The Karush-Kuhn-Tucker optimality conditions for multi-objective programming problems with fuzzy-valued objective functions

The KKT optimality conditions for multiobjective programming problems with fuzzy-valued objective functions are derived in this paper. The solution concepts are proposed by defining an ordering relation on the class of all fuzzy numbers. Owing to this ordering relation being a partial ordering, the solution concepts proposed in this paper will follow from the similar solution concept, called Pareto optimal solution, in the conventional multiobjective programming problems. In order to consider the differentiation of fuzzy-valued function, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the KKT optimality conditions are elicited naturally by introducing the Lagrange function multipliers.     

The Karush-Kuhn-Tucker optimality conditions for the optimization problem with fuzzy-valued objective function

The Karush-Kuhn-Tucker (KKT) conditions for an optimization problem with fuzzy-valued objective function are derived in this paper. A solution concept of this optimization problem is proposed by considering an ordering relation on the class of all fuzzy numbers. The solution concept proposed in this paper will follow from the similar solution concept, called non-dominated solution, in the multiobjective programming problem. In order to consider the differentiation of a fuzzy-valued function, we use the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the KKT optimality conditions are elicited naturally by introducing the Lagrange function multipliers.     

The Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function

The KKT conditions in an optimization problem with interval-valued objective function are derived in this paper. Two solution concepts of this optimization problem are proposed by considering two partial orderings on the set of all closed intervals. In order to consider the differentiation of an interval-valued function, we invoke the Hausdorff metric to define the distance between two closed intervals and the Hukuhara difference to define the difference of two closed intervals. Under these settings, we derive the KKT optimality conditions.     

The optimality conditions for optimization problems with convex constraints and multiple fuzzy-valued objective functions

The optimality conditions for multiobjective programming problems with fuzzy-valued objective functions are derived in this paper. The solution concepts for these kinds of problems will follow the concept of nondominated solution adopted in the multiobjective programming problems. In order to consider the differentiation of fuzzy-valued functions, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the optimality conditions for obtaining the (strongly, weakly) Pareto optimal solutions are elicited naturally by introducing the Lagrange multipliers.     

Duality theory in fuzzy optimization problems formulated by the Wolfe’s primal and dual pair

The weak and strong duality theorems in fuzzy optimization problem based on the formulation of Wolfe’s primal and dual pair problems are derived in this paper. The solution concepts of primal and dual problems are inspired by the nondominated solution concept employed in multiobjective programming problems, since the ordering among the fuzzy numbers introduced in this paper is a partial ordering. In order to consider the differentiation of a fuzzy-valued function, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the Wolfe’s dual problem can be formulated by considering the gradients of differentiable fuzzy- valued functions. The concept of having no duality gap in weak and strong sense are also introduced, and the strong duality theorems in weak and strong sense are then derived naturally.     

Wolfe Duality for Interval-Valued Optimization

Weak and strong duality theorems in interval-valued optimization problem based on the formulation of the Wolfe primal and dual problems are derived. The solution concepts of the primal and dual problems are based on the concept of nondominated solution employed in vector optimization problems. The concepts of no duality gap in the weak and strong sense are also introduced, and strong duality theorems in the weak and strong sense are then derived.     

Necessary optimality conditions for bilevel set optimization problems

Bilevel programming problems are hierarchical optimization problems where in the upper level problem a function is minimized subject to the graph of the solution set mapping of the lower level problem. In this paper necessary optimality conditions for such problems are derived using the notion of a convexificator by Luc and Jeyakumar. Convexificators are subsets of many other generalized derivatives. Hence, our optimality conditions are stronger than those using e.g., the generalized derivative due to Clarke or Michel-Penot. Using a certain regularity condition Karush-Kuhn-Tucker conditions are obtained.      

Approximate optimality conditions for minimax programming problems

In this article we study non-smooth Lipschitz programming problems with set inclusion and abstract constraints. Our aim is to develop approximate optimality conditions for minimax programming problems in absence of any constraint qualification. The optimality conditions are worked out not exactly at the optimal solution but at some points in a neighbourhood of the optimal solution. For this reason, we call the conditions as approximate optimality conditions. Later we extend the results in terms of the limiting subdifferentials in presence of an appropriate constraint qualification thereby leading to the optimality conditions at the exact optimal point.     

Qualification conditions for multivalued functions in Banach spaces with applications to nonsmooth vector optimization problems

In this paper we introduce qualification conditions for multivalued functions in Banach spaces involving the A-approximate subdifferential, and we show that these conditions guarantee metric regularity of multivalued functions. The results are then applied for deriving Lagrange multipliers of Fritz—John type and Kuhn—Tucker type for infinite non-smooth vector optimization problems.     

