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.     

Multiobjective programming problems with (G,C,ρ)-convexity

In this paper, we deal with multiobjective programming problems involving functions which are not necessarily differential. A new concept of generalized convexity, which is called (G,C,??)-convexity, is introduced. We establish not only sufficient but also necessary optimality conditions for multiobjective programming problems from a viewpoint of the new generalized convexity. When the sufficient conditions are utilized, the corresponding duality theorems are derived for general Mond-Weir type dual program.     

On second-order Fritz John type optimality conditions in nonsmooth multiobjective programming

We study a multiobjective optimization program with a feasible set defined by equality constraints and a generalized inequality constraint. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point considered. We provide necessary second order optimality conditions and also sufficient conditions via a Fritz John type Lagrange multiplier rule and a set-valued second order directional derivative, in such a way that our sufficient conditions are close to the necessary conditions. Some consequences are obtained for parabolic directionally differentiable functions and C ^1,1 functions, in this last case, expressed by means of the second order Clarke subdifferential. Some illustrative examples are also given.     

Sufficient global optimality conditions for weakly convex minimization problems

In this paper, we present sufficient global optimality conditions for weakly convex minimization problems using abstract convex analysis theory. By introducing (L,X)-subdifferentials of weakly convex functions using a class of quadratic functions, we first obtain some sufficient conditions for global optimization problems with weakly convex objective functions and weakly convex inequality and equality constraints. Some sufficient optimality conditions for problems with additional box constraints and bivalent constraints are then derived.      

On the Dini-Hadamard subdifferential of the difference of two functions

In this paper we first provide a general formula of inclusion for the Dini-Hadamard ε-subdifferential of the difference of two functions and show that it becomes equality in case the functions are directionally approximately starshaped at a given point and a weak topological assumption is fulfilled. To this end we give a useful characterization of the Dini-Hadamard ε-subdifferential by means of sponges. The achieved results are employed in the formulation of optimality conditions via the Dini-Hadamard subdifferential for cone-constrained optimization problems having the difference of two functions as objective.     

Optimality conditions for an isolated minimum of order two in C constrained optimization

In this paper we obtain first and second-order optimality conditions for an isolated minimum of order two for the problem with inequality constraints and a set constraint. First-order sufficient conditions are derived in terms of generalized convex functions. In the necessary conditions we suppose that the data are continuously differentiable. A notion of strongly KT invex inequality constrained problem is introduced. It is shown that each Kuhn-Tucker point is an isolated global minimizer of order two if and only if the problem is strongly KT invex. The article could be considered as a continuation of [I. Ginchev, V.I. Ivanov, Second-order optimality conditions for problems with C¹ data, J. Math. Anal. Appl. 340 (2008) 646-657].     

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.     

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.

     

Optimality and duality for nonsmooth multiobjective programming problems with V-r-invexity

In the paper, we consider a class of nonsmooth multiobjective programming problems in which involved functions are locally Lipschitz. A new concept of invexity for locally Lipschitz vector-valued functions is introduced, called V-r-invexity. The generalized Karush–Kuhn–Tuker necessary and sufficient optimality conditions are established and duality theorems are derived for nonsmooth multiobjective programming problems involving V-r-invex functions (with respect to the same function η).     

Optimality and Duality for Invex Nonsmooth Multiobjective programming problems

We consider nonsmooth multiobjective programming problems with inequality and equality constraints involving locally Lipschitz functions. Several sufficient optimality conditions under various (generalized) invexity assumptions and certain regularity conditions are presented. In addition, we introduce a Wolfe-type dual and Mond–Weir-type dual and establish duality relations under various (generalized) invexity and regularity conditions.     

Sufficiency and Duality in Multiobjective Variational Problems with Generalized Type I Functions

Recently Hachimi and Aghezzaf introduced the notion of (F,α,ρ,d)-type I functions, a new class of functions that unifies several concepts of generalized type I functions. Here, we extend the concepts of (F,α,ρ,d)-type I and generalized (F,α,ρ, d)-type I functions to the continuous case and we use these concepts to establish various sufficient optimality conditions and mixed duality results for multiobjective variational problems. Our results apparently generalize a fairly large number of sufficient optimality conditions and duality results previously obtained for multiobjective variational problems.     

