Necessary optimality conditions in nonsmooth semi-infinite multiobjective optimization under metric subregularity

Use your smartphone to scan this QR code and download this article ABSTRACT We consider nonsmooth semi-infinite multiobjective optimization problems under mixed constraints, including infinitely many mixed constraints by using Clarke subdifferential. Semi-infinite programming (SIP) is the minimization of many scalar objective functions subject to a possibly infinite systemof inequality or/and equality constraints. SIPs have been proved to be very important in optimization and applications. Semi-infinite programming problems arise in various fields of engineering such as control systems design, decision-making under competition, and multi-objective optimization. There is extensive literature on standard semi-infinite programming problems. The investigation of optimality conditions for these problems is always one of themost attractive topics and has been studied extensively in the literature. In our work, we study optimality conditions for weak efficiency of amulti-objective semi-infinite optimization problemundermixed constraints including infinitelymanyof both equality and inequality constraints in termsof Clarke sub-differential. Our conditions are the form of the Karush-Kuhn-Tucker (KKT) multiplier. To the best of our knowledge, only a few papers are dealingwith optimality conditions for SIPs subject tomixed constraints. By the Pshenichnyi-Levin-Valadire (PLV) property and the directionalmetric sub-regularity, we introduce a type of Mangasarian-Fromovitz constraint qualification (MFCQ). Then we show that (MFCQ) is a sufficient condition to guarantee the extended Abadie constraint qualification (ACQ) to satisfy. In our constraint qualifications, all functions are nonsmooth and the number of constraints is not necessarily finite. In our paper, we do not need the involved functions: convexity and differentiability. Later, we apply the extended Abadie constraint qualification to get the KKT multipliers for weak efficient solutions of SIP. Many examples are provided to illustrate some advantages of our results. The paper is organized as follows. In Section Preliminaries, we present our basic definitions of nonsmooth and convex analysis. Section Main Results prove necessary conditions for the weakly efficient solution in terms of the Karush-Kuhn-Tucker multiplier rule with the help of some constraint qualifications.


INTRODUCTION
Semi-infinite optimization (SIP) is the simultaneous minimization of finitely many scalar objective functions under an arbitrary set of inequality constraints or/and equality constraints. (SIPs) arise in many fields of applied mathematics such as robotics, control system design, etc, see for instance [1][2][3] . Investigation of optimality conditions for SIPs has been considered extensively in the literature. With linear semi-infinite systems, Goberna 4 introduced he Farkas-Minkowski property, Puenten and Vera 5 proposed the local Farkas-Minkowski property and used it as a constraint qualification to get Lagrange multipliers. For convexsemi-infinite optimization, many constraint qualifications have been studied in Lopez and Vercher 6 . With the help of the Abadie constraint qualification, optimality conditions for semi-infinite systems of convex and linear inequalities were developed in Li 7 . For smooth problems, Stein 8 proposed the Abadie and Mangasarian-Fromovitz constraint qualifications to conisder optimality conditions. By employing variational analysis, Mordukhovich and Nghia 9 obtained necessary conditions under the extended perturbed Mangasarian-Fromovitz and Farkas--Minkowski constraint qualification. For nonsmooth problem with inequality constraints, Zheng and Yang 10 employed the directional derivative to obtain Lagrange multiplier rules. Kanzi and Nobakhtian 11,12 introduced several nonsmooth analogues of the Abadie constraint qualification and the Pshenichnyi-Levin-Valadire property and applied them to obtain optimality conditions. Chuong 13 proposed the limiting constraint qualification in terms of the Mordukhovich subdifferential and applied it to optimality conditions. Kanzi 14 investigated nonsmooth semi-infinite problems with mixed constraints by the Michel-Penot subdifferential. We observe that the constraints in the above mentioned papers contain finitely many equalities. There are very few publications dealing with infinitely many equality constraints. In this paper we investigate opyimality conditions for weak efficiency of a multiobjective semi-infinite optimization problem under mixed constraints including infinitely many of both equality and inequality constraints in terms of Clarke subdifferential. By the Pshenichnyi-Levin-Valadire (PLV) property and the directional metric subregularity, we propose Mangasarian-Fromovitz constraint qualification (MFCQ) to guarantee the extend Abadie constraint qualification (ACQ) to satisfy. In our constraint qualifications, all functions are nonsmooth and the number of the equality constraints is not necessary finite. Then, we apply them to get the KKT multipliers. The paper is organized as follows. In Section Prelininaries, we present our basic definitions. Section Main results prove necessary conditions for weak efficiency in terms of Karush-Kuhn-Tucker multiplier under some constraint qualification.

