Issue 
Int. J. Simul. Multidisci. Des. Optim.
Volume 11, 2020



Article Number  17  
Number of page(s)  7  
DOI  https://doi.org/10.1051/smdo/2020011  
Published online  10 August 2020 
Research Article
Optimality and duality for nonsmooth semiinfinite Econvex multiobjective programming with support functions
^{1}
Department of Mathematics, Faculty of Science, Jouf University, P.O. Box 2014, Sakaka, Saudi Arabia
^{2}
Department of Mathematics, Faculty of Science, Suez University, P.O. Box 43533, Suez, Egypt
^{*} email: drtemam@yahoo.com
Received:
17
September
2019
Accepted:
29
June
2020
In this paper, we study a nonsmooth semiinfinite multiobjective Econvex programming problem involving support functions. We derive sufficient optimality conditions for the primal problem. We formulate MondWeir type dual for the primal problem and establish weak and strong duality theorems under various generalized Econvexity assumptions.
Key words: Nonsmooth semiinfinite multiobjective optimization / generalized econvexity / duality
© T. Emam, published by EDP Sciences, 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Semiinfinite multiobjective programming consider several conflicting objective functions have to be optimized over a feasible set described by infinite number of inequality constraints. Semiinfinite programming problems have occupied the attention of a number of mathematicians due to their applications in many areas such as in engineering, robotics, and transportation problems, see [1]. Optimality conditions and duality results for semiinfinite programming problems have been studied see, [2–10]. Optimality and duality results for semiinfinite multiobjective programming problems that involved differentiable functions were obtained by Caristi et al. [11]. Several kinds of constraints qualifications were defined by Kanzi and Nobakhtian [12] and they obtained necessary and sufficient optimality conditions for nonsmooth semiinfinite multiobjective programming problems. Mishra et al. [13] proved necessary and sufficient optimality conditions for nondifferential semiinfinite programming problems involving square root of quadratic functions, for more details see [14]. Mond and Schechter [15] have constructed symmetric duality of both Wolfe and MondWeir types for nonlinear programming problems where the objective contains the support function. Optimality and duality for a nondifferentiable nonlinear programming problem involving support function have been obtained by Husain et al. [16], see for more details [17–20]. In other hand, convexity and their generalizations play an important role in optimization theory. Youness point of view of convexity is based on the effecting of an operator E on the domain on which functions are defined [21,22]. This kind of convexity is called Econvexity and can be viewed in many fields such as in differential geometry when a manifold is deformed by an operator E. In the field of physical chemistry an Econvexity can be occurred when the binding force f between elements construct a crystal effect by a solution E. In mathematical programming, the notion of Econvexity of functions plays an important role in solving the problem of type composite model problem [23] such as the problem
This paper is organized as follows: In Section 2, we mention some definitions and preliminaries. In Section 3, the sufficient optimality conditions for multiobjective semiinfinite Econvex programming problems involving support functions are established. In Section 4, we formulate MondWeir type dual for multiobjective semiinfinite Econvex programming problems involving support functions and establish weak, strong and strictconverse duality theorems under generalized Econvexity assumptions.
2 Definitions and preliminaries
In this section, we present some definitions and results, which will be needed in this article. Let R ^{ n } be the ndimensional Euclidean space and be the nonnegative orthant of R ^{ n }. Let ⟨ . , .⟩ denotes the Euclidean inner product and  .  be Euclidean norm in R ^{ n }. Given a nonempty set D ⊆ R ^{ n }, we denote the closure of D by and convex cone (containing origin) by cone(D). The native polar cone and the strictly negative polar cone are defined respective by
Definition 1 [24] Let D ⊆ R ^{ n }. The contingent cone T(D, x) at is defined by
Definition 2 [24] A function f : R ^{ n } → R is said to be Lipschitz near x ∈ R ^{ n }, if there exist a positive constant K and a neighborhood N of x such that for any y, z ∈ N, we have
The function f is said to be locally Lipschitz on R ^{ n } if it is Lipschitz near x for every x ∈ R ^{ n }.
Definition 3 [24] The Clarke generalized directional derivative of a locally Lipschitz function f at x ∈ R ^{ n } in the direction d ∈ R ^{ n }, denoted by f ^{∘} (x, d), is defined as where y ∈ R ^{ n }.
