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

Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets

The article proves sufficient conditions and necessary conditions for SDP representability of convex sets and convex hulls by proposing a new approach to construct SDP representations. The contributions of this paper are: (i) For bounded SDP representable sets $W_1,\cdots,W_m$, we give an explicit construction of an SDP representation for $conv(\cup_{k=1}^mW_k)$. This provides a technique for … Read more

Semidefinite Representation of Convex Sets

Let $S =\{x\in \re^n:\, g_1(x)\geq 0, \cdots, g_m(x)\geq 0\}$ be a semialgebraic set defined by multivariate polynomials $g_i(x)$. Assume $S$ is convex, compact and has nonempty interior. Let $S_i =\{x\in \re^n:\, g_i(x)\geq 0\}$, and $\bdS$ (resp. $\bdS_i$) be the boundary of $S$ (resp. $S_i$). This paper, as does the subject of semidefinite programming (SDP), concerns … 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