First order optimality conditions in vector optimization involving stable functions

We study a nonsmooth vector optimization problem with an arbitrary feasible set or a feasible set defined by a generalized inequality constraint and an equality constraint. We assume that the involved functions are nondifferentiable. First, we provide some calculus rules for the contingent derivative in which the stability (a local Lipschitz property at a point) of the functions plays a crucial role. Second, another calculus rules are established for steady functions. Third, necessary optimality conditions are stated using tangent cones to the feasible set and the contingent derivative of the objective function. Finally, some necessary and sufficient conditions are presented through Lagrange multiplier rules.     

Fuzzy necessary optimality conditions for vector optimization problems

The primary aim of this article is to derive Lagrange multiplier rules for vector optimization problems using a non-convex separation technique and the concept of abstract subdifferential. Furthermore, we present a method of estimation of the norms of such multipliers in very general cases and for many particular subdifferentials.     

First-order necessary optimality conditions for general bilevel programming problems

We formulate in this paper several versions of the necessary conditions for general bilevel programming problems. The technique used is related to standard methods of nonsmooth analysis. We treat separately the following cases: Lipschitz case, differentiable case, and convex case. Many typical examples are given to show the efficiency of theoretical results. In the last section, we formulate the general multilevel programming problem and give necessary conditions of optimality in the general case. We illustrate then the application of these conditions by an example.Lecturer, Département d'Informatique et de Recherche Opérationnelle, Université de Montréal, Montreal, Canada.The author is indebted to Professor M. Florian for support and encouragement in the writing of this paper.     

Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions

In this paper, we first examine how global optimality of non-convex constrained optimization problems is related to Lagrange multiplier conditions. We then establish Lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints. We also obtain necessary global optimality conditions, which are different from the Lagrange multiplier conditions for special classes of quadratic optimization problems. These classes include weighted least squares with ellipsoidal constraints, and quadratic minimization with binary constraints. We discuss examples which demonstrate that our optimality conditions can effectively be used for identifying global minimizers of certain multi-extremal non-convex quadratic optimization problems. The work of Z. Y. Wu was carried out while the author was at the Department of Applied Mathematics, University of New South Wales, Sydney, Australia.     

Optimal control of parabolic problems with state constraints: a penalization method for optimality conditions

In this paper state constrained optimal control problems governed by parabolic evolution equations are studied. Our purpose is to obtain a (first-order) decoupled optimality system (that ensures the Lagrange multipliers existence). In a first step we are led to Slater-like assumptions and we are then allowed to extend the application field of the decoupled system we obtain. With a weaker assumption the existence of Lagrange multipliers (that are measures) for nonqualified problems may be established.     

Sequential optimality conditions for composed convex optimization problems

Using a general approach which provides sequential optimality conditions for a general convex optimization problem, we derive necessary and sufficient optimality conditions for composed convex optimization problems. Further, we give sequential characterizations for a subgradient of the precomposition of a K-increasing lower semicontinuous convex function with a K-convex and K-epi-closed (continuous) function, where K is a nonempty convex cone. We prove that several results from the literature dealing with sequential characterizations of subgradients are obtained as particular cases of our results. We also improve the above mentioned statements.     

On subdifferential calculus for set-valued mappings and optimality conditions

The aim of this paper is to establish formulas for the subdifferentials of the sum and the composition of convex set-valued mappings under the Attouch–Brézis qualification condition. An application to a general set-valued optimization problem is considered.     

Sequential optimality conditions for cardinality-constrained optimization problems with applications

Computational Optimization and Applications - Recently, a new approach to tackle cardinality-constrained optimization problems based on a continuous reformulation of the problem was proposed....     

Bounded sets of Lagrange multipliers for vector optimization problems in infinite dimension

The aim of this paper is to point out some sufficient constraint qualification conditions ensuring the boundedness of a set of Lagrange multipliers for vectorial optimization problems in infinite dimension. In some (smooth) cases these conditions turn out to be necessary for the existence of multipliers as well.     

The KKT optimality conditions in a class of generalized convex optimization problems with an interval-valued objective function

In this paper, we study the Karush–Kuhn–Tucker optimality conditions in a class of nonconvex optimization problems with an interval-valued objective function. Firstly, the concepts of preinvexity and invexity are extended to interval-valued functions. Secondly, several properties of interval-valued preinvex and invex functions are investigated. Thirdly, the KKT optimality conditions are derived for LU-preinvex and invex optimization problems with an interval-valued objective function under the conditions of weakly continuous differentiablity and Hukuhara differentiablity. Finally, the relationships between a class of variational-like inequalities and the interval-valued optimization problems are established.     

Sufficient conditions for the existence of multipliers and Lagrangian duality in abstract optimization problems

We consider the following optimization problem: in an abstract setX, find and elementx that minimizes a real functionf subject to the constraintsg(x)0 andh(x)=0, whereg andh are functions fromX into normed vector spaces. Assumptions concerning an overall convex structure for the problem in the image space, the existence of interior points in certain sets, and the normality of the constraints are formulated. A theorem of the alternative is proved for systems of equalities and inequalities, and an intrinsic multiplier rule and a Lagrangian saddle-point theorem (strong duality theorem) are obtained as consequences.