Definition 4 [24] The Clarke generalized subdifferential of f at x ∈ R ^{ n } is denoted by ∂cf(x), defined as
Definition 5 [21] A set M ⊆ R ^{ n } is said to be Econvex set with respect to an operator E : R ^{ n } → R ^{ n } if λE(x) + (1 − λ)E(y) ∈ M for each x, y ∈ M and 0 ≤ λ ≤ 1.
Every Econvex set with respect to an operator E : R ^{ n } → R ^{ n } is a convex set when E = I. If M _{1} and M _{2} are Econvex sets, then M _{1} _{⋂} M _{2} is Econvex set but M _{1} _{⋃} M _{2} is not necessarily Econvex set. If E : R ^{ n } → R ^{ n } is a linear map, and M _{1}, M _{2} ⊆ R ^{ n } are Econvex sets, then M _{1} + M _{2} is Econvex set.
Example 1 Let E : R ^{2} → R ^{2} be defined as E(x, y) = (0, y). The set is an Econvex set with respect to the operator E.
Definition 6 A locally Lipschitz function f : R ^{ n } → R is said to be Econvex with respect to an operator E : R ^{ n } → R ^{ n } at x* ∈ R ^{ n } if for each x ∈ R ^{ n } and every ξ ∈ ∂ cf(Ex ^{*}).
The function f is said to be Econvex near x ^{*} ∈ R ^{ n } if it is Econvex at each point of neighborhood of x ^{*} ∈ R ^{ n }.
Definition 7 A locally Lipschitz function f : R ^{ n } → R is said to be strictly Econvex with respect to an operator E : R ^{ n } → R ^{ n } at x* ∈ R ^{ n } if for each x ∈ R ^{ n }, x ≠ x ^{*} and every ξ ∈ ∂ cf (Ex ^{*}) .
The function f is said to be strictly Econvex near x ^{*} ∈ R ^{ n } if it is strictly Econvex at each point of neighborhood of x ^{*} ∈ R ^{ n }.
Proposition 1 [21] If g _{i } : R ^{ n } → R, i = 1, 2, ..., m is Econvex with respect to E : R ^{ n } → R ^{ n } then the set M ={x ∈ R ^{n } : g _{i } (x ) ≤ 0, i = 1, 2, ..., m } is Econvex set.
Definition 8 A locally Lipschitz function f : R ^{ n } → R is said to be pseudo Econvex with respect to E : R ^{ n } → R ^{ n } at x ^{*} ∈ R ^{ n } if for each x ∈ R ^{ n } and every ξ ∈ ∂ cf (Ex ^{*}).
Definition 9 A locally Lipschitz function f : R ^{ n } → R is said to be strictly pseudo Econvex with respect to E : R ^{ n } → R ^{ n } at x ^{*} ∈ R ^{ n } if for each x ∈ R ^{ n }, x ≠ x ^{*} and every ξ ∈ ∂ cf (Ex ^{*}).
Definition 10 A locally Lipschitz function f : R ^{ n } → R is said to be quasi Econvex with respect to E : R ^{ n } → R ^{ n } at x ^{*} ∈ R ^{ n } if for each x ∈ R ^{ n } and every ξ ∈ ∂ cf (Ex ^{*}).
The function f is said to be quasi Econvex near x ^{*} ∈ R ^{ n } if it is quasi Econvex at each point of neighborhood of x ^{*} ∈ R ^{ n }.
Proposition 2 [21] If g _{i } : R ^{ n } → R, i = 1, 2, ..., m is quasi Econvex with respect to E : R ^{ n } → R ^{ n } then the set M ={x ∈ R ^{n } : g _{i } (x ) ≤ 0, i = 1, 2, ..., m} is Econvex set.
Remark 1

Every Econvex function is also quasi Econvex with respect to same E : R ^{ n } → R ^{ n }, but not conversely.

Every Econvex function is also pseudo Econvex with respect to same E : R ^{ n } → R ^{ n }, but not conversely.

Every strictly Econvex function is also strictly pseudo Econvex with respect to same E : R ^{ n } → R ^{ n }, but not conversely.
