On semi-infinite minmax programming with generalized invexity

Here, we consider the minmax programming problem with a set of restrictions indexed in a compact. As a novelty, we obtain optimality criteria of the Kuhn--Tucker type involving a limited number of restrictions and prove both necessity and sufficiency under new weaker invexity assumptions. Also some dual problems are introduced and it is proved that the weak and strong duality properties hold within the same environment.     

Nonsmooth multiobjective programming: strong Kuhn–Tucker conditions

We consider a multiobjective optimization problem with a feasible set defined by inequality and equality constraints and a set constraint, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given in such a way that they generalize the classical ones, when the functions are differentiable. The relationships between them are analyzed. Then, we establish strong Kuhn–Tucker necessary optimality conditions in terms of the Clarke subdifferentials such that the multipliers of the objective function are all positive. Furthermore, sufficient optimality conditions under generalized convexity assumptions are derived. Moreover, the concept of efficiency is used to formulate duality for nonsmooth multiobjective problems. Wolf and Mond–Weir type dual problems are formulated. We also establish the weak and strong duality theorems.     

非凸分式优化问题的最优性充分条件和对偶

通过引入广义弧连通概念,在Rn空间中,研究极大极小非凸分式规划问题的最优性充分条件及其对偶问题.首先获得了极大极小非凸分式规划问题的最优性充分条件;然后建立分式规划问题的一个对偶模型并得到了弱对偶定理,强对偶定理和逆对偶定理.     

Probabilistic constraints in primal and dual linear programs: Duality results

We present two pairs of dually related probabilistic constrained problems as extensions of the linear programming duality concept. In the first pair, a bilinear function appears in the objectives and each objective directly depends on the feasibility set of the other problem, as in the game theoretical formulation of dual linear programs. In the second pair, we reformulate the objectives and eliminate their direct dependence on the feasibility set of the other problem. We develop conditions under which the dually related problems have no duality gap and conditions under which the two pairs of problems are equivalent as far as their optimality sets are concerned.     

Necessary and Sufficient Conditions for Minimax Fractional Programming

We establish the necessary and sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems solving generalized convex functions. Subsequently, we apply the optimality conditions to formulate one parametric dual problem and we prove weak duality, strong duality, and strict converse duality theorems.     

多目标半定规划的最优性条件及对偶理论

在不变凸的假设下来讨论多目标半定规划的最优性条件、对偶理论以及非凸半定规划的最优性条件．首先给出了非凸半定规划的一个KKT条件成立的充分必要条件, 并利用此定理证明了其最优性必要条件．其次讨论了多目标半定规划的最优性必要条件、充分条件, 并对其建立Wolfe对偶模型, 证明了弱对偶定理和强对偶定理．     

OPTIMALITY CONDITIONS AND DUALITY MODELS FOR A CLASS OF NONSMOOTH CONTINUOUS-TIME GENERALIZED FRACTIONAL PROGRAMMING PROBLEMS

Abstract

Both parametric and parameter-free stationary-point-type and saddle-point-type necessary and sufficient optimality conditions are established for a class of nonsmooth continuous-time generalized fractional programming problems with Volterra-type integral inequality and nonnegativity constraints. These optimality criteria are then utilized for constructing ten parametric and parameter-free Wolfe-type and Lagrangian-type dual problems and for proving weak, strong, and strict converse duality theorems. Furthermore, it is briefly pointed out how similar optimality and duality results can be obtained for two important special cases of the main problem containing arbitrary norms and square roots of positive semidefinite quadratic forms. All the results developed here are also applicable to continuous-time programming problems with fractional, discrete max, and conventional objective functions, which are special cases of the main problem studied in this paper.     

Sufficient optimality conditions and duality theory for interval optimization problem

This paper addresses the duality theory of a nonlinear optimization model whose objective function and constraints are interval valued functions. Sufficient optimality conditions are obtained for the existence of an efficient solution. Three type dual problems are introduced. Relations between the primal and different dual problems are derived. These theoretical developments are illustrated through numerical example.     

Nonlinear Lagrangian Theory for Nonconvex Optimization

The Lagrangian function in the conventional theory for solving constrained optimization problems is a linear combination of the cost and constraint functions. Typically, the optimality conditions based on linear Lagrangian theory are either necessary or sufficient, but not both unless the underlying cost and constraint functions are also convex.We propose a somewhat different approach for solving a nonconvex inequality constrained optimization problem based on a nonlinear Lagrangian function. This leads to optimality conditions which are both sufficient and necessary, without any convexity assumption. Subsequently, under appropriate assumptions, the optimality conditions derived from the new nonlinear Lagrangian approach are used to obtain an equivalent root-finding problem. By appropriately defining a dual optimization problem and an alternative dual problem, we show that zero duality gap will hold always regardless of convexity, contrary to the case of linear Lagrangian duality.     

On duality in problems of optimal control described by convex differential inclusions of Goursat-Darboux type

Sufficient conditions of optimality are derived for convex and non-convex problems with state constraints on the basis of the apparatus of locally conjugate mappings. The duality theorem is formulated and the conditions under which the direct and dual problems are connected by the duality relation are searched for.     

