Sufficient Optimality Theorem Research Articles

Second-Order Optimality Conditions for Multiobjective Optimization Whose Order Induced by Second-Order Cone

This paper is devoted to developing first-order necessary, second-order necessary, and second-order sufficient optimality conditions for a multiobjective optimization problem whose order is induced by a finite product of second-order cones (here named as Q-multiobjective optimization problem). For an abstract-constrained Q-multiobjective optimization problem, we derive two basic necessary optimality theorems for weak efficient solutions and a second-order sufficient optimality theorem for efficient solutions. For Q-multiobjective optimization problem with explicit constraints, we demonstrate first-order and second-order necessary optimality conditions under Robinson constraint qualification as well as second-order sufficient optimality conditions under upper second-order regularity for the explicit constraints. As applications, we obtain optimality conditions for polyhedral conic, second-order conic, and semi-definite conic Q-multiobjective optimization problems.

Second-order optimality conditions for cone constrained multi-objective optimization

The aim of this paper is to develop second-order necessary and second-order sufficient optimality conditions for cone constrained multi-objective optimization. First of all, we derive, for an abstract constrained multi-objective optimization problem, two basic necessary optimality theorems for weak efficient solutions and a second-order sufficient optimality theorem for efficient solutions. Secondly, basing on the optimality results for the abstract problem, we demonstrate, for cone constrained multi-objective optimization problems, the first-order and second-order necessary optimality conditions under Robinson constraint qualification as well as the second-order sufficient optimality conditions under upper second-order regularity for the conic constraint. Finally, using the optimality conditions for cone constrained multi-objective optimization obtained, we establish optimality conditions for polyhedral cone, second-order cone and semi-definite cone constrained multi-objective optimization problems.

On optimality conditions and duality theorems for robust semi-infinite multiobjective optimization problems

In this paper, we consider a semi-infinite multiobjective optimization problem with more than two differentiable objective functions and uncertain constraint functions, which is called a robust semi-infinite multiobjective optimization problem and give its robust counterpart \({\mathrm{(RSIMP)}}\) of the problem, which is regarded as the worst case of the uncertain semi-infinite multiobjective optimization problem. We prove a necessary optimality theorem for a weakly robust efficient solution of \({\mathrm{(RSIMP)}} \), and then give a sufficient optimality theorem for a weakly robust efficient solution of \({\mathrm{(RSIMP)}}\). We formulate a Wolfe type dual problem of \({\mathrm{(RSIMP)}}\) and give duality results which hold between \({\mathrm{(RSIMP)}}\) and its dual problem.

Optimality and mixed duality in multiobjective E-convex programming

In this paper, we consider a class of multiobjective E-convex programming problems with inequality constraints, where the objective and constraint functions are E-convex functions which were firstly introduced by Youness (J. Optim. Theory Appl. 102:439-450, 1999). Fritz-John and Kuhn-Tucker necessary and sufficient optimality theorems for the multiobjective E-convex programming are established under the weakened assumption of the theorems in Megahed et al. (J. Inequal. Appl. 2013:246, 2013) and Youness (Chaos Solitons Fractals 12:1737-1745, 2001). A mixed duality for the primal problem is formulated and weak and strong duality theorems between primal and dual problems are explored. Illustrative examples are given to explain the obtained results.

Non-differentiable multiobjective programming under generalised functions

In this paper, we consider a nonlinear multiobjective programming problem where functions involved are non-differentiable. The use of directional derivative in association with a hypothesis of an invex kind on the set has been of much interest in the recent past. Recently under preinvexity assumptions on fi′(xo;ηi(x,xo)) and g′j (xo;θj(x,xo)), necessary and sufficient optimality conditions for a non-smooth multiobjective optimisation problem under generalised class of dI-V-type I functions have been derived. Working in this direction, we introduce a new class of (φ, dI)-V-type I functions and illustrate through various non-trivial examples that this class is non-empty and extends some known classes introduced in the literature. Also, we obtain sufficient optimality conditions to enable a feasible solution of the primal problem to be its weak efficient/efficient solution. Then through an example, we illustrate the relevance of the sufficient optimality theorem obtained to get the efficient solution of the problem. Further, we formulate Wolfe type and Mond-Weir type multiobjective dual programs and establish various duality theorems under this newly introduced class of functions.

Solution Concepts and New Optimality Conditions in Bilevel Multiobjective Programming

In this paper, new sufficient optimality theorems for a solution of a differentiable bilevel multiobjective optimization problem (BMOP) are established. We start with a discussion on solution concepts in bilevel multiobjective programming; a theorem giving necessary and sufficient conditions for a decision vector to be called a solution of the BMOP and a proposition giving the relations between four types of solutions of a BMOP are presented and proved. Then, under the pseudoconvexity assumptions on the upper and lower level objective functions and the quasiconvexity assumptions on the constraints functions, we establish and prove two new sufficient optimality theorems for a solution of a general BMOP with coupled upper level constraints. Two corollary of these theorems, in the case where the upper and lower level objectives and constraints functions are convex are presented.