Let C be a nonempty compact Econvex set in R ^{ n }. The support function S(.C) : R ^{ n }→ R ∪ { +∞ } is given by
Example 2 Let E : R ^{2} → R ^{2} be defined as E(y _{1}, y _{2} ) = (0, y _{2} ). If , then the support function S(.C) : R→ R ∪ {+∞} is given by i.e.
The support function, being convex and everywhere finite, has a Clark subdifferential [24], in the sense of convex analysis. Its subdifferential is given by
In this paper, we consider the following nonsmooth semiinfinite multiobjective Econvex programming problem:
subject to where I is an index set which is possibly infinite, f _{ j } (x), j = 1, … , p and g _{ i } (x), i ∈ I are locally Lipschitz Econvex functions from R ^{ n } to R∪ { +∞ }. Let M denote the Econvex feasible set of (P).
Let x ^{*} ∈ M. We denote I (x ^{*}) ={i ∈ I (g _{ i } ∘ E)x ^{*} = 0 }, the index set of active constraints and let
The following constraint qualifications are generalization of constraint qualifications from [12] for multiobjective Econvex programming problem with support functions (P).
Definition 11 We say that:

The Abedie constraint qualification (ACQ) holds at if .

The Basic constraint qualification (BCQ) holds at if .

The Regular constraint qualification (RCQ) holds at x ∈ M if .
Definition 12 A feasible point x ^{*} ∈ M is said to be weakly efficient solution for (P) if there is no x ∈ M such that
3 Optimality conditions
In this section, we prove the sufficient optimality conditions for considered nonsmooth semiinfinite multiobjective Econvex programming problem (P) as follows:
Theorem 1 (Necessary optimality conditions) Let E : R ^{ n } → R ^{ n } and x ^{*} be a feasible solution of (P). Assume that Ex^{*} be a weakly efficient solution of (P) and a suitable constraints qualification from Definition (11) holds at E(x ^{*} ). If cone(G (Ex ^{*})) is closed, then there exist τ _{j } ≥ 0, z _{j } ∈ C _{j } (for j = 1, 2, ..., p) and λ _{i } ≥ 0 (for i ∈ I(x ^{*} )) with λ _{i } ≠ 0 for finitely many indices i, such that (1) (2) (3)
Proof: See Theorem 3.4 (ii) of Kanzi and Nobakhtian [12].
Theorem 2 (Sufficient optimality conditions) Let E : R ^{ n } → R ^{ n } and x ^{*} be a feasible solution of (P). Assume that there exist τ _{j } ≥ 0, z _{j } ∈ C _{j } (for j = 1, 2, ..., p) and λ _{i } ≥ 0 (for i ∈ I(x ^{*} )) with λ _{i } ≠ 0 for finitely many indices i, such that necessary optimality conditions (1)–(3) hold at x ^{*}. If τ _{j } (f _{j } (.) + ⟨ z _{ j }, . ⟩) for j = 1, 2, ..., p are pseudo Econvex at x ^{*} and λ _{i } g _{i } (.), i ∈ I(x ^{*}) are quasi Econvex at x ^{*} with respect to the same E and f _{j } (Ex) ≤ f _{j } (x), j = 1, … , p, ∀ x ∈ M. Then, Ex^{*} is a weakly efficient solution for (P).
Proof: Suppose, contrary to the result, that Ex ^{*} ∈ M, is not a weakly efficient solution for (P). Then, there exists a feasible point x ∈ M for (P) such thatbut f _{ j } (Ex) ≤ f _{ j }(x) and τ _{ j } ≥ 0, for j = 1, 2,..., p, so we have(4)
Since ⟨z _{ j }, Ex ⟩ ≤ S(xC _{ j }), j = 1, 2,..., p and the assumption ⟨z _{ j }, Ex ^{*} ⟩ = S(x ^{*}C _{ j }), j = 1, 2, ..., p, we have(5)
Now, from equation (1), there exist ξ _{ j } ∈ ∂ cf _{ j } (Ex ^{*}) and ζ _{ i } ∈ ∂ cg _{ i } (Ex ^{*}) such that(6)
Since Ex is a feasible point for (P) where M is Econvex set and λ _{ i } g _{ i } (Ex ^{*}) = 0, i ∈ I(x ^{*}), we have(7)and from quasi Econvexity of g _{ i }, i ∈ I(x ^{*}), we getby using (6), we have
Thus, from pseudo Econvexity of τ _{ j }(f _{ j }(.) + ⟨ z _{ j }, . ⟩), for j = 1, 2, ..., p, we getwhich contradicts (5). Thus Ex ^{*} is a weakly efficient solution for (P).
