Nonsmooth multiobjective optimization involving generalized univex functions

In this paper, nonsmooth univex, nonsmooth quasiunivex, and nonsmooth pseudounivex functions are introduced. By utilizing these new concepts, sufficient optimality conditions for a weakly efficient solution of the nonsmooth multiobjective programming problem are established. Weak and strong duality theorems are also derived for Mond-Weir type multiobjective dual programs.

In this paper, a class of nonsmooth multiobjective programming problems is considered. We introduce the new concept of invex of order type II for nondifferentiable locally Lipschitz functions using the tools of Clarke subdifferential. The new functions are used to derive the sufficient optimality condition for a class of nonsmooth multiobjective programming problems. Utilizing the sufficient optimality conditions, weak and strong duality theorems are established for Wolfe type duality model.

In this paper, we have considered a nonsmooth multiobjective optimization problem where the objective and constraint functions involved are directionally differentiable. A new class of generalized functions (d − ρ − η − θ)-type I univex is introduced which generalizes many earlier classes cited in literature. Based upon these generalized functions, we have derived weak, strong, converse and strict converse duality theorems for mixed type multiobjective dual program in order to relate the efficient and weak efficient solutions of primal and dual problem.

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.

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.

Duality for nondifferentiable minimax programming in complex spaces

Optimality and Duality for Nonsmooth Multiobjective Fractional Programming with Generalized Invexity

In this paper, a new generalised class of higher order (F, α, ρ, d)-V-type I function is introduced for a non-smooth multi-objective fractional programming problem involving support functions. The newly defined class extends several known classes in the literature has been justified through a non-trivial example. In the framework of new concept, we determine conditions under which a fractional function becomes higher order (F, α, ρ, d)-V-type I function and do some computational work to substantiate the analysis. Further, we establish Karush-Kuhn-Tucker type sufficient optimality conditions and derive various duality results for higher order Mond-Weir type and Schaible type dual programs.

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.

This paper considers the following nonsmooth multiobjective programming problem: (P) weak min f(x)=(f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> (x), hellip, f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (x)), s.t. -g(x)isinS, h(x)=0, where xisinR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , f(x)isinR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> , g(x)isinR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> , h(x)isinR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sup> , SsubR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> is a closed convex cone with interior, and f, g, h satisfy the local Lipschitz condition. The necessary Lagrangian condition is established for a weak minimum of (P), and the proof is different from the proofs given by B.D.Craven, Ying Meiqian and other authors. Necessary conditions of Kuhn-Tucker type are obtained and sufficient conditions are discussed.

We consider non-smooth multiobjective programming problems with inequality and equality constraints involving locally Lipschitz functions. Several optimality conditions under various (generalized) V-invexity assumptions and certain regularity conditions are presented. Further, we introduce a Mond–Weir type dual and establish duality relations under the aforesaid assumptions.

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