Arbitrary Feasible Set Research Articles

Inexact restoration for derivative-free expensive function minimization and applications

The Inexact Restoration approach has proved to be an adequate tool for handling the problem of minimizing an expensive function within an arbitrary feasible set by using different degrees of precision. This framework allows one to obtain suitable convergence and complexity results for an approach that rationally combines low- and high-precision evaluations. In this paper we consider the case where the domain of the optimization problem is an abstract metric space. Assumptions about differentiability or even continuity will not be used in the general algorithm based on Inexact Restoration. Although optimization phases that rely on smoothness cannot be used in this case, basic convergence and complexity results are recovered. A new derivative-free optimization phase is defined and the subproblems that arise at this phase are solved using a regularization approach that takes advantage of different notions of stationarity. The new methodology is applied to the problem of reproducing a controlled experiment that mimics the failure of a dam.

First order optimality conditions in vector optimization involving stable functions

We study a nonsmooth vector optimization problem with an arbitrary feasible set or a feasible set defined by a generalized inequality constraint and an equality constraint. We assume that the involved functions are nondifferentiable. First, we provide some calculus rules for the contingent derivative in which the stability (a local Lipschitz property at a point) of the functions plays a crucial role. Second, another calculus rules are established for steady functions. Third, necessary optimality conditions are stated using tangent cones to the feasible set and the contingent derivative of the objective function. Finally, some necessary and sufficient conditions are presented through Lagrange multiplier rules.

Optimality Conditions in Differentiable Vector Optimization via Second-Order Tangent Sets

We provide second-order necessary and sufficient conditions for a point to be an efficient element of a set with respect to a cone in a normed space, so that there is only a small gap between necessary and sufficient conditions. To this aim, we use the common second-order tangent set and the asymptotic second-order cone utilized by Penot. As an application we establish second-order necessary conditions for a point to be a solution of a vector optimization problem with an arbitrary feasible set and a twice Frechet differentiable objective function between two normed spaces. We also establish second-order sufficient conditions when the initial space is finite-dimensional so that there is no gap with necessary conditions. Lagrange multiplier rules are also given.

Second order necessary conditions in set constrained differentiable vector optimization

We state second order necessary optimality conditions for a vector optimization problem with an arbitrary feasible set and an order in the final space given by a pointed convex cone with nonempty interior. We establish, in finite-dimensional spaces, second order optimality conditions in dual form by means of Lagrange multipliers rules when the feasible set is defined by a function constrained to a set with convex tangent cone. To pass from general conditions to Lagrange multipliers rules, a generalized Motzkin alternative theorem is provided. All the involved functions are assumed to be twice Frechet differentiable.

First and second order sufficient conditions for strict minimality in nonsmooth vector optimization

In this paper we present first and second order sufficient conditions for strict local minima of orders 1 and 2 to vector optimization problems with an arbitrary feasible set and a twice directionally differentiable objective function. With this aim, the notion of support function to a vector problem is introduced, in such a way that the scalar case and the multiobjective case, in particular, are contained. The obtained results extend the multiobjective ones to this case. Moreover, specializing to a feasible set defined by equality, inequality, and set constraints, first and second order sufficient conditions by means of Lagrange multiplier rules are established.

Strict Minimality Conditions in Nondifferentiable Multiobjective Programming

In this paper, sufficient conditions for superstrict minima of order m to nondifferentiable multiobjective optimization problems with an arbitrary feasible set are provided. These conditions are expressed through the Studniarski derivative of higher order. If the objective function is Hadamard differentiable, a characterization for strict minimality of order 1 (which coincides with superstrict minimality in this case) is obtained.

FIRST AND SECOND ORDER SUFFICIENT CONDITIONS FOR STRICT MINIMALITY IN MULTIOBJECTIVE PROGRAMMING

ABSTRACT In this paper, first and second order sufficient conditions are established for strict local Pareto minima of orders 1 and 2 to multiobjective optimization problems with an arbitrary feasible set and a twice differentiable objective function are provided. For this aim, the concept of support function to a multiobjective problem is introduced, so that the scalar case in particular is contained. The obtained results generalize the classical ones of this case. Furthermore, particularizing to a feasible set defined by equality and inequality constraints, first and second order optimality conditions in primal form as well as dual form (by means of a Lagrange multiplier rule) are obtained.

Strict Efficiency in Vector Optimization

The notion of strict minimum of order m for real optimization problems is extended to vector optimization. Its properties and characterization are studied in the case of finite-dimensional spaces (multiobjective problems). Also the notion of super-strict efficiency is introduced for multiobjective problems, and it is proved that, in the scalar case, all of them coincide. Necessary conditions for strict minimality and for super-strict minimality of order m are provided for multiobjective problems with an arbitrary feasible set. When the objective function is Fréchet differentiable, necessary and sufficient conditions are established for the case m=1, resulting in the situation that the strict efficiency and super-strict efficiency notions coincide.

Distribution of coalitional power under probabilistic voting procedures

Pattanaik and Peleg (1986) imposed regularity, ex-post Pareto optimality and independence of irrelevant alternatives on a probabilistic voting procedure and showed that: (i) the distribution of coalitional power for decisiveness in two-alternative feasible sets is subadditive in general, but additive if the universal set has at least four alternatives; and (ii) the distribution of coalitional power in an arbitrary feasible set is almost complete random dictatorship, and becomes complete random dictatorship under certain additional conditions. This paper formulates the problem in terms of citizens’ sovereignty and monotonicity conditions (which are in line with Arrow’s original work) instead of ex-post Pareto optimality and proves that: (i) Pattanaik and Peleg’s coalitional weights for decisiveness in two-alternative feasible sets become additive even with only three alternatives in the universal set; and (ii) the distribution of coalitional power in an arbitrary feasible set is completely characterized by random dictatorship without the additional conditions of Pattanaik and Peleg.