The following corollary is a direct consequence of Remark 1 and Theorem 2.
Corollary 1 Let E : R ^{ n } → R ^{ n } and x ^{*} be a feasible solution of (P). Assume that there exist τ _{j } ≥ 0, z _{j } ∈ C _{j } (for j = 1, 2, ..., p) and λ _{i } ≥ 0(for i ∈ I(x ^{*} )) with λ _{ i } ≠ 0 for finitely many indices i, such that necessary optimality conditions (1)–(3) hold at x ^{*}. If τ _{j } (f _{j } (.) + ⟨ z _{j }, . ⟩) for j = 1, 2, ..., p and λ _{i } g _{i } (.), ∈ I(x^{*}) are Econvex at x ^{*} with respect to the same E and f _{j } (Ex) ≤ f _{j } (x), j = 1, … , p, ∀ x ∈ M. Then, Ex^{*} is a weakly efficient solution for (P).
Example 3 Let E : R ^{2} → R ^{2} be defined as and let M be given by
Consider the bicriteria Econvex programming problem where and f _{2}(x _{1}, x _{2}) = (x _{2} − x _{1})^{3} and where x = (x _{1}, x _{2})for C _{1} = C _{2} ={(0, x _{2}) : − 12 ≤ x _{2} ≤ 0}. It is clear that M is Econvex with respect to E and
E(M) = { (x _{1}, x _{2}) ∈ R ^{2} : x _{1} + x _{2} − 3 ≤ 0, x _{1} − 2x _{2} ≤ 0, x _{2} ≥ 0, x _{1} ≥ 0 } .
By choosing as the active constraint of (P) then I (x ^{*}) = 1. It is clear that all defined functions are locally Lipschitz functions at Ex ^{*} and ∂f _{1}(Ex ^{*}) = (12α ^{2}, 0), ∂ f _{2}(Ex ^{*}) = (− 3α ^{2}, 3α ^{2}), ∂ g _{1}(Ex ^{*}) = (1, −2) where α ∈ [0, 1]. Since τ _{ j } (f _{ j }(x) + ⟨ z _{ j }, x ⟩)for j = 1, 2 are pseudo Econvex and λ _{1} g _{1}(x) are quasi Econvex at x ^{*} with respect to same E and conditions (1)–(3) of theorem (1) holds at x ^{*} ∈ M as there exist where α ∈ [0, 1] such that
Then there is no x ∈ M such thatand hence where and are weakly efficient solutions for (P).
4 Duality criteria
Many authors have formulated MondWeir type dual and established duality results in various optimization problems with support functions; see [10,15,17,20,21,25] and the references therein. Following the above mentioned works, we formulate MondWeir type dual for nonsmooth semiinfinite Econvex programming problem with support function (P) and establish duality theorems.
We now discuss the weak, strong and strict converse duality for (P) and (D).
