Optimality conditions for vector optimization problem governed by the cone constrained generalized equations

ABSTRACT

The paper considers a class of vector optimization problems with cone constrained generalized equations. By virtue of advanced tools of variational analysis and generalized differentiation, a limiting normal cone of the graph of the normal cone constrained by the second-order cone is obtained. Based on the calmness condition, we derive an upper estimate of the coderivative for a composite set-valued mapping. The necessary optimality condition for the problem is established under the linear independent constraint qualification. As a special case, the obtained optimality condition is simplified with the help of strict complementarity relaxation conditions. The numerical results on different examples are given to illustrate the efficiency of the optimality conditions.     

Sufficient Optimality Conditions for a Bilevel Semivectorial D.C. Problem

Abstract

The paper is devoted to the study of a bilevel multiobjective optimization problems with objectives and constraints given as differences of convex functions. The main attention is paid to deriving sufficient optimality conditions. Several intermediate optimization problems are introduced to help us in our investigation.     

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.     

Some kind of Pareto stationarity for multiobjective problems with equilibrium constraints

ABSTRACT

In this paper, we derive some optimality and stationarity conditions for a multiobjective problem with equilibrium constraints (MOPEC). In particular, under a generalized Guignard constraint qualification, we show that any locally Pareto optimal solution of MOPEC must satisfy the strong Pareto Kuhn-Tucker optimality conditions. We also prove that the generalized Guignard constraint qualification is the weakest constraint qualification for the strong Pareto Kuhn-Tucker optimality. Furthermore, under certain convexity or generalized convexity assumptions, we show that the strong Pareto Kuhn-Tucker optimality conditions are also sufficient for several popular locally Pareto-type optimality conditions for MOPEC.     

OPTIMALITY CONDITIONS AND APPROXIMATE OPTIMALITY CONDITIONS IN LOCALLY LIPSCHITZ VECTOR OPTIMIZATION

Abstract

In this paper, we study constrained locally Lipschitz vector optimization problems in which the objective and constraint spaces are Hilbert spaces, the decision space is a Banach space, the dominating cone and the constraint cone may be with empty interior. Necessary optimality conditions for this type of optimization problems are derived. A sufficient condition for the existence of approximate efficient solutions to a general vector optimization problem is presented. Necessary conditions for approximate efficient solutions to a constrained locally Lipschitz optimization problem is obtained.     

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.     

Global Optimization of Nonlinear Bilevel Programming Problems

A novel technique that addresses the solution of the general nonlinear bilevel programming problem to global optimality is presented. Global optimality is guaranteed for problems that involve twice differentiable nonlinear functions as long as the linear independence constraint qualification condition holds for the inner problem constraints. The approach is based on the relaxation of the feasible region by convex underestimation, embedded in a branch and bound framework utilizing the basic principles of the deterministic global optimization algorithm, BB [2, 4, 5, 11]. Epsilon global optimality in a finite number of iterations is theoretically guaranteed. Computational studies on several literature problems are reported.     

Optimal time delays in a class of reaction-diffusion equations

ABSTRACT

A class of semilinear parabolic reaction diffusion equations with multiple time delays is considered. These time delays and corresponding weights are to be optimized such that the associated solution of the delay equation is the best approximation of a desired state function. The differentiability of the mapping is proved that associates the solution of the delay equation to the vector of weights and delays. Based on an adjoint calculus, first-order necessary optimality conditions are derived. Numerical test examples show the applicability of the concept of optimizing time delays.     

Global minimization algorithms for concave quadratic programming problems

Abstract: In this article, we consider the concave quadratic programming problem which is known to be NP hard. Based on the improved global optimality conditions by [Dür, M., Horst, R. and Locatelli, M., 1998, Necessary and sufficient global optimality conditions for convex maximization revisited, Journal of Mathematical Analysis and Applications, 217, 637–649] and [Hiriart-Urruty, J.B. and Ledyav, J.S., 1996, A note in the characterization of the global maxima of a convex function over a convex set, Journal of Convex Analysis, 3, 55–61], we develop a new approach for solving concave quadratic programming problems. The main idea of the algorithms is to generate a sequence of local minimizers either ending at a global optimal solution or at an approximate global optimal solution within a finite number of iterations. At each iteration of the algorithms we solve a number of linear programming problems with the same constraints of the original problem. We also present the convergence properties of the proposed algorithms under some conditions. The efficiency of the algorithms has been demonstrated with some numerical examples.     

Optimality Conditions for the Difference of Convex Set-Valued Mappings

Using the concept of subdifferential of cone-convex set valued mappings recently introduced by Baier and Jahn J. Optimiz. Theory Appl. 100 (1999), 233–240, we give necessary optimality conditions for nonconvex multiobjective optimization problems. An example illustrating the usefulness of our results is also given. Mathematics Subject classification: Primary 90C29, 90C26; Secondary 49K99.     

Optimum experimental design for a regression on a hypercube-generalization of Hoel's result

In the note Hoel's result (1965, Ann. Math. Statist., 36, 1097–1106) is generalized to a large family of experimental design optimality criterions. Sufficient conditions for optimality criterion are given, which ensure existence of the optimum experimental design measure which is a product of design measures on lower dimensional domains.     

