The global weak sharp minima with explicit exponents in polynomial vector optimization problems

In this paper we discuss the global weak sharp minima property for vector optimization problems with polynomial data. Exploiting the imposed polynomial structure together with tools of variational analysis and a quantitative version of \L ojasiewicz’s gradient inequality due to D’Acunto and Kurdyka, we establish the H\”older type global weak sharp minima with explicitly calculated … Read more

Discriminants and Nonnegative Polynomials

For a semialgebraic set K in R^n, let P_d(K) be the cone of polynomials in R^n of degrees at most d that are nonnegative on K. This paper studies the geometry of its boundary. When K=R^n and d is even, we show that its boundary lies on the irreducible hypersurface defined by the discriminant of … Read more

Sum of Squares Method for Sensor Network Localization

We formulate the sensor network localization problem as finding the global minimizer of a quartic polynomial. Then sum of squares (SOS) relaxations can be applied to solve it. However, the general SOS relaxations are too expensive to implement for large problems. Exploiting the special features of this polynomial, we propose a new structured SOS relaxation, … Read more

Computing the stability number of a graph via linear and semidefinite programming

We study certain linear and semidefinite programming lifting approximation schemes for computing the stability number of a graph. Our work is based on, and refines De Klerk and Pasechnik’s approach to approximating the stability number via copositive programming (SIAM J. Optim. 12 (2002), 875–892). We provide a closed-form expression for the values computed by the … Read more

PHoM – a Polyhedral Homotopy Continuation Method for Polynomial Systems

PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedral-linear homotopy functions, based on the polyhedral homotopy theory, from input data for a given system of polynomial equations $\f(\x) … Read more