Theorem 3 (Weak Duality) Let x be feasible for (P) and (y, τ, λ, z _{1}, … , z _{ p }) be feasible for (D). If τ _{j } (f _{j } (.) + ⟨ z _{j }, . ⟩) for j = 1, 2, ..., p are pseudo Econvex at y and λ _{i } g _{i } (.), i ∈ I are quasi Econvex at y with respect to the same E and f _{j } (Ex) ≤ f _{j } (x), j = 1, … , p, ∀ x ∈ M. Then the following cannot hold:
Proof: Let x be feasible for (P) and (y, τ, λ, z _{1}, … , z _{ p }) be feasible for (D), then from (8), there exist ξ _{ j } ∈ ∂ cf _{ j }(Ey) and ζ _{ i } ∈ ∂ cg _{ i }(Ey) such that(10)
We proceed to the result of the theorem by contradiction. Assume that
But f _{ j } (Ex) ≤ f _{ j }(x) and τ _{ j } ≥ 0, for j = 1, 2, ..., p, so we have(11)and by using the inequality ⟨z, Ex ⟩ ≤ S(xC), we get(12)
Now, since Ex is feasible for (P) where M is Econvex set and (y, τ, λ, z _{1}, … , z _{ p }) is feasible for (D), we haveand from definition of quasi Econvexity of g _{ i }(x), i ∈ I at y, we have(13)for each x ∈ M and every and ζ _{ i } ∈ ∂ cg _{ i } (Ex). From (10) in (13), we getfor each x ∈ M and some ξ _{ j } ∈ ∂ cf _{ j } (Ey). Thus, from the definition of pseudo Econvexity of τ _{ j } (f _{ j }(.) + ⟨ z _{ j } , .⟩) for j = 1, 2, ..., p, we havewhich contradicts (12). Hence,cannot hold.
The following corollary is a direct consequence of Remark 1 and Theorem 3.
Corollary 2 Let x be feasible for (P) and (y, τ, λ, z _{1}, … , z _{p } ) be feasible for (D). If τ _{j } (f _{j } (.) + ⟨ z _{j }, . ⟩) for j = 1, 2, ..., p are Econvex at y and λ _{i } g _{i } (.), i ∈ I are Econvex at y with respect to the same E and f _{j } (Ex) ≤ f _{j } (x), j = 1,… , p, ∀ x ∈ M. Then the following cannot hold:
The following example shows that the generalized Binvexity imposed in the above theorem is essential.
Example 4 [26] We consider the following problem:
Subject to where f_{1}(x) = − 2x, f_{2}(x) = x^{2}, S(x C_{1}) = S(x C_{2}) = x for C _{1} = C_{2} = [ − 1, 1] and g_{i}(x) = − i x , for i ∈ I : = N. It is clear that the feasible set of (P) is M : = R and for y = 1 ∈ M, I(y) = I. Let us formulate MondWeir dual of (P) as follow:
Subject to where with λ = (λ _{ i })_{ i∈I } ≠ 0 for finitely many indices i ∈ N and z _{ j } ∈ C _{ j } for j = 1, 2. By choosing We have (y, τ, λ, z _{1}, z _{2}) be feasible for (D) .
Note that λ _{i }g _{i } (.) is not quasi Econvex at y with respect to E(y) = y and that f_{1}(x) + S(x C _{1}) = − 1 < f _{1}(Ey) + ⟨ z _{1}, Ey ⟩ = 0 holds. This means that pseudo Econvexity and quasi Econvexity assumptions are essential for weak duality.
The following theorem gives strong duality relation between the primal problem (P) and the dual problem (D).
Theorem 4 (Strong Duality) Let E : R ^{ n } → R ^{ n } and x ^{*} be a feasible solution of (P). Assume that Ex ^{*} be a weakly efficient solution of (P) and a suitable constraints qualification from Definition (11) holds at x ^{*} and cone (G(x ^{*})) is closed. If the pseudo Econvexity and quasi Econvexity assumptions of the weak duality theorem are satisfied, and f_{j}(Ex) ≤ f _{ j }(x), j = 1, … , p, ∀ x ∈ M. Then there exists (τ, λ, z _{1}, … , z _{ p } ) such that (x ^{*}, τ, λ, z _{1}, … , z _{p } ) is a weakly efficient solution for (D) and the respective objective values are equal.
Proof: Since Ex ^{*} is a weakly efficient solution for (P) at which the suitable constraints qualification holds and cone(G(x ^{*})) is closed, from the KuhnTucker necessary conditions, there exists (τ, λ, z _{1}, … , z _{ p }) such that (x ^{*}, τ, λ, z _{1}, … , z _{ p }) is feasible for (D).
From weak duality theorem (3), the following cannot hold:
Since ⟨z, Ex ⟩ ≤ S(x C), and f _{ j } (Ex) ≤ f _{ j } (x), j = 1, … , p, we havecannot hold, and hence (x ^{*}, τ, λ, z _{1}, … , z _{ p }) is a weakly efficient solution for (D) and the objective values of (P) and (D) are equal at x.
