相关论文: Noncommutative polynomial optimization under symme…

The Quadratic Assignment Problem (QAP) is an important discrete optimization instance that encompasses many well-known combinatorial optimization problems, and has applications in a wide range of areas such as logistics and computer vision.…

Non-commutative polynomial optimization (NPO) problems seek to minimize the state average of a polynomial of some operator variables, subject to polynomial constraints, over all states and operators, as well as the Hilbert spaces where…

We establish new convergence rates for the Moment-Sum-of-Squares (Moment-SoS) relaxations for the Generalized Moment Problem (GMP) with countable moment constraints on vectors of measures, under dual optimum attainment, $S$-fullness and…

Understanding and approximating extremal energy states of local Hamiltonians is a central problem in quantum physics and complexity theory. Recent work has focused on developing approximation algorithms for local Hamiltonians, and in…

We present Ca$\Sigma$oS, the first MATLAB software specifically designed for nonlinear sum-of-squares optimization. A symbolic polynomial algebra system allows to formulate parametrized sum-of-squares optimization problems and facilitates…

This paper focuses on the study of a mathematical program with equilibrium constraints, where the objective and the constraint functions are all polynomials. We present a method for finding its global minimizers and global minimum using a…

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…

We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on $n$-vertex graphs are not the…

This paper establishes new Positivstellens\"atze for polynomials that are positive on sets defined by polynomial matrix inequalities (PMIs). We extend the classical Handelman and Krivine-Stengle theorems from the scalar inequality setting…

The minimum sum-of-squares clustering (MSSC), or k-means type clustering, has been recently extended to exploit prior knowledge on the cardinality of each cluster. Such knowledge is used to increase performance as well as solution quality.…

We present a systematic computational framework for generating positive quadrature rules in multiple dimensions on general geometries. A direct moment-matching formulation that enforces exact integration on polynomial subspaces yields…

We give new algorithms based on the sum-of-squares method for tensor decomposition. Our results improve the best known running times from quasi-polynomial to polynomial for several problems, including decomposing random overcomplete…

We present an efficient framework for solving algebraically-constrained global non-convex polynomial optimization problems over subsets of the hypercube. We prove the existence of an equivalent nonlinear reformulation of such problems that…

Noncommutative constraint satisfaction problems (NC-CSPs) are higher-dimensional operator extensions of classical CSPs. Despite their significance in quantum information, their approximability remains largely unexplored. A notable example…

The goal of this paper is to present a systematic method to compute reference dependent positively invariant sets for systems subject to constraints. To this end, we first characterize these sets as level sets of reference dependent…

Model averaging (MA) and ensembling play a crucial role in statistical and machine learning practice. When multiple candidate models are considered, MA techniques can be used to weight and combine them, often resulting in improved…

We consider polynomial optimization problems on Cartesian products of basic compact semialgebraic sets. The solution of such problems can be approximated as closely as desired by hierarchies of semidefinite programming relaxations, based on…

Certifying function nonnegativity is a ubiquitous problem in computational mathematics, with especially notable applications in optimization. We study the question of certifying nonnegativity of signomials based on the recently proposed…

Convexification is a core technique in global polynomial optimization. Currently, there are two main approaches competing in theory and practice: the approach of nonlinear programming and the approach based on positivity certificates from…

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…