A New Condition and Approach for Zero-Sum Stochastic Games with Average Payoffs

Abstract

This article deals with discrete-time two-person zero-sum stochastic games with Borel state and action spaces. The optimality criterion to be studied is the long-run expected average payoff criterion, and the (immediate) payoff function may have neither upper nor lower bounds. We first replace the optimality equation widely used in the previous literature with two so-called optimality inequalities, and give a new set of conditions for the existence of solutions to the optimality inequalities. Then, from the optimality inequalities we ensure the existence of a pair of average optimal stationary strategies. Our new condition is slightly weaker than those in the previous literature, and as a byproduct some interesting results such as the convergence of a value iteration scheme to the value of the discounted payoff game is obtained. Finally, we first apply the main results in this article to generalized inventory systems, and then further provide an example of controlled population processes for which all of our conditions are satisfied, while some of conditions in some of previous literature fail to hold.     

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.      

Duality results for extended multifacility location problems

Abstract

Duality statements are presented for multifacility location problems as suggested by Drezner Hiu 1991, where for each given point the sum of weighted distances to all facilities plus set-up costs is determined and the maximal value of these sums is to be minimized. We develop corresponding dual problems for the cases with and without set-up costs and present associated optimality conditions. In the concluding part of this note we use these optimality conditions for a geometrical characterization of the set of optimal solutions and consider for an illustration corresponding examples.     

Higher-order conditions for strict efficiency

Luu and Kien (On higher order conditions for strict efficiency, Soochow J. Math. 33 (2007), pp. 17–31), proposed higher-order conditions for strict efficiency of vector optimization problems based on the derivatives introduced in Ginchev (Higher order optimality conditions in nonsmooth optimization, Optimization 51 (2002), pp. 47–72). These derivatives are defined for scalar functions and in their terms necessary and sufficient conditions are obtained a point to be strictly efficient (isolated) minimizer of a given order for quite arbitrary scalar function. Passing to vector functions, Luu and Kien lose the peculiarity that the optimality conditions work with arbitrary functions. In this article, applying the mentioned derivatives for the scalarized problem and restoring the original idea, optimality conditions for strict efficiency of a given order are proposed, which work with quite arbitrary vector functions. It is shown that the results of Luu and Kien are corollaries of the given conditions and generalizations are discussed.     

Mixed duality without a constraint qualification for minimax fractional programming

Without the need of a constraint qualification, we establish the necessary and sufficient optimality conditions for minimax fractional programming. Using these optimality conditions, we construct a mixed dual model which unifies the Mond–Weir dual, Wolfe dual and a parameter dual models. Several duality theorems are established. Consequently, this article partly solves the problem posed by Lai et al. [H.C. Lai, J.C. Liu and K. Tanaka (1999). Duality without a constraint qualification for minimax fractional programming. Journal of Optimization Theory and Applications, 101, 109–125.].     

Pontryagin principles for bounded discrete-time processes

Abstract

We establish necessary conditions and sufficient conditions of optimality in the form of Pontryagin principles for infinite-horizon discrete-time optimal control problems governed by a difference inequation.     

Risk-sensitive average continuous-time Markov decision processes with unbounded rates

Abstract

In this paper we study the risk-sensitive average cost criterion for continuous-time Markov decision processes in the class of all randomized Markov policies. The state space is a denumerable set, and the cost and transition rates are allowed to be unbounded. Under the suitable conditions, we establish the optimality equation of the auxiliary risk-sensitive first passage optimization problem and obtain the properties of the corresponding optimal value function. Then by a technique of constructing the appropriate approximating sequences of the cost and transition rates and employing the results on the auxiliary optimization problem, we show the existence of a solution to the risk-sensitive average optimality inequality and develop a new approach called the risk-sensitive average optimality inequality approach to prove the existence of an optimal deterministic stationary policy. Furthermore, we give some sufficient conditions for the verification of the simultaneous Doeblin condition, use a controlled birth and death system to illustrate our conditions and provide an example for which the risk-sensitive average optimality strict inequality occurs.     

Generalized Higher-Order Optimality Conditions for Set-Valued Optimization under Henig Efficiency

In this paper, generalized mth-order contingent epiderivative and generalized mth-order epiderivative of set-valued maps are introduced, respectively. By virtue of the generalized mth-order epiderivatives, generalized necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a fixed set. Generalized Kuhn–Tucker type necessary and sufficient optimality conditions are also obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map.     

Generalized nonsmooth invexity over cones in vector optimization

In this paper K-nonsmooth quasi-invex and (strictly or strongly) K-nonsmooth pseudo-invex functions are defined. By utilizing these new concepts, the Fritz–John type and Kuhn–Tucker type necessary optimality conditions and number of sufficient optimality conditions are established for a nonsmooth vector optimization problem wherein Clarke’s generalized gradient is used. Further a Mond Weir type dual is associated and weak and strong duality results are obtained.