A new look at nonnegativity on closed sets and polynomial optimization

We first show that a continuous function f is nonnegative on a closed set $K\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $d\nu =fd\mu$ with support equal to K, are all positive semidefinite (if $K$ is compact $\mu$ is an arbitrary finite Borel measure with support equal to K. In particular, we obtain a convergent explicit hierarchy of semidefinite (outer) approximations with {\it no} lifting, of the cone of nonnegative polynomials of degree at most $d$. Wen used in polynomial optimization on certain simple closed sets \K (like e.g., the whole space $\R^n$, the positive orthant, a box, a simplex, or the vertices of the hypercube), it provides a nonincreasing sequence of upper bounds which converges to the global minimum by solving a hierarchy of semidefinite programs with only one variable. This convergent sequence of upper bounds complements the convergent sequence of lower bounds obtained by solving a hierarchy of semidefinite relaxations

ACP-EEC CONSULTATIVE ASSEMBLY JOINT COMMITTEE RESOLUTION on cultural cooperation between the ACP States and the EEC.

We consider the problem of minimizing a continuous function f over a compact set K. We compare the hierarchy of upper bounds proposed by Lasserre [Lasserre JB (2011) A new look at nonnegativity on closed sets and polynomial optimization. SIAM J. Optim. 21(3):864–885] to bounds that may be obtained from simulated annealing. We show that, when f is a polynomial and K a convex body, this comparison yields a faster rate of convergence of the Lasserre hierarchy than what was previously known in the literature

New approximations for the cone of copositive matrices and its dual

We provide convergent hierarchies for the cone C of copositive matrices and its dual, the cone of completely positive matrices. In both cases the corresponding hierarchy consists of nested spectrahedra and provide outer (resp. inner) approximations for C (resp. for its dual), thus complementing previous inner (resp. outer) approximations for C (for the dual). In particular, both inner and outer approximations have a very simple interpretation. Finally, extension to K-copositivity and K-complete positivity for a closed convex cone K, is straightforward.Comment: 8

On the complexity of Putinar's Positivstellensatz

We prove an upper bound on the degree complexity of Putinar's Positivstellensatz. This bound is much worse than the one obtained previously for Schm\"udgen's Positivstellensatz but it depends on the same parameters. As a consequence, we get information about the convergence rate of Lasserre's procedure for optimization of a polynomial subject to polynomial constraints

Nonlinear control synthesis by convex optimization

A stability criterion for nonlinear systems, recently derived by the third author, can be viewed as a dual to Lyapunov's second theorem. The criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state-space and flows along the system trajectories toward the equilibrium. The new criterion has a remarkable convexity property, which in this note is used for controller synthesis via convex optimization. Recent numerical methods for verification of positivity of multivariate polynomials based on sum of squares decompositions are used

Strong nonnegativity and sums of squares on real varieties

Motivated by scheme theory, we introduce strong nonnegativity on real varieties, which has the property that a sum of squares is strongly nonnegative. We show that this algebraic property is equivalent to nonnegativity for nonsingular real varieties. Moreover, for singular varieties, we reprove and generalize obstructions of Gouveia and Netzer to the convergence of the theta body hierarchy of convex bodies approximating the convex hull of a real variety.Comment: 11 pages, 4 figure