Parametric Duality Models for Semi-infinite Discrete Minmax Fractional Programming Problems Involving Generalized （η,ρ）-Invex Functions

A semi-infinite programming problem is a mathematical programming problem with a finite number of variables and infinitely many constraints. Duality theories and generalized convexity concepts are important research topics in mathematical programming. In this paper, we discuss a fairly large number of paramet- ric duality results under various generalized （η,ρ）-invexity assumptions for a semi-infinite minmax fractional programming problem.     

Optimality and invexity in optimization problems in Banach algebras (spaces)

In this paper we introduce into nonsmooth optimization theory in Banach algebras a new class of mathematical programming problems, which generalizes the notion of smooth KT-(p,r)-invexity. In fact, this paper focuses on the optimality conditions for optimization problems in Banach algebras, regarding the generalized KT-(p,r)-invexity notion and Kuhn–Tucker points.     

Continuous-time optimization problems involving invex functions

In [D.H. Martin, The essence of invexity, J. Optim. Theory Appl. 47 (1985) 65-76] Martin introduced the notions of KKT-invexity and WD-invexity for mathematical programming problems. These notions are relaxations of invexity. In this work we generalize these concepts for continuous-time nonlinear optimization problems. We prove that the notion of KKT-invexity is a necessary and sufficient condition for global optimality of a Karush-Kuhn-Tucker point and that the notion of WD-invexity is a necessary and sufficient condition for weak duality.     

On <Emphasis Type="Italic">G</Emphasis>-invex multiobjective programming. Part I. Optimality

In this paper, a generalization of convexity, namely G-invexity, is considered in the case of nonlinear multiobjective programming problems where the functions constituting vector optimization problems are differentiable. The modified Karush-Kuhn-Tucker necessary optimality conditions for a certain class of multiobjective programming problems are established. To prove this result, the Kuhn-Tucker constraint qualification and the definition of the Bouligand tangent cone for a set are used. The assumptions on (weak) Pareto optimal solutions are relaxed by means of vector-valued G-invex functions.     

Necessary and sufficient conditions for weak efficiency in non-smooth vectorial optimization problems

By using the concepts of local cone approximations and K-directional derivatives, we obtain necessary conditions of optimality for the weak efficiency of the vectorial optimization problems with the inequalities and abstract constraints. We introduce the notion of stationary point (weak and strong) and we prove that, under suitable hypotheses of K-invexity, the stationary points are weakly efficient solutions (global).     

Generalized convexity for non-regular optimization problems with conic constraints

In non-regular problems the classical optimality conditions are totally inapplicable. Meaningful results were obtained for problems with conic constraints by Izmailov and Solodov (SIAM J Control Optim 40(4):1280–1295, 2001). They are based on the so-called 2-regularity condition of the constraints at a feasible point. It is well known that generalized convexity notions play a very important role in optimization for establishing optimality conditions. In this paper we give the concept of Karush–Kuhn–Tucker point to rewrite the necessary optimality condition given in Izmailov and Solodov (SIAM J Control Optim 40(4):1280–1295, 2001) and the appropriate generalized convexity notions to show that the optimality condition is both necessary and sufficient to characterize optimal solutions set for non-regular problems with conic constraints. The results that exist in the literature up to now, even for the regular case, are particular instances of the ones presented here.     

Multiobjective programming under d-invexity

In this paper, a generalization of convexity is considered in the case of nonlinear multiobjective programming problem where the functions involved are nondifferentiable. By considering the concept of Pareto optimal solution and substituting d-invexity for convexity, the Fritz John type and Karush–Kuhn–Tucker type necessary optimality conditions and duality in the sense of Mond–Weir and Wolfe for nondifferentiable multiobjective programming are given.     

Sufficient Conditions for Nonsmooth r-Invexity

In this paper, some sufficient conditions are derived for nonsmooth r-invexity with respect to Clarke's generalized directional derivatives. Some notions including nearly convex-like mappings and nearly convex sets are used to obtain the results.     

On the convexity of integrals of multivalued mappings: Applications in control theory

Using the notion of the local convexity index, we characterize in a quantitative way the local convexity of a set in then-dimensional Euclidean space, defined by an integral of a multivalued mapping. We estimate the rate of convergence of the conditional gradient method for solving an abstract optimization problem by means of the convexity index of the constraining set at the solution point. These results are applied to the qualitative analysis of the solutions of time-optimal and Mayer problems for linear control systems, as well as for estimating the convergence rate of algorithms solving these problems.     

Generalized Convexity of Functions and Generalized Monotonicity of Set-Valued Maps

We establish connections between some concepts of generalized monotonicity for set-valued maps introduced earlier and some notions of generalized convexity. Moreover, a notion of pseudomonotonicity for set-valued maps is introduced; it is shown that, if a function f is continuous, then its pseudoconvexity is equivalent to the pseudomonotonicity of its generalized subdifferential in the sense of Clarke and Rockafellar.     

Application of continuous-time programming problems to a class of variational-type inequalities

In this paper, we establish sufficiency criteria under generalized ρ−(η,θ)-invexity conditions for general continuous-time programming problems with nonlinear equality/inequality constraints. Using this we establish some existence criteria for solutions of a class of variational-type inequalities.     

On Optimality Conditions for Henig Efficiency and Superefficiency in Vector Equilibrium Problems

Abstract

Necessary optimality conditions for local Henig efficient and superefficient solutions of vector equilibrium problems involving equality, inequality, and set constraints in Banach space with locally Lipschitz functions are established under a suitable constraint qualification via the Michel–Penot subdifferentials. With assumptions on generalized convexity, necessary conditions for Henig efficiency and superefficiency become sufficient ones. Some applications to vector variational inequalities and vector optimization problems are given as well.     

