Uniqueness for Quasi-variational Inequalities

This paper presents a uniqueness result for a quasi-variational inequality QVI(1) that, in contrast to existing results, does not require the projection mapping on a variable closed and convex set to be a contraction. Our basic idea is to find a simple QVI(0), for example a variational inequality, for which we can show the existence of a unique solution. Further, exploiting some nonsingularity condition, we will guarantee the existence of a continuous solution path from the unique solution of QVI(0) to a solution of QVI(1). Finally, we can show that the existence of a second different solution of QVI(1) contradicts the nonsingularity condition. Moreover, we present some matrix-based sufficient conditions for our nonsingularity assumption, and we discuss these assumptions in the context of generalized Nash equilibrium problems with quadratic cost and affine linear constraint functions.     

A smoothing Newton method for mathematical programs governed by second-order cone constrained generalized equations

In this paper, we consider a class of mathematical programs governed by second-order cone constrained parameterized generalized equations. We reformulate the necessary optimality conditions as a system of nonsmooth equations under linear independence constraint qualification and the strict complementarity condition. A set of second order sufficient conditions is proposed, which is proved to be sufficient for the second order growth of the stationary point. The smoothing Newton method in [40] is employed to solve the system of nonsmooth equations whose strongly BD-regularity at a solution point is demonstrated under the second order sufficient conditions. Several illustrative examples are provided and discussed.     

广义多目标数学规划非支配解的二阶条件

§1.引言在不等式约束规划中,解的二阶条件是十分重要的课题.关于解的二阶条件,在单目标规划中已经得到了一些很重要的结果,如文献[1—4]等,都从各个不同的方面,引进不同的约束规格来讨论单目标数学规划解的二阶条件.在多目标数学规划中,有关“有效解”、“弱有效解”及“真有效解”的性质及一阶条件,已在不少书及文章中出现,如文献[5—9]等.本文试图就广义多目标数学规划相对于一般凸锥及某个多面体锥的局部和整体非支配解的二阶条件进行讨论.     

Optimality conditions in mathematical programming and composite optimization

New second order optimality conditions for mathematical programming problems and for the minimization of composite functions are presented. They are derived from a general second order Fermat's rule for the minimization of a function over an arbitrary subset of a Banach space. The necessary conditions are more accurate than the recent results of Kawasaki (1988) and Cominetti (1989); but, more importantly, in the finite dimensional case they are twinned with sufficient conditions which differ by the replacement of an inequality by a strict inequality. We point out the equivalence of the mathematical programming problem with the problem of minimizing a composite function. Our conditions are especially important when one deals with functional constraints. When the cone defining the constraints is polyhedral we recover the classical conditions of Ben-Tal—Zowe (1982) and Cominetti (1990).     

On the Coderivative of the Projection Operator onto the Second-order Cone

The limiting (Mordukhovich) coderivative of the metric projection onto the second-order cone $\mathbb{R}^{n}$ is computed. This result is used to obtain a sufficient condition for the Aubin property of the solution map of a parameterized second-order cone complementarity problem and to derive necessary optimality conditions for a mathematical program with a second-order cone complementarity problem among the constraints.     

Revisiting Generalized Nash Games and Variational Inequalities

Generalized Nash games with shared constraints represent an extension of Nash games in which strategy sets are coupled across players through a shared or common constraint. The equilibrium conditions of such a game can be compactly stated as a quasi-variational inequality (QVI), an extension of the variational inequality (VI). In (Eur. J. Oper. Res. 54(1):81–94, 1991), Harker proved that for any QVI, under certain conditions, a solution to an appropriately defined VI solves the QVI. This is a particularly important result, given that VIs are generally far more tractable than QVIs. However Facchinei et al. (Oper. Res. Lett. 35(2):159–164, 2007) suggested that the hypotheses of this result are difficult to satisfy in practice for QVIs arising from generalized Nash games with shared constraints. We investigate the applicability of Harker’s result for these games with the aim of formally establishing its reach. Specifically, we show that if Harker’s result is applied in a natural manner, its hypotheses are impossible to satisfy in most settings, thereby supporting the observations of Facchinei et al. But we also show that an indirect application of the result extends the realm of applicability of Harker’s result to all shared-constraint games. In particular, this avenue allows us to recover as a special case of Harker’s result, a result provided by Facchinei et al. (Oper. Res. Lett. 35(2):159–164, 2007), in which it is shown that a suitably defined VI provides a solution to the QVI of a shared-constraint game.     

Gap Function Approach to the Generalized Nash Equilibrium Problem

We consider an optimization reformulation approach for the generalized Nash equilibrium problem (GNEP) that uses the regularized gap function of a quasi-variational inequality (QVI). The regularized gap function for QVI is in general not differentiable, but only directionally differentiable. Moreover, a simple condition has yet to be established, under which any stationary point of the regularized gap function solves the QVI. We tackle these issues for the GNEP in which the shared constraints are given by linear equalities, while the individual constraints are given by convex inequalities. First, we formulate the minimization problem involving the regularized gap function and show the equivalence to GNEP. Next, we establish the differentiability of the regularized gap function and show that any stationary point of the minimization problem solves the original GNEP under some suitable assumptions. Then, by using a barrier technique, we propose an algorithm that sequentially solves minimization problems obtained from GNEPs with the shared equality constraints only. Further, we discuss the case of shared inequality constraints and present an algorithm that utilizes the transformation of the inequality constraints to equality constraints by means of slack variables. We present some results of numerical experiments to illustrate the proposed approach.     

Sufficient conditions to compute any solution of a quasivariational inequality via a variational inequality

We define the concept of reproducible map and show that, whenever the constraint map defining the quasivariational inequality (QVI) is reproducible then one can characterize the whole solution set of the QVI as a union of solution sets of some variational inequalities (VI). By exploiting this property, we give sufficient conditions to compute any solution of a generalized Nash equilibrium problem (GNEP) by solving a suitable VI. Finally, we define the class of pseudo-Nash equilibrium problems, which are (not necessarily convex) GNEPs whose solutions can be computed by solving suitable Nash equilibrium problems.     

Stochastic programs without duality gaps

This paper studies dynamic stochastic optimization problems parameterized by a random variable. Such problems arise in many applications in operations research and mathematical finance. We give sufficient conditions for the existence of solutions and the absence of a duality gap. Our proof uses extended dynamic programming equations, whose validity is established under new relaxed conditions that generalize certain no-arbitrage conditions from mathematical finance.     

Dissipative and Non-Dissipative Evolutionary Quasi-Variational Inequalities with Gradient Constraints

Set-Valued and Variational Analysis - Evolutionary quasi-variational inequality (QVI) problems of dissipative and non-dissipative nature with pointwise constraints on the gradient are studied. A...