Polyhedral Constraint Set Research Articles

Solving Euclidean Max-Sum problems exactly with cutting planes

This paper studies binary quadratic programs in which the objective is defined by the maximisation of a Euclidean distance matrix, subject to a general polyhedral constraint set. This class of nonconcave maximisation problems, which we refer to as the Euclidean Max-Sum problem, includes the capacitated, generalised and max-sum diversity problems as special cases. Due to the nonconcave objective, traditional cutting plane algorithms are not guaranteed to converge globally. In this paper, we introduce two exact cutting plane algorithms to address this limitation. The new algorithms remove the need for a concave reformulation, which is known to significantly slow down convergence. We establish exactness of the new algorithms by examining the concavity of the quadratic objective in a given direction, a concept we refer to as directional concavity. Numerical results show that the algorithms outperform other exact methods for benchmark diversity problems (capacitated, generalised and max-sum), and can easily solve problems of up to three thousand variables.

On the risk of convex-constrained least squares estimators under misspecification

We consider the problem of estimating the mean of a noisy vector. When the mean lies in a convex constraint set, the least squares projection of the random vector onto the set is a natural estimator. Properties of the risk of this estimator, such as its asymptotic behavior as the noise tends to zero, have been well studied. We instead study the behavior of this estimator under misspecification, that is, without the assumption that the mean lies in the constraint set. For appropriately defined notions of risk in the misspecified setting, we prove a generalization of a low noise characterization of the risk due to [Found. Comput. Math. 16 (2016) 965–1029] in the case of a polyhedral constraint set. An interesting consequence of our results is that the risk can be much smaller in the misspecified setting than in the well-specified setting. We also discuss consequences of our result for isotonic regression.

Solving bilevel problems with polyhedral constraint set

Solving bilevel problems with polyhedral constraint set

Continuity Results and Error Bounds on Pseudomonotone Vector Variational Inequalities via Scalarization

Continuity (both lower and upper semicontinuities) results of the Pareto/efficient solution mapping for a parametric vector variational inequality with a polyhedral constraint set are established via scalarization approaches, within the framework of strict pseudomonotonicity assumptions. As a direct application, the continuity of the solution mapping to a parametric weak Minty vector variational inequality is also discussed. Furthermore, error bounds for the weak vector variational inequality in terms of two known regularized gap functions are also obtained, under strong pseudomonotonicity assumptions.

Semicontinuity Results on Parametric Vector Variational Inequalities with Polyhedral Constraint Sets

New results on the lower semicontinuity and upper semicontinuity properties of the Pareto solutions to a parametric vector variational inequality with a polyhedral constraint set are obtained by scalarization approaches.

A Result on Vector Variational Inequalities with Polyhedral Constraint Sets

In this note, by using some well-known results on properly efficient solutions of vector optimization problems, we show that the Pareto solution set of a vector variational inequality with a polyhedral constraint set can be expressed as the union of the solution sets of a family of (scalar) variational inequalities.

Lipschitz Continuity of Solutions of Variational Inequalities with a Parametric Polyhedral Constraint

It is proved that the metric projection from a point onto a moving polyhedron is Lipschitz continuous with respect to the perturbations on the right-hand sides of the linear inequalities defining the polyhedron. The property leads to a simple sufficient condition for Lipschitz continuity of a locally unique solution of parametric variational inequalities with a moving polyhedral constraint set. Applications of these results to traffic network equilibrium problems are given in detail.

Admissible sets and feedback control for discrete-time linear dynamical systems with bounded controls and states

For linear discrete-time dynamical systems, necessary and sufficient conditions are given for a control sequence belonging to a polyhedral constraint set, to stabilize the system under the condition that the state is restricted to a polyhedral state constraint set. A variable structure linear state feedback controller, given in terms of the controls at the vertices of the polyhedral state constraint set, is presented.

Optimality conditions for the bilevel programming problem

AbstractThe bilevel programming problem (BLPP) is a sequence of two optimization problems where the constraint region of the first is determined implicitly by the solution to the second. In this article it is first shown that the linear BLPP is equivalent to maximizing a linear function over a feasible region comprised of connected faces and edges of the original polyhedral constraint set. The solution is shown to occur at a vertex of that set. Next, under assumptions of differentiability, first‐order necessary optimality conditions are developed for the more general BLPP, and a potentially equivalent mathematical program is formulated. Finally, the relationship between the solution to this problem and Pareto optimality is discussed and a number of examples given.

Piecewise convex programs

A piecewise convex program is a convex program such that the constraint set can be decomposed in a finite number of closed convex sets, called the cells of the decomposition, and such that on each of these cells the objective function can be described by a continuously differentiable convex function. In a first part, a cutting hyperplane method is proposed, which successively considers the various cells of the decomposition, checks whether the cell contains an optimal solution to the problem, and, if not, imposes a convexity cut which rejects the whole cell from the feasibility region. This elimination, which is basically a dual decomposition method but with an efficient use of the specific structure of the problem is shown to be finitely convergent. The second part of this paper is devoted to the study of some special cases of piecewise convex program and in particular the piecewise quadratic program having a polyhedral constraint set. Such a program arises naturally in stochastic quadratic programming with recourse, which is the subject of the last section.

Technical Note—A Vertex Ranking Procedure for Solving the Linear Fixed-Charge Problem

Murty has suggested that the linear fixed-charge problem can be solved by ranking the vertices of the polyhedral constraint set according to their continuous objective values and then adding the fixed charges to determine an optimal solution. Several problems involving degeneracy and inequality constraints arise when one attempts to implement Murty's method directly. This note describes a method that uses the vertex ranking approach in conjunction with procedures that resolve these difficulties. We also give computational results.