Optimality Criteria and Duality in Multiobjective Programming Involving Nonsmooth Invex Functions

In this paper a generalization of invexity is considered in a general form, by means of the concept of K-directional derivative. Then in the case of nonlinear multiobjective programming problems where the functions involved are nondifferentiable, we established sufficient optimality conditions without any convexity assumption of the K-directional derivative. Then we obtained some duality results.     

Characterizations of linear suboptimality for mathematical programs with equilibrium constraints

The paper is devoted to the study of a new notion of linear suboptimality in constrained mathematical programming. This concept is different from conventional notions of solutions to optimization-related problems, while seems to be natural and significant from the viewpoint of modern variational analysis and applications. In contrast to standard notions, it admits complete characterizations via appropriate constructions of generalized differentiation in nonconvex settings. In this paper we mainly focus on various classes of mathematical programs with equilibrium constraints (MPECs), whose principal role has been well recognized in optimization theory and its applications. Based on robust generalized differential calculus, we derive new results giving pointwise necessary and sufficient conditions for linear suboptimality in general MPECs and its important specifications involving variational and quasivariational inequalities, implicit complementarity problems, etc. Research was partially supported by the National Science Foundation under grant DMS-0304989 and by the Australian Research Council under grant DP-0451168.     

Duality for minmax programming involving V-invex functions

Sufficient optimality conditions and duality results for a class of minmax programming problems are obtained under V-invexity type assumptions on objective and constraint functions. Applications of these results to certain fractional and generalized fractional programming problems are also presented     

Generalized convex functions and vector variational inequalities

In this paper, (, ,Q)-invexity is introduced, where :X ×X intR _m ⁺ , :X ×X X,X is a Banach space,Q is a convex cone ofR ^m . This unifies the properties of many classes of functions, such asQ-convexity, pseudo-linearity, representation condition, null space condition, andV-invexity. A generalized vector variational inequality is considered, and its equivalence with a multi-objective programming problem is discussed using (, ,Q)-invexity. An existence theorem for the solution of a generalized vector variational inequality is proved. Some applications of (, ,Q)-invexity to multi-objective programming problems and to a special kind of generalized vector variational inequality are given.The author is indebted to Dr. V. Jeyakumar for his constant encouragement and useful discussion and to Professor P. L. Yu for encouragement and valuable comments about this paper.     

Mathematical programming in rough environment

Classical mathematics is usually crisp while most real-life problems are not; therefore, classical mathematics is usually not suitable for dealing with real-life problems. In this article, we present a systematic and focused study of the application of rough sets (Z. Pawlak, Rough sets, In. J. Comput. Informa. Sci. 11 (1982), pp. 341–356.) to a basic area of decision theory, namely ‘mathematical programming’. This new framework concerns mathematical programming in a rough environment and is called ‘rough programming’ (L. Baoding, Theory and Practice of Uncertain Programming, 1st ed., Physica-Verlag, Heidelberg, 2002; E.A. Youness, Characterizing solutions of rough programming problems, Eut. J. Oper. Res. 168 (2006), pp. 1019–1029). It implies the existence of the roughness in any part of the problem as a result of the leakage, uncertainty and vagueness in the available information. We classify rough programming problems into three classes according to the place of the roughness. In rough programming, wherever roughness exists, new concepts like rough feasibility and rough optimality come to the front of our interest. The study of convexity for rough programming problems plays a key role in understanding global optimality in a rough environment. For this, a theoretical framework of convexity in rough programming and conceptualization of the solution is created on the lines of their crisp counterparts.     

On <Emphasis Type="Italic">G</Emphasis>-invex multiobjective programming. Part II. Duality

This paper represents the second part of a study concerning the so-called G-multiobjective programming. A new approach to duality in differentiable vector optimization problems is presented. The techniques used are based on the results established in the paper: On G-invex multiobjective programming. Part I. Optimality by T.Antczak. In this work, we use a generalization of convexity, namely G-invexity, to prove new duality results for nonlinear differentiable multiobjective programming problems. For such vector optimization problems, a number of new vector duality problems is introduced. The so-called G-Mond–Weir, G-Wolfe and G-mixed dual vector problems to the primal one are defined. Furthermore, various so-called G-duality theorems are proved between the considered differentiable multiobjective programming problem and its nonconvex vector G-dual problems. Some previous duality results for differentiable multiobjective programming problems turn out to be special cases of the results described in the paper.     

A new method of solving nonlinear mathematical programming problems involving r-invex functions

A new approach to a solution of a nonlinear constrained mathematical programming problem involving r-invex functions with respect to the same function η is introduced. An η-approximated problem associated with an original nonlinear mathematical programming problem is presented that involves η-approximated functions constituting the original problem. The equivalence between optima points for the original mathematical programming problem and its η-approximated optimization problem is established under r-invexity assumption.     

Generalized convexity and concavity of the optimal value function in nonlinear programming

In this paper we consider generalized convexity and concavity properties of the optimal value functionf ^* for the general parametric optimization problemP(_ε) of the form min _x f(x, ε) s.t.x∈R(ε). Many results on convexity and concavity characterizations off ^* were presented by the authors in a previous paper. Such properties off ^* and the solution set mapS ^* form an important part of the theoretical basis for sensitivity, stability and parametric analysis in mathematical optimization. We give sufficient conditions for several types of generalized convexity and concavity off ^*, in terms of respective generalized convexity and concavity assumptions onf and convexity and concavity assumptions on the feasible region point-to-set mapR. Specializations of these results to the parametric inequality-equality constrained nonlinear programming problem are provided. Research supported by Grant ECS-8619859, National Science Foundation and Contract N00014-86-K-0052, Office of Naval Research.