Generalized fractional programming problems containing locally subdifferentiable and ρ-Univex functions

Parametric and nonparametric sufficient optimality conditions are established for a class of nonsmooth generalized fractional programming problems containing ρ-univex functions. Subsequently, these optimality criteria are utilized as a basis for construction of two parametric and four parameter–free duality models and proving appropriate duality theorems     

Convex composite non–Lipschitz programming

In this paper necessary, and sufficient optimality conditions are established without Lipschitz continuity for convex composite continuous optimization model problems subject to inequality constraints. Necessary conditions for the special case of the optimization model involving max-min constraints, which frequently arise in many engineering applications, are also given. Optimality conditions in the presence of Lipschitz continuity are routinely obtained using chain rule formulas of the Clarke generalized Jacobian which is a bounded set of matrices. However, the lack of derivative of a continuous map in the absence of Lipschitz continuity is often replaced by a locally unbounded generalized Jacobian map for which the standard form of the chain rule formulas fails to hold. In this paper we overcome this situation by constructing approximate Jacobians for the convex composite function involved in the model problem using ε-perturbations of the subdifferential of the convex function and the flexible generalized calculus of unbounded approximate Jacobians. Examples are discussed to illustrate the nature of the optimality conditions. Received: February 2001 / Accepted: September 2001?Published online February 14, 2002     

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.     

Using copositivity for global optimality criteria in concave quadratic programming problems

In this note we specify a necessary and sufficient condition for global optimality in concave quadratic minimization problems. Using this condition, it follows that, from the perspective of worst-case complexity of concave quadratic problems, the difference between local and global optimality conditions is not as large as in general. As an essential ingredient, we here use the-subdifferential calculus via an approach of Hiriart-Urruty and Lemarechal (1990).     

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     

Sufficient optimality conditions for nonconvex control problems with state constraints

The sufficient optimality conditions of Zeidan for optimal control problems (Refs. 1 and 2) are generalized such that they are applicable to problems with pure state-variable inequality constraints. We derive conditions which neither assume the concavity of the Hamiltonian nor the quasiconcavity of the constraints. Global as well as local optimality conditions are presented.     

On second-order optimality conditions for continuously Fréchet differentiable vector optimization problems

ABSTRACT

In this paper, we study a vector optimization problem (VOP) with both inequality and equality constraints. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point of study. By virtue of a second-order constraint qualification of Abadie type, we provide second-order Karush–Kuhn–Tucker type necessary optimality conditions for the VOP. Moreover, we also obtain second-order sufficient optimality conditions for a kind of strict local efficiency. Both the necessary conditions and the sufficient conditions are shown in equivalent pairs of primal and dual formulations by using theorems of the alternative for the VOP.     

Optimality and duality in fractional multiple objective programming involving semilocally preinvex and related functions

Necessary and sufficient optimality conditions are obtained for a nonlinear fractional multiple objective programming problem involving η-semidifferentiable functions. Also, a general dual is formulated and duality results are proved using concepts of generalized semilocally preinvex functions.     

Necessary Conditions in Nonsmooth Minimization via Lower and Upper Subgradients

The paper concerns first-order necessary optimality conditions for problems of minimizing nonsmooth functions under various constraints in infinite-dimensional spaces. Based on advanced tools of variational analysis and generalized differential calculus, we derive general results of two independent types called lower subdifferential and upper subdifferential optimality conditions. The former ones involve basic/limiting subgradients of cost functions, while the latter conditions are expressed via Fréchet/regular upper subgradients in fairly general settings. All the upper subdifferential and major lower subdifferential optimality conditions obtained in the paper are new even in finite dimensions. We give applications of general optimality conditions to mathematical programs with equilibrium constraints deriving new results for this important class of intrinsically nonsmooth optimization problems.