Related papers: Moment/Sum-of-Squares Hierarchy for Complex Polyno…
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…
The Lasserre hierarchy of semidefinite programming (SDP) relaxations is an effective scheme for finding computationally feasible SDP approximations of polynomial optimization over compact semi-algebraic sets. In this paper, we show that,…
We consider min-max optimization problems for polynomial functions, where a multivariate polynomial is maximized with respect to a subset of variables, and the resulting maximal value is minimized with respect to the remaining variables.…
This paper investigates the minimization of the expectation of piecewise polynomial loss functions over Wasserstein balls. This optimization problem often appears as a key sub-problem of distributionally robust optimization problems. We…
Lasserre's hierarchy is a sequence of semidefinite relaxations for solving polynomial optimization problems globally. This paper studies the relationship between optimality conditions in nonlinear programming theory and finite convergence…
We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several…
The behaviour of the moment-sums-of-squares (moment-SOS) hierarchy for polynomial optimal control problems on compact sets has been explored to a large extent. Our contribution focuses on the case of non-compact control sets. We describe a…
Global polynomial optimization is an important tool across applied mathematics, with many applications in operations research, engineering, and physical sciences. In various settings, the polynomials depend on external parameters that may…
In this paper, we propose a new convergent conic programming hierarchy of relaxations involving both semi-definite cone and second-order cone constraints for solving nonconvex polynomial optimization problems to global optimality. The…
Lasserre's moment-SOS hierarchy consists of approximating instances of the generalized moment problem (GMP) with moment relaxations and sums-of-squares (SOS) strenghtenings that boil down to convex semidefinite programming (SDP) problems.…
The prevalence of neural networks in society is expanding at an increasing rate. It is becoming clear that providing robust guarantees on systems that use neural networks is very important, especially in safety-critical applications. A…
We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], for the special case when the feasible set is the unit (hyper)sphere. The…
We present a hierarchy of tractable relaxations to obtain lower bounds on the minimum value of a polynomial over a constraint set defined by polynomial equations. In contrast to previous convex relaxation techniques for this problem, our…
The Lasserre hierarchy is a systematic procedure for constructing a sequence of increasingly tight relaxations that capture the convex formulations used in the best available approximation algorithms for a wide variety of optimization…
We exhibit a convex polynomial optimization problem for which the diagonally-dominant sum-of-squares (DSOS) and the scaled diagonally-dominant sum-of-squares (SDSOS) hierarchies, based on linear programming and second-order conic…
Disjointly constrained multilinear programming concerns the problem of maximizing a multilinear function on the product of finitely many disjoint polyhedra. While maximizing a linear function on a polytope (linear programming) is known to…
This paper studies matrix constrained polynomial optimization. We investigate how to get explicit expressions for Lagrange multiplier matrices from the first order optimality conditions. The existence of these expressions can be shown under…
This paper considers polynomial optimization with unbounded sets. We give a homogenization formulation and propose a hierarchy of Moment-SOS relaxations to solve it. Under the assumptions that the feasible set is closed at infinity and the…
Given a compact parameter set $Y\subset R^p$, we consider polynomial optimization problems $(P_y$) on $R^n$ whose description depends on the parameter $y\inY$. We assume that one can compute all moments of some probability measure $\phi$ on…
We consider optimization problems with polynomial inequality constraints in non-commuting variables. These non-commuting variables are viewed as bounded operators on a Hilbert space whose dimension is not fixed and the associated polynomial…