Optimality Conditions in Nondifferentiable G-Invex Multiobjective Programming

The purpose of this paper is to consider a class of nondifferentiable multiobjective fractional programming problems in which every component of the objective function contains a term involving the support function of a compact convex set. Based on the (C,α,ρ,d)-convexity, sufficient optimality conditions and duality results for weakly efficient solutions of the nondifferentiable multiobjective fractional programming problem are established. The results extend and improve the corresponding results in the literature.

In this paper, a nondifferentiable multiobjective programming problem is considered where every component of objective and constraint functions contain a term involving the support function of a compact convex set. A new class of higher order (F,α,ρ,d)-type I function is introduced. Necessary optimality conditions and the duality theorems for Wolfe and unified higher order dual problems are established. Several known results can be deduced as special cases.

This paper deals with non-differentiable multi-objective semi-infinite programming problem. It is a problem of simultaneous minimisation of finitely many scalar valued functions subject to an arbitrary (possibly infinite) set of constraints. Non-differentiability enters, due to the square root of a quadratic form which appears in the objective functional. Concept of efficiency of order m has been extended to the above stated problem. In order to study this new solution concept, the notion of ρ-invexity of order m is also proposed which is utilised to establish sufficient optimality conditions for the non-differentiable multi-objective semi-infinite programming problem. Mond-Weir type of dual is proposed for which weak, strong and strict converse duality theorems are established.

We consider a multi-objective semi-infinite programming problem with a feasible set defined by inequality constraints. First, we present a Fritz–John type necessary optimality condition. Then, we introduce two constraint qualifications and derive the weak and strong Karush–Kuhn–Tucker types necessary conditions for (weakly) efficient solution of the considered problem. Finally, an extension of a Caristi–Ferrara–Stefanescu result for the (\(\Phi\), \(\rho\))-invexity is proved, and some sufficient conditions are presented under this weak assumption. All results are given in term of Clarke subdifferential.

In this article, the second-order optimality conditions for nonlinear programming with multiple interval-valued objective functions (in brief, NPMIOF) are studied. In the first part, the second- and first-order necessary optimality conditions for some types of efficient solutions of NPMIOF are established under of Abadie second- and first-order constraint qualifications. In the next part, we investigate both primal and dual second-order sufficient optimality conditions. Our second-order necessary conditions enhance the previous results. The second-order sufficient optimality conditions are new.

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.

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.

In this paper, we point out an inconsistency between assumptions andresults on the second order strong and converse duality in a recentpaper of I. Ahmad ( Information Sciences 173 (2005) 23-34). We thenprovide appropriate modifications to rectify this deficiency.

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.

In this paper, a new class of nonconvex differentiable multiobjective programming problems involving n-set functions with both inequality and equality constraints is considered.Then, under V-r-convexity and/or generalized V-r-convexity hypotheses, several sufficient optimality conditions, saddle point criteria and various mixed duality theorems are proved for such not necessarily convex vector optimization problems involving n-set functions.

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.

A generalized nondifferentiable fractional optimization problem (GFP), which consists of a maximum objective function defined by finite fractional functions with differentiable functions and support functions, and a constraint set defined by differentiable functions, is considered. Recently, Kim et al. [Journal of Optimization Theory and Applications 129 (2006), no. 1, 131-146] proved optimality theorems and duality theorems for a nondifferentiable multiobjective fractional programming problem (MFP), which consists of a vector-valued function whose components are fractional functions with differentiable functions and support functions, and a constraint set defined by differentiable functions. In fact if <TEX>$\overline{x}$</TEX> is a solution of (GFP), then <TEX>$\overline{x}$</TEX> is a weakly efficient solution of (MFP), but the converse may not be true. So, it seems to be not trivial that we apply the approach of Kim et al. to (GFP). However, modifying their approach, we obtain optimality conditions and duality results for (GFP).

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.

The composite vector function is introduced, which is applied to multiobjective programming with equality constraints and inequality constraints, three theorems on the optimality sufficient condition are established in this article. on the Banach space,optimality of composite function and pseudoconvex and strictly pseudo-convex of composite function are redefined. We introduce Jacobian matrix and knowledge of norm approximation. Several optimality sufficient conditions are given on pseudoconvex and strictly pseudo-convex in some part of the objective function and constraints.

We consider optimal control problems that can have singular or sliding modes as solutions or parts of solutions [1; 2, p. 26]; in these modes, the Pontryagin function is independent of some control variables or has a nonunique maximum on the set of control variables. For such problems with specific degeneration properties [3, p. 47], the classical optimality conditions (if they apply at all) are noneffective: the Lagrange–Pontryagin necessary conditions are satisfied and can produce infinitely many spurious solutions, while the sufficient conditions fail to hold. One needs effective conditions, which are lacking in the theory yet. The aim of the present paper is to fill the gap to a certain extent by suggesting a reasonably complete set of necessary and sufficient local optimality conditions obtained as a generalization of the classical second-order conditions in terms of the corresponding Clebsch and Riccati inequalities and the maximum principle. To this end, we use a special construction (known as the method of multiple maxima [3–5]) of the Krotov function in the Krotov sufficient conditions.