}_{i} H_{i}({x})+ \sum_{i=1}^{l} \eta^{G}_{i} G_{i}({x})\) is a strictly generalized convex function at x̄ on M. Hence,
Therefore, from (3.1), (3.4) and (3.5), we obtain
Thus, we arrive at a contradiction and hence the result. □
The following result is a direct consequence of Theorem 3.1, where the MOSIPVC Mstationary point is replaced by a MOSIPVC Sstationary point.
Corollary 3.1
Let x̄ be a MOSIPVC Sstationary 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 Mstationary 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
Consider the following problem in \(\mathbb{R}^{2}\):
Note that \(f_{1}(x)=\vert x_{1}\vert \), \(f_{2}(x)=\vert x_{1}\vert +\vert x_{2}\vert \) and the feasible region of the MOSIPVC (3.6) is given by
which is represented by the shaded region in Figure 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 Mstationary 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 Mstationary 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.
Proof Suppose on the contrary that x̄ is not a weakly efficient solution for the MOSIPVC, that is, there exists a feasible point x̃ such that
By strictly generalized convexity of \(f_{i}\), we have
From the Mstationary condition, we have \(\lambda_{i} > 0\), \(i=1,\ldots,m\). Thus, we get
Since x̄ is a MOSIPVC Mstationary point, from (3.1) and (3.8), we have
From (3.3), we have
From the generalized convexity of \(\sum_{t\in T(\bar{x})} \mu _{t}g_{t}({x})\sum_{i=1}^{l} \eta^{H}_{i} H_{i}({x})+ \sum_{i=1}^{l} \eta^{G}_{i} G_{i}({x})\), at x̄ on M, we get
which contradicts (3.9). Hence, x̄ is a weakly efficient solution of the MOSIPVC and the proof is complete.
The following result is a direct consequence of Theorem 3.3, where the MOSIPVC Mstationary point is replaced by a MOSIPVC Sstationary point.
Corollary 3.2
Let x̄ be a MOSIPVC Sstationary 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
Consider the following problem in \(\mathbb{R}^{2}\):
Note that \(f_{1}(x)=\vert x_{1}\vert \), \(f_{2}(x)=\vert x_{2}\vert \) and the feasible region of the MOSIPVC (3.12) is given by
which is represented by the shaded region in Figure 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 Mstationary 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.
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Acknowledgements
The research of SMG was partially supported by MOST 1062221E182038MY2 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)/2012EMR1.
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YS conceived of the study and drafted the manuscript initially. SMG 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.
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Guu, SM., Singh, Y. & Mishra, S.K. On strong KKT type sufficient optimality conditions for multiobjective semiinfinite programming problems with vanishing constraints. J Inequal Appl 2017, 282 (2017). https://doi.org/10.1186/s136600171558x
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DOI: https://doi.org/10.1186/s136600171558x