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 HamiltonJacobi-Bellman inequalities is also characterized. The LPs are then solved numer...

A multivariate polynomial p(x) = p(x1, . . . , xn) is sos-convex if its Hessian H(x) can befactored as H(x) = M (x)M(x) with a possibly nonsquare polynomial matrix M(x). It iseasy to see that sos-convexity is a sufficient condition for convexity of p(x). Moreover, theproblem of deciding sos-convexity of a polynomial can be cast as the feasibility of a semidefiniteprogram, wh...

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 HamiltonJacobi-Bellman inequalities is also characterized. The LPs are then solved numer...

This paper discusses how to find the global minimum of functions that are summations of small polynomials (“small” means involving a small number of variables). Some sparse sum of squares (SOS) techniques are proposed. We compare their computational complexity and lower bounds with prior SOS relaxations. Under certain conditions, we also discuss how to extract the global minimizers from these s...

The Lasserre/Sum-of-Squares (SoS) hierarchy is a systematic procedure for constructing a sequence of increasingly tight semidefinite relaxations. It is known that the hierarchy converges to the 0/1 polytope in n levels and captures the convex relaxations used in the best available approximation algorithms for a wide variety of optimization problems. In this paper we characterize the set of 0/1 ...

In this paper, nonlinear controllers are designed for buck converters via the Sum of Squares (SOS) method based on their discretized bilinear model. We first consider a class control laws in rational funciton. The condition Lyapunov stability theory is converted to an SOS condition, and by using method, stabilizing controller discrete-time system derived ensure that input constraints can also b...

Recently the Mathematical Programming community showed a renewed interest in Hilbert’s Positivstellensatz. The reason for this is that global optimization of polynomials in IR[x1, . . . , xn] is NPhard, while the question whether a polynomial can be written as a sum of squares has tractable aspects. This is due to the fact that Semidefinite Programming can be used to decide in polynomial time (...

Abstract We show that adjustable robust linear programs with affinely box data uncertainties under separable polynomial decision rules admit exact sums of squares (SOS) reformulations. These problems share the same optimal values and a one-to-one correspondence between solutions. A sum representation non-negativity non-convex over plays key role in reformulation. This reformulation allows us to...

Given a k-ary predicate P, a random instance of a constraint satisfaction problem with predicate P (CSP(P)) is a set of m constraints on n variables. Each constraint is P applied to k random literals. Such an instance is unsatisfiable with high probability when m >> n. Refutation is certifying that a given randomly chosen instance is unsatisfiable. This task has applications in cryptography, ha...

Direction of arrival (DOA) estimation using a uniform linear array (ULA) is a classical problem in array signal processing. In this paper, we focus on DOA estimation based on the maximum likelihood (ML) criterion, transform the estimation problem into a novel formulation, named as sum-of-squares (SOS), and then solve it using semidefinite programming (SDP). We first derive the SOS and SDP metho...