Conjugate Duality and the Control of Linear Discrete Systems

In this paper we deal with the minimization of a convex function over the solution set of a range inclusion problem determined by a multivalued operator with convex graph. We attach a dual problem to it, provide regularity conditions guaranteeing strong duality and derive for the resulting primal–dual pair necessary and sufficient optimality conditions. We also discuss the existence of optimal solutions for the primal and dual problems by using duality arguments. The theoretical results are applied in the context of the control of linear discrete systems.     

Some Farkas-type results for fractional programming problems with DC functions

For a kind of fractional programming problem that the objective functions are the ratio of two DC (difference of convex) functions with finitely many convex constraints, in this paper, its dual problems are constructed, weak and strong duality assertions are given, and some sufficient and necessary optimality conditions which characterize their optimal solutions are obtained. Some recently obtained Farkas-type results for fractional programming problems that the objective functions are the ratio of a convex function to a concave function with finitely many convex constraints are the special cases of the general results of this paper.     

Nonsmooth minimax programming under locally Lipschitz (Φ, ρ)-invexity

Minimax programming problems involving locally Lipschitz (Φ, ρ)-invex functions are considered. The parametric and non-parametric necessary and sufficient optimality conditions for a class of nonsmooth minimax programming problems are obtained under nondifferentiable (Φ, ρ)-invexity assumption imposed on objective and constraint functions. When the sufficient conditions are utilized, parametric and non-parametric dual problems in the sense of Mond-Weir and Wolfe may be formulated and duality results are derived for the considered nonsmooth minimax programming problem. With the reference to the said functions we extend some results of optimality and duality for a larger class of nonsmooth minimax programming problems.     

Kuhn-Tucker sufficiency for global minimum of multi-extremal mathematical programming problems

The Kuhn-Tucker Sufficiency Theorem states that a feasible point that satisfies the Kuhn-Tucker conditions is a global minimizer for a convex programming problem for which a local minimizer is global. In this paper, we present new Kuhn-Tucker sufficiency conditions for possibly multi-extremal nonconvex mathematical programming problems which may have many local minimizers that are not global. We derive the sufficiency conditions by first constructing weighted sum of square underestimators of the objective function and then by characterizing the global optimality of the underestimators. As a consequence, we derive easily verifiable Kuhn-Tucker sufficient conditions for general quadratic programming problems with equality and inequality constraints. Numerical examples are given to illustrate the significance of our criteria for multi-extremal problems.     

Optimality and duality in nonsmooth multiobjective fractional programming problem with constraints

In this paper, we establish some quotient calculus rules in terms of contingent derivatives for the two extended-real-valued functions defined on a Banach space and study a nonsmooth multiobjective fractional programming problem with set, generalized inequality and equality constraints. We define a new parametric problem associated with these problem and introduce some concepts for the (local) weak minimizers to such problems. Some primal and dual necessary optimality conditions in terms of contingent derivatives for the local weak minimizers are provided. Under suitable assumptions, sufficient optimality conditions for the local weak minimizers which are very close to necessary optimality conditions are obtained. An application of the result for establishing three parametric, Mond–Weir and Wolfe dual problems and several various duality theorems for the same is presented. Some examples are also given for our findings.

     

一类非光滑优化问题的最优性与对偶

本文研究了一类带等式和不等式约束的非光滑多目标优化问题,给出了该类问题的Karush-Kuhn-Tucker最优性必要条件和充分条件,建立了该类规划问题的一类混合对偶模型的弱对偶定理、强对偶定理、逆对偶定理、严格逆对偶定理和限制逆对偶定理.     

Fritz john second-order duality for nonlinear programming

A second-order dual to a nonlinear programming problem is formulated. This dual uses the Fritz John necessary optimality conditions instead of the Karush-Kuhn-Tucker necessary optimality conditions, and thus, does not require a constraint qualification. Weak, strong, strict-converse, and converse duality theorems between primal and dual problems are established.     

On duality theorems for a nondifferentiable minimax fractional programming

The optimality conditions of [Lai et al. (J. Math. Anal. Appl. 230 (1999) 311)] can be used to construct two kinds of parameter-free dual models of nondifferentiable minimax fractional programming problems which involve pseudo-/quasi-convex functions. In this paper, the weak duality, strong duality, and strict converse duality theorems are established for the two dual models.     

一类非光滑优化问题的最优性与对偶（英文）

本文研究了一类带等式和不等式约束的非光滑多目标优化问题,给出了该类问题的Karush-Kuhn-Tucker最优性必要条件和充分条件,建立了该类规划问题的一类混合对偶模型的弱对偶定理、强对偶定理、逆对偶定理、严格逆对偶定理和限制逆对偶定理.     

Optimality conditions and duality models for generalized fractional programming problems containing locally subdifferentiable and ρ:-convex functions

Both parametric and nonparametric necessary and sufficient optimality conditions are established for a class of nonsmooth generalized fractional programming problems containing ρ-convex functions. Subsequently, these optimality criteria are utilized as a basis for constructing two parametric and four parameter-free duality models and proving appropriate duality theorems. Several classes of generalized fractional programming problems, including those with arbitrary norms, square roots of positive semidefinite quadratic forms, support functions, continuous max functions, and discrete max functions, which can be viewed as special cases of the main problem are briefly discussed. The optimality and duality results developed here also contain, as special cases, similar results for nonsmooth problems with fractional, discrete max, and conventional objective functions which are particular cases of the main problem considered in this paper