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Optimality conditions are established in terms of Lagrange–Fritz–John multipliers as well as Lagrange–Kuhn–Tucker multipliers for set optimization problems (without any convexity assumption) by using new scalarization techniques. Additionally, we indicate how these results may be applied to some particular weak vector equilibrium problems.

This paper is devoted to the investigation of optimality conditions for approximate quasi-weakly efficient solutions to a class of nonsmooth Vector Equilibrium Problem (VEP) via convexificators. First, a necessary optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is presented by making use of the properties of convexificators. Second, the notion of approximate pseudoconvex function in the form of convexificators is introduced, and its existence is verified by a concrete example. Under the introduced generalized convexity assumption, a sufficient optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is also established. Finally, a scalar characterization for approximate quasi-weakly efficient solutions to problem (VEP) is obtained by taking advantage of Tammer’s function.

In this paper, our purpose is to use the improvement set to investigate the scalarization and optimality conditions of E-globally proper efficient solution for the set-valued equilibrium problems with constraints. First, the notion of E-globally proper efficient solution for set-valued equilibrium problems with constraints is introduced in locally convex Hausdorff topological spaces. Second, the linear scalarization theorems of E-globally proper efficient solution are derived. Finally, under the assumption of nearly E-subconvexlikeness, the Kuhn–Tucker and Lagrange optimality conditions for set-valued equilibrium problems with constraints are obtained in the sense of E-globally proper efficiency. Meanwhile, we give some examples to illustrate our results. The results obtained in this paper improve and generalize some known results in the literature.

In this paper, we establish several proper separation theorems for an element and a convex set and for two convex sets in terms of their quasi-relative interiors. Then, we prove that the separation theorem given by [Cammaroto, F and B Di Bella (2007). On a separation theorem involving the quasi-relative. Proceedings of the Edinburgh Mathematical Society, 50(3), 605–610] in Theorem 2.5, is in fact a proper separation theorem for two convex sets in which the classical interior is replaced by the quasi-relative interior. Besides, we extend some known results in the literature, such as [Adán, M and V Novo (2004). Proper efficiency in vector optimization on real linear spaces. Journal of Optimization Theory and Applications, 121, 515–540] in Theorem 2.1 and [Edwards, R (1965). Functional Analysis: Theory and Applications. New York: Reinhart and Winston] in Corollary 2.2.2, through the quasi-relative interior and the quasi-interior, respectively. As an application, we provide Karush–Kuhn–Tucker multipliers for quasi-relative solutions of vector optimization problems. Several examples are given to illustrate the obtained results.

Interval programming is one of main approaches treating imprecise and uncertain elements involved in optimization problems. In this paper, an interval linear fractional bilevel program is considered, which is characterized in that both objective coefficients and the right-hand side vector are interval numbers, and an evolutionary algorithm (EA) is proposed to solve the problem. First, the interval parameter space of the follower’s problem is taken as the search space of the proposed EA. For each individual, one can evaluate it by dealing with a simplified interval bilevel program in which only the leader’s objective involves interval parameters. In addition, the optimality conditions of linear fractional programs are applied to convert and solve the simplified problem. Finally, some computational examples were solved and the results show that the proposed algorithm is efficient and robust.

Certain classes of optimal boundary control problems for the Boussinesq equations with variable density are studied. Controls for the velocity vector and temperature are applied on parts of the boundary of the domain, while Dirichlet and Navier friction boundary conditions for the velocity and Dirichlet and Robin boundary conditions for the temperature are assumed on the remaining parts of the boundary. As a first step, we prove a result on the existence of weak solution of the dynamical equations; this is done by first expressing the fluid density in terms of the stream-function. Then, the boundary optimal control problems are analyzed, and the existence of optimal solutions are proved; their corresponding characterization in terms of the first-order optimality conditions are obtained. Such optimality conditions are rigorously derived by using a penalty argument since the weak solutions are not necessarily unique neither isolated, and so standard methods cannot be applied.

This paper derives first order necessary and sufficient conditions for unconstrained cone d.c. programming problems where the underlined space is partially ordered with respect to a cone. These conditions are given in terms of directional derivatives and subdifferentials of the component functions. Moreover, conjugate duality for cone d.c. optimization is discussed and weak duality theorem is proved in a more general partially ordered linear topological vector space (generalizing the results in [11]).

A distributed control problem for cooperative parabolic systems governed by Schrödinger operator is considered. The performance index is more general than the quadratic one and has an integral form. Constraints on controls are imposed. Making use of the Dubovitskii-Milyutin Theorem given by Walczak (1984, On some control problems Acta Univ. Lod. Folia Math., 1, 187-196), the optimality conditions are derived for the Neumann problem. Finally, several mathematical examples for derived optimality conditions are presented.

In this paper we consider the optimization problem with a multiobjective composed convex function as objective function, namely, being a composite of a convex and componentwise increasing vector function with a convex vector function. By the conjugacy approach, we obtain a dual problem for it. The existence of weak and strong duality is proved.

Using this general result, we introduce the dual problem for the multiobjective location problem in a general normed space, in which the existing facilities are represented by sets of points.

The biobjective Weber-minimax problem, the multiobjective Weber problem and the multiobjective minimax problem with demand sets are studied as particular cases of this problem.

No abstract received.

The aim of this short note is to give a linear programming approach to optimality conditions (expressed through support criteria) in control problems with piecewise deterministic Markov dynamics. Two classes are considered: classical, discounted control problems and asymptotic problems associated to two-scales (perturbed) systems.