LIPSCHITZ r-INVEX FUNCTIONS AND NONSMOOTH PROGRAMMING

In this paper, a new class of nonconvex nonsmooth multiobjective programming problems with both inequality and equality constraints is considered. Namely, the sufficient optimality conditions and Mond-Weir duality results are established for such nondifferentiable multicriteria optimization problems in which the involved functions are nondifferentiable (b,?,?,?)w-univex functions (not necessarly with respect to the same functions b and ? and ?). Then the aforesaid results developed here under (b,?,?, ?)w-uinvexity are applicable for a larger class of nonsmooth vector optimization problems than under other generalized convexity notions existing in the literature.

This article deals with a class of non-smooth semi-infinite programming (SIP) problems in which the index set of the inequality constraints is an arbitrary set not necessarily finite. We introduce several kinds of constraint qualifications for these non-smooth SIP problems and we study the relationships between them. Finally, necessary and sufficient optimality conditions are investigated.

A class of nonsmooth multiobjective fractional programming is formulated. We establish the necessary and sufficient optimality conditions without the need of a constraint qualification. Then a mixed dual is introduced for a class of nonsmooth fractional programming problems, and various duality theorems are established without a constraint qualification.

The purpose of this paper is to consider a class of nonsmooth minimax semi-infinite fractional programming problem. Based on the concept of \( H- \) tangent derivative, a new generalization of convexity, namely generalized uniform \( (B_{H} ,\rho ) - \) invexity, is defined for this problem. For such semi-infinite programming problems, several sufficient optimality conditions are established and proved by utilizing the above defined new classes of functions. The results extend and improve the corresponding results in the literature.

In the research of optimization problems, optimality conditions play an important role. By using some derivatives, various types of necessary and/or sufficient optimality conditions have been introduced by many researchers. Especially, in convex programming, necessary and sufficient optimality conditions in terms of the subdifferential have been studied extensively. Recently, necessary and sufficient optimality conditions for quasiconvex programming have been investigated by the authors. However, there are not so many results concerned with Karush–Kuhn–Tucker type optimality conditions for non-differentiable quasiconvex programming. In this paper, we study a Karush–Kuhn–Tucker type optimality condition for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. We show some closedness properties for Greenberg–Pierskalla subdifferential. Under the Slater constraint qualification, we show a necessary and sufficient optimality condition for essentially quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. Additionally, we introduce a necessary and sufficient constraint qualification of the optimality condition. As a corollary, we show a necessary and sufficient optimality condition for convex programming in terms of the subdifferential.

Strong duality for a trust-region type relaxation of the quadratic assignment problem

The sufficient optimality conditions and duality results have recently been given for the following generalised convex programming problem:where the funtion f and g satisfyfor some η: X0 × X0 → ℝnIt is shown here that a relaxation defining the above generalised convexity leads to a new class of multi-objective problems which preserves the sufficient optimality and duality results in the scalar case, and avoids the major difficulty of verifying that the inequality holds for the same function η(. , .). Further, this relaxation allows one to treat certain nonlinear multi-objective fractional programming problems and some other classes of nonlinear (composite) problems as special cases.

Invited Papers.- Combined Relaxation Methods for Generalized Monotone Variational Inequalities.- Abstract Convexity and the Monge-Kantorovich Duality.- Optimality Conditions and Duality for Multiobjective Programming Involving (C, ?, ?, d) type-I Functions.- Contributed Papers.- Partitionable Variational Inequalities with Multi-valued Mappings.- Almost Convex Functions: Conjugacy and Duality.- Pseudomonotonicity of a Linear Map on the Interior of the Positive Orthant.- An Approach to Discrete Convexity and Its Use in an Optimal Fleet Mix Problem.- A Unifying Approach to Solve a Class of Parametrically-Convexifiable Problems.- Mathematical Programming with (?, ?)-invexity.- Some Classes of Pseudoconvex Fractional Functions via the Charnes-Cooper Transformation.- Equilibrium Problems Via the Palais-Smale Condition.- Points of Efficiency in Vector Optimization with Increasing-along-rays Property and Minty Variational Inequalities.- Higher Order Properly Efficient Points in Vector Optimization.- Higher-order Pseudoconvex Functions.- Sufficient Optimality Conditions and Duality in Nonsmooth Multiobjective Optimization Problems under Generalized Convexity.- Optimality Conditions for Tanaka's Approximate Solutions in Vector Optimization.- On the Work of W. Oettli in Generalized Convexity and Nonconvex Optimization - a Review and Some Perspectives.- Local and Global Consumer Preferences.- Optimality Conditions for Convex Vector Functions by Mollified Derivatives.- On Arcwise Connected Convex Multifunctions.- A Sequential Method for a Class of Bicriteria Problems.- Decomposition of the Measure in the Integral Representation of Piecewise Convex Curves.- Rambling Through Local Versions of Generalized Convex Functions and Generalized Monotone Operators.- Monotonicity and Dualities.- On Variational-like Inequalities with Generalized Monotone Mappings.- Almost Pure Nash Equilibria in Convex Noncooperative Games.- A Spectral Approach to Solve Box-constrained Multi-objective Optimization Problems.

This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is finitely PC2 (continuous and C2 apart from finitely many points). We prove that the control-to-state operator is continuously differentiable even though the nonlinear coefficient is non-smooth. This enables us to establish “no-gap” second-order necessary and sufficient optimality conditions in terms of an abstract curvature functional, i.e., for which the sufficient condition only differs from the necessary one in the fact that the inequality is strict. A condition that is equivalent to the second-order sufficient optimality condition and could be useful for error estimates in, e.g., finite element discretizations is also provided.

In this paper, we are concerned with one of the difficult class of optimization problems called the interval-valued optimization problem with vanishing constraints. Sufficient optimality conditions for a LU optimal solution are derived under generalized convexity assumptions. Moreover, appropriate duality results are discussed for a Mond-Weir type dual problem. In addition, numerical examples are given to support the sufficient optimality conditions and weak duality theorem.

Optimality Conditions and Duality Models for a Class of Nonsmooth Constrained Fractional Optimal Control Problems

Based upon Ben-Tal's generalized algebraic operations, new classes of functions, namely (h,?)-type-I, quasi (h,?)-type-I, and pseudo (h,?)-type-I, are defined for a multi-objective programming problem. Sufficient optimality conditions are obtained for a feasible solution to be a Pareto efficient solution for this problem. Some duality results are established by utilizing the above defined classes of functions, considering the concept of a Pareto efficient solution.

Necessary and sufficient optimality conditions are obtained for a nonlinear fractional programming problem involving semidifferentiable functions introduced by Ewing, Kaul and Kaur. A dual is formulated and duality results are proved using concepts of semilocally quasiconvex and semilocally pseudoconvex functions. An equivalent program using generalized Charnes Cooper transformation is given

A new class of convex functions, namely, (b, F)-convex functions is introduced. Sufficient optimality conditions and duality results are established for nonlinear programming problems under (b, F)-convexity assumptions on the functions involved. Further, we extend these results to fractional programming problems.

Multiple objective fractional programming involving semilocally type I-preinvex and related functions