相关论文: Rounding Sum-of-Squares Relaxations

We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by…

We present a technique for reducing the computational requirements by several orders of magnitude in the evaluation of semidefinite relaxations for bounding the set of quantum correlations arising from finite-dimensional Hilbert spaces. The…

We introduce the concept of disjunctive sum of squares for certifying nonnegativity of polynomials. Unlike the popular sum of squares approach where nonnegativity is certified by a single algebraic identity, the disjunctive sum of squares…

In this paper, we explore the merits of various algorithms for polynomial optimization problems, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation…

We consider the optimization of pairwise objective functions, i.e., objective functions of the form $H(\mathbf{x}) = H(x_1,\ldots,x_N) = \sum_{1\leq i<j \leq N} H_{ij}(x_i,x_j)$ for $x_i$ in some continuous state spaces $\mathcal{X}_i$.…

We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre hierarchy. For binary polynomial optimization problems of degree $2d$ and an odd number of variables $n$, we prove that $\frac{n+2d-1}{2}$ levels of the…

We consider semidefinite programs (SDPs) of size n with equality constraints. In order to overcome scalability issues, Burer and Monteiro proposed a factorized approach based on optimizing over a matrix Y of size $n$ by $k$ such that $X =…

Semidefinite programs (SDPs) are standard convex problems that are frequently found in control and optimization applications. Interior-point methods can solve SDPs in polynomial time up to arbitrary accuracy, but scale poorly as the size of…

We consider the global optimization of nonconvex quadratic programs and mixed-integer quadratic programs. We present a family of convex quadratic relaxations which are derived by convexifying nonconvex quadratic functions through…

The problem of minimizing a polynomial over a set of polynomial inequalities is an NP-hard non-convex problem. Thanks to powerful results from real algebraic geometry, one can convert this problem into a nested sequence of…

Consider the problem of minimizing a polynomial $f$ over a compact semialgebraic set ${\mathbf{X} \subseteq \mathbb{R}^n}$. Lasserre introduces hierarchies of semidefinite programs to approximate this hard optimization problem, based on…

In this work, we propose a trajectory optimization approach for robot navigation in cluttered 3D environments. We represent the robot's geometry as a semialgebraic set defined by polynomial inequalities such that robots with general shapes…

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…

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…

We present a new algorithm for solving a polynomial program P based on the recent "joint + marginal" approach of the first author for, parametric optimization. The idea is to first consider the variable x1 as a parameter and solve the…

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…

Motivated by stability analysis of large scale power systems, we describe how the Lasserre (moment-sums of squares, SOS) hierarchy can be used to generate outer approximations of the region of attraction (ROA) of sparse polynomial…

For many applications in signal processing and machine learning, we are tasked with minimizing a large sum of convex functions subject to a large number of convex constraints. In this paper, we devise a new random projection method (RPM) to…

We propose a novel non-negative spherical relaxation for optimization problems over binary matrices with injectivity constraints, which in particular has applications in multi-matching and clustering. We relax respective binary matrix…

We present a novel iterative algorithm for approximating the linear least squares solution with low complexity. After a motivation of the algorithm we discuss the algorithm's properties including its complexity, and we present theoretical…