On optimality and duality for nonsmooth multiobjective fractional optimization problems

In this paper, we consider a nonsmooth multiobjective fractional program. For sufficient optimality conditions, we define the nearly invex functions for locally Lipschitz vector-valued functions. We obtain generalized sufficient optimality theorems and prove weak and strong duality theorems for the multiobjective fractional optimization problem involving nearly invex functions.

Wolfe type duality involving nonsmooth (B, η) -invex functions for a minmax programming problem

A nonsmooth minmax programming problem, under the assumption of proper (b, η) – invexity conditions on the functions involved in the objective and in the constraints, is considered. For such a problem, a sufficient optimality theorem is proved, a Wolfe type dual is presented and duality theorems relating the primal and the dual are proved. Applications of these results to certain nonsmooth fractional and nonsmooth generalized fractional programming duality are also presented.

Minmax programming : optimality and Wolfe type duality for a problem involving bectovex functions

A number of sufficient optimality theorems are proved for a certain minmax programming problem under a variety of bectovexity (generalized bectovexity) assumptions on the functions involved in the objective and the constraints. A dual is presented for such a problem and duality theorems relating the primal and the dual are proved. It is shown that the dual for certain minmax (generalized) fractional programming problem is a special case of the main problem considered in the paper.

Complex Fractional Programming Involving Generalized Quasi/Pseudo Convex Functions

(P) minimize Re'f(z, z) + (z H Az) 1/2 ]/Re[g(z, z) - (z H Bz)1/2] subject to h(z, z) ∈ S ⊂ C m , z ∈ C m , where f, g: C 2n → C and h: C 2n → C m are analytic functions, A, B ∈ C m×n are positive semidefinite Hermitian matrices, S is a polyhedral cone in C m . In this paper, we establish conditions for the existence of an optimal solution in (P) involving (£, p, θ)-quasiconvex/-pseudoconvex analytic functions. Based on the sufficient optimality theorem, we construct a duality model and then establish weak/strong, and strict converse duality theorems.

SUFFICIENT OPTIMALITY CONDITIONS AND DUALITY IN MULTIOBJECTIVE OPTIMIZATION INVOLVING GENERALIZED CONVEXITY

We are concerned with nonlinear multiobjective programming problem with inequality constraints. We introduce some classes of nonconvex functions by relaxing the definitions of invex and preinvex functions. Examples are given to shows relationship between them. Various first and second order sufficient optimality theorems are obtained. Moreover, we prove duality and converse duality results for Mond-Weir type dual.

Counterexample and Optimality Conditions in Differentiable Multiobjective Programming

Recently, sufficient optimality theorems for (weak) Pareto-optimal solutions of a multiobjective optimization problem (MOP) were stated in Theorems 3.1 and 3.3 of Ref. 1. In this note, we give a counterexample showing that the theorems of Ref. 1 are not true. Then, by modifying the assumptions of these theorems, we establish two new sufficient optimality theorems for (weak) Pareto-optimal solutions of (MOP); moreover, we give generalized sufficient optimality theorems for (MOP).

A sufficient optimality theoriem for protoconvex programs

We establish two first order sufficient optimality theorems; one for unconstrained nonlinear minimization problem, and the other for constrained nonlinear minimization problems, both with non-differentiable protoconvex or quasiconvex data functions that are not necessarily locally Lipschitz

Composite Nonsmooth Multiobjective Programs with V-ρ-Invexity

In this paper, some problems consisting of nonsmooth composite multiobjective programs have been treated with V-ρ-invexity type of conditions. In particular, we prove the generalized Karush–Kuhn–Tucker sufficient optimality theorem and duality theorems for nonsmooth composite multiobjective programs. Also, weak vector saddle point theorems are obtained for the composite programs under V-ρ-invexity conditions. The results obtained generalize the results of Hun Kuk et al. to some extent.

Duality for multiobjective variational problems with generalized invexity

In this paper, we establish a sufficient optimality theorem for a multiobjective variational problems under generalized invexity conditions. Using the equivalence relation of Weir and Mond, and the concept of proper efficiency, we study Wolfe type duals for multiobjective variational problems under pre-invexity and generalized invexity conditions on the functions.

Duality for MinmaxB-vex Programming Involvingn-Set Functions

A minmax programming problem involving severalB-vexn-set functions is considered. Necessary and sufficient optimality theorems and Wolfe type duality results are established. Duality for then-set generalized (minmax) fractional programming problem is derived as a special case of the main problem.

Wolfe-Type Duality Involving (B, η)-Invex Functions For a Minmax Programming Problem

A sufficient optimality theorem is proved for a certain minmax programming problem under the assumptions of proper (b,η)-invexity conditions on the functions involved in the objective and in the constraints. Next a dual is presented for such a problem and duality theorems relating the primal and the dual are proved. The dual for a minimax (generalized) fractional programming problem is presented as special case of the main problem considered in the paper.

Multiobjective control problems with invexity

Abstract We present a sufficient optimality theorem for a multiobjective control problem and establish the weak duality theorem and the strict converse duality theorem for the Mond-Weir type dual problem of our multiobjective control problem, under invexity conditions.

A Fritz John type sufficient optimality theorem in complex space

A Fritz John type sufficient optimality theorem is proved for nonlinear programming problems in finite dimensional complex space over polyhedral cones, which may include equality as well as inequality constraints.