Stronger Kuhn-Tucker Conditions In Multiobjective Optimization

For a multiobjective optimization problem, stronger K-T conditions, unlikeK-T conditions, are necessary optimality conditions where all Lagrange multipliers associated with the components the objective function are positive.For multiobjective optimization problem, stronger K-T conditions play an important role both in theory and computational methods. Therefore, it's important to derive stronger K-T conditions. In 1994, Maeda [1] first establisheda stronger K-T condition for a inequality constrained multiobjective optimization problem where the objective and constraint functions are continuouslydifferentiate. Later, many researchers studied stronger K-T conditions for nonsmooth vector optimization problem and extended Maeda's results.This is a survey of obtained results of stronger K-T necessary optimality conditions for multiobjective optimization problem. We present stronger K-T type necessary conditions for efficiency for the problem in which the functions are involved are, respectively, continuously differentiable, semidifferentiable, and locally Lipschitz, etc. We also present the constraint qualifications which ensure these stronger K-T conditions.We consider the following vector minimization problem:(P) min f(x)s.t. g(x)(?)0,where f:Rⁿ→R^p,g:Rⁿ→R^m.We denoted byX = {x∈Rⁿ| g(x)(?) 0}the feasible set of problem (P).1. In Section 3.1, we present results of vector optimization problem where the objective and constraint functions are continuously differentiable. For problem (P), under the assumption of continuous differentiability of objective and constraint functions, Maeda proposed a generalized Guignard constraintqualification:where C(Q(x⁰); x⁰)) is called linearizing cone to Q(x⁰) at x⁰, and obtained an important theorem of stronger K-T type necessary conditions as follows:Theorem 1 Let x⁰∈X be any feasible solution to problem (P), and suppose that (GGCQ) holds at x⁰∈X. In order that x⁰∈X be an efficient solution to problem (P), it is necessary that there exist vectorsλ∈R^p andμ∈R^m such that2. In Section 3.2, we present the stronger K-T optimality conditions of nonsmooth multiobjective optimization problem in which the function's are involved are, respectively, semidifferentiable and locally Lipschitz, etc.2.1. Preda and Chitescu [2] extended the results obtained by Maeda in continuously case of the optimization problem to directionally diffentiable case. In Subsection 3.2.1, we present the following main result they obtained under the analogue of generalized Guignard constraint qualification:Theorem 2 Let x⁰∈X be a feasible solution to problem (P). Suppose that p > 1 and assume that:(1) constraint qualification (GGCQ) holds at x⁰; (2) f_k, i = 1,2,…,p and k≠i, and g_j, j∈I(x⁰) are quasiconvex functions at x⁰, f_i quasiconcave at x⁰;(3) f_i⁺(x⁰,·) is a concave and convex function on Rⁿ;(4) f_k⁺(x⁰,·), k = 1,2,…,p and k≠i, and g_j⁺(x⁰,·), j∈I(x⁰), are conxex functions on Rⁿ.Then, there exist vectorsλ∈R^p andμ∈R^m such that, for any d∈Rⁿ,2.2. In§.2.2, we present Li's results [4] of stronger K-T optimality conditions in terms of Clarke subdifferential under the assumption that the objective and the constraint functions of the optimization problem are locally Lipschitz. The following theorem are the main result:Theorem 3 Let x⁰ be a efficient solution of problem (P), and suppose that for all i = 1, 2,…,p and j∈I(x⁰), the Clarke subdifferentials dfi(x⁰) and (?)_gj(x⁰) are polytopes. If either constraint qualification (CQ2) or (CQ2') holds at x⁰, then stronger K-T condition (SKTC) holds at x⁰.2.3. For problem (P), Li [3] propose a generalized Guignard constrained qualification in terms of Dini directional derivative, and obtained a stronger K-T necessary optimality conditions in terms of upper convexificator. In Subsection3.2.3, we present these results. The following is the main of them:Theorem 4 Let x⁰∈X be a solution of problem (P). Suppose that, at (1) f₁ and g_j admit, respectively, upper convexificator (?)^*f_i(x⁰) and (?)^*g_j(x⁰), i =1,2,…,p and j∈I(x⁰).(2) f_i is directionally differentiable, i = 1,2,…,p, with f'_i0(x⁰;.) beinglinear for some i₀∈{1,2,…,p}, and f'_k(x⁰;.) is sublinear for all k = 1, 2,…,p and k≠i₀.(3) g_j^-(x⁰; .) is sublinear, j∈I(x⁰).If (GGCQ) holds at x⁰, then there exist real numbersλ_i > 0, i = 1,2,…,p, andμ_j≥0, j∈I(x⁰), such that2.4. For problem (P), Bigi and Pappalardo Ref [5] proposed a very generalMangasarian-Fromovitz constraint qualification. In Subsection 3.2.3, we presented the constraint qualification and the stonger K-T necessary conditionsresults obtained in [5] for efficiency and weak efficiency in terms of Dini directional derivatives.Theorem 5 Let the function g_j be upper semicontinuous at x⁰ for each j(?) I(x⁰), and let f_i^-(x⁰,·-), i = 1,2,…,p, and g_j^-(x⁰,·), j∈I(x⁰) be con-vexlike. Suppose that (3.6) holds at x⁰. If x⁰ is a weak efficient solution of problem (P), then existλ_i > 0, i = 1,2,…,p, andμ_j≥0, j∈I(x⁰), such thatMoreover, for all Lagrange multiplier (λ,μ), we haveλ_i > 0, (?)i = 1,2,…,p.