© T. Emam, published by EDP Sciences, 2020

1 Introduction

Semi-infinite multi-objective programming consider several conflicting objective functions have to be optimized over a feasible set described by infinite number of inequality constraints. Semi-infinite 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 semi-infinite programming problems have been studied see, [2–10]. Optimality and duality results for semi-infinite multi-objective 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 semi-infinite multi-objective programming problems. Mishra et al. [13] proved necessary and sufficient optimality conditions for nondifferential semi-infinite 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 Mond-Weir 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 E-convexity 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 E-convexity can be occurred when the binding force f between elements construct a crystal effect by a solution E. In mathematical programming, the notion of E-convexity 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 multi-objective semi-infinite E-convex programming problems involving support functions are established. In Section 4, we formulate Mond-Weir type dual for multi-objective semi-infinite E-convex programming problems involving support functions and establish weak, strong and strict-converse duality theorems under generalized E-convexity 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 n-dimensional 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 E-convex 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 E-convex 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 E-convex sets, then M _{1   ⋂} M ₂ is E-convex set but M _{1   ⋃} M ₂ is not necessarily E-convex set. If E : R ⁿ  → R ⁿ is a linear map, and M ₁, M ₂ ⊆ R ⁿ are E-convex sets, then M ₁ + M ₂ is E-convex set.

Example 1 Let E : R ² → R ² be defined as E(x, y) = (0, y). The set is an E-convex set with respect to the operator E.

Definition 6 A locally Lipschitz function f : R ⁿ  → R is said to be E-convex 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 E-convex near x ^* ∈ R ⁿ if it is E-convex at each point of neighborhood of x ^* ∈ R ⁿ .

Definition 7 A locally Lipschitz function f : R ⁿ  → R is said to be strictly E-convex 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 E-convex near x ^* ∈ R ⁿ if it is strictly E-convex at each point of neighborhood of x ^* ∈ R ⁿ .

Proposition 1 [21] If g _i  : R ⁿ  → R, i = 1, 2, ..., m is E-convex with respect to E : R ⁿ  → R ⁿ then the set M ={x ∈ R ⁿ  : ⁡ g _i (x ) ≤ 0, i = 1, 2, ..., m } is E-convex set.

Definition 8 A locally Lipschitz function f : R ⁿ  → R is said to be pseudo E-convex 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 E-convex 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 E-convex 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 E-convex near x ^* ∈ R ⁿ if it is quasi E-convex at each point of neighborhood of x ^* ∈ R ⁿ .

Proposition 2 [21] If g _i  : R ⁿ  → R, i = 1, 2, ..., m is quasi E-convex with respect to E : R ⁿ  → R ⁿ then the set M ={x ∈ R ⁿ  : ⁡ g _i (x ) ≤ 0, i = 1, 2, ..., m} is E-convex set.

Remark 1

• Every E-convex function is also quasi E-convex with respect to same E : R ⁿ  → R ⁿ , but not conversely.

• Every E-convex function is also pseudo E-convex with respect to same E : R ⁿ  → R ⁿ , but not conversely.

• Every strictly E-convex function is also strictly pseudo E-convex with respect to same E : R ⁿ  → R ⁿ , but not conversely.

Let C be a nonempty compact E-convex 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 semi-infinite multi-objective E-convex 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 E-convex functions from R ⁿ to R∪ { +∞ }. Let M denote the E-convex 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 multi-objective E-convex 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 semi-infinite multi-objective E-convex 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 E-convex at x ^* and λ _i g _i (.), i ∈ I(x ^*) are quasi E-convex 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(x|C _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 E-convex set and λ _i g _i (Ex ^*) = 0, i ∈ I(x ^*), we have(7)and from quasi E-convexity of g _i , i ∈ I(x ^*), we getby using (6), we have

Thus, from pseudo E-convexity 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 E-convex 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 E-convex 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 E-convex 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 E-convex and λ ₁ g ₁(x) are quasi E-convex 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 Mond-Weir 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 Mond-Weir type dual for nonsmooth semi-infinite E-convex programming problem with support function (P) and establish duality theorems.

subject to(8) (9)

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 E-convex at y and λ _i g _i (.), i ∈ I are quasi E-convex 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(x|C), we get(12)

Now, since Ex is feasible for (P) where M is E-convex set and (y, τ, λ, z ₁, … , z _p ) is feasible for (D), we haveand from definition of quasi E-convexity 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 E-convexity 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 E-convex at y and λ _i g _i (.), i ∈ I are E-convex 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 B-invexity imposed in the above theorem is essential.

Example 4 [26] We consider the following problem:

Subject to where f₁(x) = − 2x, f₂(x) = x², S(x |C₁) = S(x |C₂) = |x |for C ₁ = C₂ = [ − 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 Mond-Weir 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 E-convex at y with respect to E(y) = y and that f₁(x) + S(x |C ₁) = − 1 < f ₁(Ey) + ⟨ z ₁, Ey ⟩ = 0 holds. This means that pseudo E-convexity and quasi E-convexity 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 E-convexity and quasi E-convexity 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 Kuhn-Tucker 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 E-convexity 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 E-convex and λ _ig_i(.), i ∈ I be quasi E-convex 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 E-convex 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 E-convexity 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 E-convexity 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 semi-infinite multi-objective programming problems very naturally lend to a highly disaggregated formulation. Computation of economic equilibria is a very promising area of application for nonsmooth semi-infinite multi-objective programming problems. A paper [27] give example evidence of the solving-power of ACCPM (analytical center cutting plane method) on these reputedly difficult problems. At the end, we would like to mention applications to nonsmooth semi-infinite multi-objective programming problems. In the first application [28], ACCPM (analytical center cutting plane method) is used to solve a Lagrangian relaxation of the capacitate multi-item lot sizing problem with set-up times. A full integration of ACCPM in a column generation, or Lagrangian relaxation, framework, for structured nonsmooth semi-infinite multi-objective programming problems, shows that the reliability and robustness of ACCPM in applications where a non-differentiable 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 semi-infinite E-convex multi-objective programming with support functions. The obtained results extended and improved corresponding results of [26,21] to nonsmooth E-convex case. By applying the obtained results, one can study fractional programming, set-valued optimization and variational inequalities and so on. For instance, we can apply the obtained Kuhn-Tucker necessary and sufficient conditions to study the optimality conditions and duality for nonsmooth multiobjective programming problems with generalized E-convexity. We can also apply the Kuhn-Tucker sufficient conditions to consider the solvability of some vector variational inequalities.

References