In 2004 Newman suggested a semidefinite programming relaxation for the Linear Ordering Problem (LOP) that is related to the semidefinite program used in the Goemans-Williamson algorithm to approximate the Max Cut problem. Her model is based on the observation that linear orderings can be fully described by a series of cuts. Newman shows that her … Read more
We propose an exact algorithm for the Target Visitation Problem (TVP). The (TVP) is a composition of the Linear Ordering Problem and the Traveling Salesman Problem. It has several military and non-military applications, where two important, often competing factors are the overall distance traveled (e.g. by an unmanned aerial vehicle) and the visiting sequence of … Read more
The study of matrix inequalities in a dimension-free setting is in the realm of free real algebraic geometry (RAG). In this paper we investigate constrained trace and eigenvalue optimization of noncommutative polynomials. We present Lasserre’s relaxation scheme for trace optimization based on semidefinite programming (SDP) and demonstrate its convergence properties. Finite convergence of this relaxation … Read more
The use of graph databases for social networks, images, web links, pathways and so on, has been increasing at a fast pace and promotes the need for efficient graph query processing on such databases. In this study, we discuss graph query processing — referred to as graph matching — and an inherent optimization problem known … Read more
We consider the problem of minimizing a continuous function f over a compact set K. We analyze a hierarchy of upper bounds proposed by Lasserre in [SIAM J. Optim. 21(3) (2011), pp. 864-885], obtained by searching for an optimal probability density function h on K which is a sum of squares of polynomials, so that … Read more
We study the Lov\'{a}sz-Schrijver lift-and-project operator ($\LS_+$) based on the cone of symmetric, positive semidefinite matrices, applied to the fractional stable set polytope of graphs. The problem of obtaining a combinatorial characterization of graphs for which the $\LS_+$-operator generates the stable set polytope in one step has been open since 1990. We call these graphs … Read more
We propose a trust-region method for nonlinear semidefinite programs with box-constraints. The penalty barrier method can handle this problem, but the size of variable matrices available in practical time is restricted to be less than 500. We develop a trust-region method based on the approach of Coleman and Li (1996) that utilizes the distance to … Read more
We address the inverse problem of Lagrangian identification based on trajectories in the context of nonlinear optimal control. We propose a general formulation of the inverse problem based on occupation measures and complementarity in linear programming. The use of occupation measures in this context offers several advantages from the theoretical, numerical and statistical points of … Read more
Converging hierarchies of finite-dimensional semi-definite relaxations have been proposed for state-constrained optimal control problems featuring oscillation phenomena, by relaxing controls as Young measures. These semi-definite relaxations were later on extended to optimal control problems depending linearly on the control input and typically featuring concentration phenomena, interpreting the control as a measure of time with a … Read more
Given a compact semialgebraic set S of R^n and a polynomial map f from R^n to R^m, we consider the problem of approximating the image set F = f(S) in R^m. This includes in particular the projection of S on R^m for n greater than m. Assuming that F is included in a set B … Read more