RELIMINARIES
N, R n and R n + stand for the set of the natural numbers, a n-dimensional vector space and its nonnegative orthant, respectively (resp).
there is a neighborhood U of x 0 and a real number The following properties will be useful in the sequel ( 15 ).
is finite, positivel homogeneous and subadditive on R n , and ∂ ( , where ∂ denotes the subdifferential in the sense of convex analysis. (ii) ∂ C f (x 0 ) is a nonempty, convex and compact subset of R n and, for every . If in addition both f and g are regular at x 0 , then the equality holds.
Besides single-valued directional derivatives, we need the following set-valued directional derivatives.

Definition 2.2
The Hadamard set-valued directional derivative of f : which contradicts the assumption. □ The following example present that the sufficient condition given in Proposition 2.2 is not necessary.

MAIN RESULTS
We investigate the fmultiobjective semi-infinite optimization problem under mixed constraints: The index sets I and J are arbitrary. The feasible set of problem (P) is Definition 3.1 For the problem (P) and x 0 ∈ Ω. x 0 is called a local weak efficient solution of (P), written as If I is finite and g i are locally Lipschitz around x 0 for i ∈ I , obviously the problem (P) has the Pshenichnyi-Levin-Valadire (PLV) property at x 0 ∈ Ω wrt G. Sufficient conditions for G to be locally Lipschitz were considered 17 .
. Theorem 3.1 If (P) has the (PLV) property at x 0 ∈ Ω wrt G and the (MFCQ) satisfies at x 0 , then the (ACQ) satisfies at x 0 .

Proof.
By the (MFCQ), there is < 0, which implies there are β and ε such that , there exist t n → 0 + , d n → _ d, and z n → 0 such that t n z n ∈ H (x 0 + t n d n ). The metric subregularity of H gives a ≥ 0 such that, for large n, Hence, there exist _ d n and ε with t −1 n ε n → 0 + such that From (1), one has , for large n, This implies that g i ≤ 0 for all i ∈ I . By combining this and (2), one has Since h j is regular at x 0 and d ∈ L (Ω, x 0 ), one gets for all j ∈ J, h

Science & Technology Development Journal -Engineering and Technology, 3(SI3):SI52-SI57
Then, there exists t n → 0 + such that lim n→∞ (3) and (4), similar to the above arguments, one has d n ∈ T (Ω, x 0 ). As d n → d and is a closed cone, d ∈ T (Ω, x 0 ). The proof is complete. □ Remark 3.1 Nonsmooth SIPs involving mixed constraints 9,14 , the (MFCQ) was used to consider a number of equality constraints. In these paper, the functions were continuously differentiable with the linearly independent gradients such that ⟨∇ f j (x 0 ) , _ d⟩ = 0 for j ∈ J. The inequality constraints were continuously differentiable and the equalities werestrictly differentiable. By employing directional metric subregularity, out (MFCQ) can be used to nonsmooth infinite mixed constraint systems and the condition 0 ∈ DH can be applied in many cases.. The next example provides a case where Theorem 3.1 can be employed, while many Mangasarian-Fromovitz-type constraint qualifications cannot.
. Thus, (P) has the (PLV) property at x 0 wrt G. Now, we check that the (MFCQ) is fulfilled at d 2 ) ∈ X and so DH (x 0 , ·) is concave, and 0 ∈ DH(x 0 , _ d). Therefore, the (MFCQ) holds at x 0 . By Theorem 3.1, the (ACQ) holds at x 0 . (We can check the (ACQ) by direct calculations as follows.
The following example shows the essentialness of the directional metric subregularity of H. Example 3.2 Let g i be the same as in Example 3.1 and , j ∈ (0, 1). 2} for x 0 = (0, 0). Similar to Example 3.1, (P) has the (PLV) property at x 0 wrt G. We check that the (MFCQ) holds at x 0 for d 2 ) ∈ X and so DH (x 0 , ·) is concave, and 0 ∈ DH .
On the other hand, as g 0 . Then, the subregularity means that for any a, r > 0, t ∈ (0, r), and But, this does not hold. Now, by employ the extend ACQ, we present a necessary optimality condition for weak efficiency of problem (P), as follows. Theorem 3.2 Let x 0 be a local weak efficiency of (P). If the (ACQ) holds at x 0 , △ is closed, and f k is regular and Lipschitz around x 0 , for k = 1, ...m, then there exist Step 1. We claim that the system , one has y ∈ −intR m + . As d ∈ T (Ω, x 0 ), there exist t n → 0 + and d n → d such that x 0 + t n d n ∈ Ω for all n ∈ N. Since f k is regular and locally Lipschitz at x 0 , one has lim n→∞ Hence, lim n→∞

Science & Technology Development Journal -Engineering and Technology, 3(SI3):SI52-SI57
As y ∈ −intR m + , for large n, one has f (x 0 + t n d n ) − f (x 0 ) ∈ −intR m + , which is a contradiction. Therefore, the mentioned system has no solution.