Generalized Univex Functions in Nonsmooth Multiobjective Optimization

In this paper, we have considered a class of constrained nonsmooth multi objective programming problem involving semi-directionally differentiable functions from a view point of generalized convexity. A new generalized class of (dI—ρ—σ)-V-type I univex functions is introduced under which various weak, strong, converse and strict converse duality theorems are established for Mond-Weir type dual program in order to relate the efficient and weak efficient solutions of primal and dual problem. Also, we have illustrated through various non-trivial examples that this class extends many earlier studied classes in literature.

Using the idea of upper convexificators, we propose constraint qualifications and study existence and boundedness of the Kuhn-Tucker multipliers for a nonsmooth multiobjective optimization problem with inequality constraints and an arbitrary set constraint. We show that, at locally weak efficient solutions where the objective and constraint functions are locally Lipschitz, the constraint qualifications are necessary and sufficient conditions for the Kuhn-Tucker multiplier sets to be nonempty and bounded under certain semiregularity assumptions on the upper convexificators of the functions.

This paper is devoted to study interval-valued optimization problems. Sufficient optimality conditions are established for LU optimal solution concept under generalized -invexity. Weak, strong and strict converse duality theorems for Wolfe and Mond-Weir type duals are derived in order to relate the LU optimal solutions of primal and dual problems. MSC: 90C46, 90C26, 90C30.

In this paper, we investigate a robust nonsmooth multiobjective optimization problem related to a multiobjective optimization with data uncertainty. We firstly introduce two kinds of generalized convex functions, which are not necessary to be convex. Robust necessary optimality conditions for weakly robust efficient solutions and properly robust efficient solutions of the problem are established by a generalized alternative theorem and the robust constraint qualification. Further, robust sufficient optimality conditions for weakly robust efficient solutions and properly robust efficient solutions of the problem are also derived. The Mond–Weir-type dual problem and Wolfe-type dual problem are formulated. Finally, we obtain the weak, strong and converse robust duality results between the primal one and its dual problems under the generalized convexity assumptions.

Nonlinear optimization problems have two different representations, the primal problem and the dual problem. The relation between the primal and the dual problem is provided by an elegant duality theory. This chapter presents the basics of duality theory. Section 4.1 discusses the primal problem and the perturbation function. Section 4.2 presents the dual problem. Section 4.3 discusses the weak and strong duality theorems, while section 4.4 discusses the duality gap. This section presents the formulation of the primal problem, the definition and properties of the perturbation function, the definition of stable primal problem, and the existence conditions of optimal multiplier vectors.

Generalized polyhedral convex optimization problems in locally convex Hausdorff topological vector spaces are studied systematically in this paper. We establish solution existence theorems, necessary and sufficient optimality conditions, weak and strong duality theorems. In particular, we show that the dual problem has the same structure as the primal problem, and the strong duality relation holds under three different sets of conditions.

In this paper, we consider a class of multiobjective E-convex programming problems with inequality constraints, where the objective and constraint functions are E-convex functions which were firstly introduced by Youness (J. Optim. Theory Appl. 102:439-450, 1999). Fritz-John and Kuhn-Tucker necessary and sufficient optimality theorems for the multiobjective E-convex programming are established under the weakened assumption of the theorems in Megahed et al. (J. Inequal. Appl. 2013:246, 2013) and Youness (Chaos Solitons Fractals 12:1737-1745, 2001). A mixed duality for the primal problem is formulated and weak and strong duality theorems between primal and dual problems are explored. Illustrative examples are given to explain the obtained results.

Duality for nondifferentiable minimax programming in complex spaces

ABSTRACTIn this paper, a second-order dual problem for a non-differentiable minimax complex programming problem is formulated. The weak, strong, and strict converse duality theorems are constructed under generalized Θ-bonvexity assumptions. This means that there are no duality gaps between primary problem and its dual problem.

In this paper, a pair of non-differentiable second-order multi-objective symmetrical dual fractional variational programs over arbitrary cones is constructed. We have introduced the concept of second-order (F, α, ρ, d)-pseudoconvexity and an example of a function which is second-order (F, α, ρ, d)-pseudoconvex but not second-order F-pseudoconvex has been constructed. Further, to relate the objective function value of primal and dual, we proved weak, strong, and converse duality theorems under second-order (F, α, ρ, d)-pseudoconvexity suppositions. Moreover, to validate the results of the weak duality theorem, an example of a multi-objective variational fractional problem and its dual model has been constructed. Several known results are also discussed as special cases.

This paper deals with non-differentiable multi-objective semi-infinite programming problem. It is a problem of simultaneous minimisation of finitely many scalar valued functions subject to an arbitrary (possibly infinite) set of constraints. Non-differentiability enters, due to the square root of a quadratic form which appears in the objective functional. Concept of efficiency of order m has been extended to the above stated problem. In order to study this new solution concept, the notion of ρ-invexity of order m is also proposed which is utilised to establish sufficient optimality conditions for the non-differentiable multi-objective semi-infinite programming problem. Mond-Weir type of dual is proposed for which weak, strong and strict converse duality theorems are established.

Higher order symmetric duality in nondifferentiable multi-objective programming problems involving generalized cone convex functions

A new notion of higher-order weakly generalized adjacent epiderivative for a set-valued map is introduced. By virtue of the epiderivative and weak minimality, a higher-order Mond-Weir type dual problem and a higher-order Wolfe type dual problem are introduced for a constrained set-valued optimization problem, respectively. Then, corresponding weak duality, strong duality, and converse duality theorems are established.

In this work, we introduce a pair of multiobjective second-order symmetric dual variational problems. Weak, strong, and converse duality theorems for this pair are established under the assumption of ?-bonvexity/?-pseudobonvexity. At the end, the static case of our problems has also been discussed.

On symmetric gH-derivative: Applications to dual interval-valued optimization problems