Related papers: Strengthening SONC Relaxations with Constraints De…
The second-order cone is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a…
We develop a general and unconditional framework for certifying the global nonnegativity of multivariate integer polynomials; based on rewriting them as sum of squares modulo their gradient ideals. We remove the two structural assumptions…
We consider a variational convex relaxation of a class of optimal partitioning and multiclass labeling problems, which has recently proven quite successful and can be seen as a continuous analogue of Linear Programming (LP) relaxation…
In this paper, we introduce a significant extension, called scenario with certificates (SwC), of the so-called scenario approach for uncertain optimization problems. This extension is motivated by the observation that in many control…
We propose an approach to trajectory optimization for piecewise polynomial systems based on the recently proposed graphs of convex sets framework. We instantiate the framework with a convex relaxation of optimal control based on occupation…
We study the problem of minimizing a multivariate polynomial function over the unit hypercube. By representing the polynomial through a hypergraph and exploiting its sparsity structure, we establish a new sufficient condition under which…
This paper studies how to solve semi-infinite polynomial programming (SIPP) problems by semidefinite relaxation method. We first introduce two SDP relaxation methods for solving polynomial optimization problems with finitely many…
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 studies the copositive optimization problem whose objective is a sparse polynomial, with linear constraints over the nonnegative orthant. We propose sparse Moment-SOS relaxations to solve it. Necessary and sufficient conditions…
Neural networks offer a computationally efficient approximation of model predictive control, but they lack guarantees on the resulting controlled system's properties. Formal certification of neural networks is crucial for ensuring safety,…
This paper offers a direct data-driven approach for learning robust control barrier certificates (R-CBCs) and robust safety controllers (R-SCs) for discrete-time input-affine polynomial systems with unknown dynamics under…
In this paper, we present a computational approach to certify almost sure reachability for discrete-time polynomial stochastic systems by turning drift--variant criteria into sum-of-squares (SOS) programs solved with standard semidefinite…
This work proposes a method for solving linear stochastic optimal control (SOC) problems using sum of squares and semidefinite programming. Previous work had used polynomial optimization to approximate the value function, requiring a high…
We study the maximization of sums of heterogeneous quadratic forms over the Stiefel manifold, a nonconvex problem that arises in several modern signal processing and machine learning applications such as heteroscedastic probabilistic…
Estimating spectral gaps of quantum many-body Hamiltonians is a highly challenging computational task, even under assumptions of locality and translation-invariance. Yet, the quest for rigorous gap certificates is motivated by their broad…
We provide two hybrid numeric-symbolic optimization algorithms, computing exact sums of nonnegative circuits (SONC) and sums of arithmetic-geometric-exponentials (SAGE) decompositions. Moreover, we provide a hybrid numeric-symbolic decision…
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…
The reduction of constraints to obtain minimal representations of sets is a very common problem in many engineering applications. While well-established methodologies exist for the case of linear constraints, the problem of how to detect…
Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. The computational complexity of VCSPs depends on the set of allowed cost functions in the input. Recently, the computational…
Certifying nonnegativity of polynomials is a well-known NP-hard problem with direct applications spanning non-convex optimization, control, robotics, and beyond. A sufficient condition for nonnegativity is the Sum of Squares (SOS) property,…