A Novel Unified Approach to Invariance in Control

In this paper, we propose a novel, unified, general approach to investigate sufficient and necessary conditions under which four types of convex sets, polyhedra, polyhedral cones, ellipsoids and Lorenz cones, are invariant sets for a linear continuous or discrete dynamical system. In proving invariance of ellipsoids and Lorenz cones for discrete systems, instead of the … Read more

Active-set prediction for interior point methods using controlled perturbations

We propose the use of controlled perturbations to address the challenging question of optimal active-set prediction for interior point methods. Namely, in the context of linear programming, we consider perturbing the inequality constraints/bounds so as to enlarge the feasible set. We show that if the perturbations are chosen appropriately, the solution of the original problem … Read more

Coordinate shadows of semi-definite and Euclidean distance matrices

We consider the projected semi-definite and Euclidean distance cones onto a subset of the matrix entries. These two sets are precisely the input data defining feasible semi-definite and Euclidean distance completion problems. We characterize when these sets are closed, and use the boundary structure of these two sets to elucidate the Krislock-Wolkowicz facial reduction algorithm. … Read more

Two-Term Disjunctions on the Second-Order Cone

Balas introduced disjunctive cuts in the 1970s for mixed-integer linear programs. Several recent papers have attempted to extend this work to mixed-integer conic programs. In this paper we study the structure of the convex hull of a two-term disjunction applied to the second-order cone, and develop a methodology to derive closed-form expressions for convex inequalities … Read more

Copositivity-based approximations for mixed-integer fractional quadratic optimization

We propose a copositive reformulation of the mixed-integer fractional quadratic problem (MIFQP) under general linear constraints. This problem class arises naturally in many applications, e.g., for optimizing communication or social networks, or studying game theory problems arising from genetics. It includes several APX-hard subclasses: the maximum cut problem, the $k$-densest subgraph problem and several of … Read more

An LP-based Algorithm to Test Copositivity

A symmetric matrix is called copositive if it generates a quadratic form taking no negative values over the nonnegative orthant, and the linear optimization problem over the set of copositive matrices is called the copositive programming problem. Recently, many studies have been done on the copositive programming problem (see, for example, \cite{aDUR10, aBOMZE12}). Among others, … Read more

Approximating Pareto Curves using Semidefinite Relaxations

We consider the problem of constructing an approximation of the Pareto curve associated with the multiobjective optimization problem $\min_{x \in S} \{(f_1(x),f_2(x))\}$, where $f_1$ and $f_2$ are two conflicting positive polynomial criteria and $S \subset R^n$ is a compact basic semialgebraic set. We provide a systematic numerical scheme to approximate the Pareto curve. We start … Read more

Modal occupation measures and LMI relaxations for nonlinear switched systems control

This paper presents a linear programming approach for the optimal control of nonlinear switched systems where the control is the switching sequence. This is done by introducing modal occupation measures, which allow to relax the problem as a primal linear programming (LP) problem. Its dual linear program of Hamilton-Jacobi-Bellman inequalities is also characterized. The LPs … Read more

Considering Copositivity Locally

Let $A$ be an element of the copositive cone $\mathcal{COP}^n$. A zero $\mathbf{u}$ of $A$ is a nonnegative vector whose elements sum up to one and such that $\mathbf{u}^TA\mathbf{u} = 0$. The support of $\mathbf{u}$ is the index set $\mathrm{supp}\mathbf{u} \subset \{1,\dots,n\}$ corresponding to the nonzero entries of $\mathbf{u}$. A zero $\mathbf{u}$ of $A$ is … Read more

Robust Stable Payoff Distribution in Stochastic Cooperative Games

Cooperative games with transferable utilities belong to a branch of game theory where groups of players can enter into binding agreements and form coalitions in order to jointly achieve some objectives. In a cooperative setting, one of the most important questions to address is how to establish a payoff distribution among the players in such … Read more