Related papers: Block Factor-width-two Matrices and Their Applicat…
Many fundamental low-rank optimization problems, such as matrix completion, phase synchronization/retrieval, power system state estimation, and robust PCA, can be formulated as the matrix sensing problem. Two main approaches for solving…
We exhibit families of $4$-CNF formulas over $n$ variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) $d$ but require SOS proofs of size $n^{\Omega(d)}$ for values of $d = d(n)$ from constant all the…
In this paper we study the fundamental problems of maximizing a continuous non-monotone submodular function over the hypercube, both with and without coordinate-wise concavity. This family of optimization problems has several applications…
We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert--Artin, Reznick, Putinar, and Putinar--Vasilescu Positivstellens\"atze. First, we…
In optimization problems involving smooth functions and real and matrix variables, that contain matrix semidefiniteness constraints, consider the following change of variables: Replace the positive semidefinite matrix $X \in \mathbb{S}^d$,…
The dynamics of many systems from physics, economics, chemistry, and biology can be modelled through polynomial functions. In this paper, we provide a computational means to find positively invariant sets of polynomial dynamical systems by…
In this work, we show how the eigenstructures of summands are related to that of the sum. In particular, we show that the sum of two positive semidefinite matrices can be written as the inner product of two block matrices $\mathbf{C} =…
We extend the existing skew polynomial representations of matrix algebras which are direct sum of matrix spaces over division rings. In this representation, the sum-rank distance between two tuples of matrices is captured by a weight…
This paper studies the matrix Moment-SOS hierarchy for solving polynomial matrix optimization. Our first result is to show the finite convergence of this hierarchy, if the nondegeneracy condition, strict complementarity condition and second…
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…
Finding the $r\times r$ submatrix of maximum volume of a matrix $A\in\mathbb R^{n\times n}$ is an NP hard problem that arises in a variety of applications. We propose a new greedy algorithm of cost $\mathcal O(n)$, for the case $A$…
Advanced optimization algorithms such as Newton method and AdaGrad benefit from second order derivative or second order statistics to achieve better descent directions and faster convergence rates. At their heart, such algorithms need to…
We consider several classes of highly important semidefinite optimization problems that involve both a convex objective function (smooth or nonsmooth) and additional linear or nonlinear smooth and convex constraints, which are ubiquitous in…
In this paper, we propose a new Semi-Nonnegative Matrix Factorization method for 2-dimensional (2D) data, named TS-NMF. It overcomes the drawback of existing methods that seriously damage the spatial information of the data by converting 2D…
To prove that a polynomial is nonnegative on R^n one can try to show that it is a sum of squares of polynomials (SOS). The latter problem is now known to be reducible to a semidefinite programming (SDP) computation much faster than…
Some sum of squares (SOS) polynomials admit decomposition certificates, or positive semidefinite Gram matrices, with additional structure. In this work, we use the structure of Gram matrices to relate the representation theory of $SL(2)$ to…
This paper is devoted to the stability analysis of spatially interconnected systems (SISs) via the sum-of-squares (SOS) decomposition of positive trigonometric polynomials. For each spatial direction of SISs, three types of interconnected…
This work presents a sum-of-squares (SOS) based framework to perform data-driven stabilization and robust control tasks on discrete-time linear systems where the full-state observations are corrupted by L-infinity bounded input,…
The second-order cone (SOC) 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…
Polynomial optimization encompasses a broad class of problems in which both the objective function and constraints are polynomial functions of the decision variables. In recent years, a substantial body of research has focused on…