The following corollary is a direct consequence of Remark 1 and Theorem 4.
Corollary 3 Let E : R ^{ n } → R ^{ n } and x^{*} be a feasible solution of (P). Assume that Ex^{*} be a weakly efficient solution of (P) and a suitable constraints qualification from Definition (11) holds at x^{*} and cone(G(x ^{*})) is closed. If the Econvexity assumptions of the weak duality theorem are satisfied, and f_{j}(Ex) ≤ f_{j}(x), j = 1, … , p, ∀ x ∈ M. Then there exists (τ, λ, z _{1}, … , z _{ p }) such that (x ^{*}, τ, λ, z _{1}, … , z _{ p}) is a weakly efficient solution for (D) and the respective objective values are equal.
The following theorem gives strict converse duality relation between the primal problem (P) and the dual problem (D).
Theorem 5 (Strict converse duality) Let Ex ^{*} be a weakly efficient solution for (P) at which a suitable constraints qualification from Definition 11 holds at x ^{*} and cone(G(x ^{*})) is closed.
Let τ_{j}(f _{ j }(.) + ⟨ z _{ j }, . ⟩) for j = 1, 2, ..., p be pseudo Econvex and λ _{i}g_{i}(.), i ∈ I be quasi Econvex with respect to the same E. If is a weak efficient solution for (D) and τ_{j}(f_{j}(.) + ⟨ z _{ j }, . ⟩) for j = 1, 2, ..., p are strictly pseudo Econvex at , then
Proof: We prove the result of theorem by contradiction. Assume that Then by strong duality Theorem (4) there exists (τ, λ, z _{1}, … , z _{ p }) such that (Ex ^{*}, τ, λ, z _{1}, … , z _{ p }) is a weakly efficient solution for (P) and the inequalitycannot be hold. i.e.(14)Now, since Ex ^{*} is a weakly efficient solution for (P), λ _{ i } ≥ 0 and is a weakly efficient solution for (D), we haveand from the definition of quasi Econvexity of λ _{ i } g _{ i }(.), i ∈ I (15)for every x ^{*} ∈ M and every By substituting from (10) in (15), we getfor each x ^{*} ∈ M and some Thus from strict pseudo Econvexity of τ _{ j }(f _{ j }(.) + ⟨ z _{ j }, . ⟩) for j = 1, 2, ..., p at , we get(16)which contradicts (1). Therefore, .
Some Applications
Let us briefly review a few interesting applications. Nonsmooth semiinfinite multiobjective programming problems very naturally lend to a highly disaggregated formulation. Computation of economic equilibria is a very promising area of application for nonsmooth semiinfinite multiobjective programming problems. A paper [27] give example evidence of the solvingpower of ACCPM (analytical center cutting plane method) on these reputedly difficult problems. At the end, we would like to mention applications to nonsmooth semiinfinite multiobjective programming problems. In the first application [28], ACCPM (analytical center cutting plane method) is used to solve a Lagrangian relaxation of the capacitate multiitem lot sizing problem with setup times. A full integration of ACCPM in a column generation, or Lagrangian relaxation, framework, for structured nonsmooth semiinfinite multiobjective programming problems, shows that the reliability and robustness of ACCPM in applications where a nondifferentiable problem must be solved repeatedly makes it a very powerful alternative to sub gradient optimization [29].
5 Conclusions
This paper investigates the optimality conditions and duality for nonsmooth semiinfinite Econvex multiobjective programming with support functions. The obtained results extended and improved corresponding results of [26,21] to nonsmooth Econvex case. By applying the obtained results, one can study fractional programming, setvalued optimization and variational inequalities and so on. For instance, we can apply the obtained KuhnTucker necessary and sufficient conditions to study the optimality conditions and duality for nonsmooth multiobjective programming problems with generalized Econvexity. We can also apply the KuhnTucker sufficient conditions to consider the solvability of some vector variational inequalities.
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Cite this article as: Tarek Emam, Optimality and duality for nonsmooth semiinfinite econvex multiobjective programming with support functions, Int. J. Simul. Multidisci. Des. Optim. 11, 17 (2020)
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