Related papers: Lasserre hierarchy for large scale polynomial opti…
We consider the problem of approximating numerically the moments and the supports of measures which are invariant with respect to the dynamics of continuous- and discrete-time polynomial systems, under semialgebraic set constraints. First,…
We study quantum algorithms for approximating Lasserre's hierarchy values for polynomial optimization. Let $f,g_1,\ldots,g_m$ be real polynomials in $n$ variables and $f^\star$ the infimum of $f$ over the semialgebraic set $S(g)=\{x:…
We revisit the problem of minimizing a given polynomial $f$ on the hypercube $[-1,1]^n$. Lasserre's hierarchy (also known as the moment- or sum-of-squares hierarchy) provides a sequence of lower bounds $\{f_{(r)}\}_{r \in \mathbb N}$ on the…
This paper proposes a real moment-HSOS hierarchy for complex polynomial optimization problems with real coefficients. We show that this hierarchy provides the same sequence of lower bounds as the complex analogue, yet is much cheaper to…
In this paper, we describe a new method to compute the minimum of a real polynomial function and the ideal defining the points which minimize this polynomial function, assuming that the minimizer ideal is zero-dimensional. Our method is a…
We consider the general polynomial optimization problem $P: f^*=\min \{f(x)\,:\,x\in K\}$ where $K$ is a compact basic semi-algebraic set. We first show that the standard Lagrangian relaxation yields a lower bound as close as desired to the…
The Moment/Sum-of-squares hierarchy provides a way to compute the global minimizers of polynomial optimization problems (POP), at the cost of solving a sequence of increasingly large semidefinite programs (SDPs). We consider large-scale…
In this paper, we address the effective degree bound problem for Lasserre's hierarchy of moment-sum-of-squares (SOS) relaxations in polynomial optimization involving $n$ variables. We assume that the first $n$ equality constraint…
This work proposes a new moment-SOS hierarchy, called CS-TSSOS, for solving large-scale sparse polynomial optimization problems. Its novelty is to exploit simultaneously correlative sparsity and term sparsity by combining advantages of two…
A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set $K$ described by polynomial inequalities and equations. In its full generality it is a non-convex, multi-extremal,…
A polynomial optimization problem (POP) asks for minimizing a polynomial function given a finite set of polynomial constraints (equations and inequalities). This problem is well-known to be hard in general, as it encodes many hard…
A basic closed semialgebraic subset of $\mathbb{R}^{n}$ is defined by simultaneous polynomial inequalities $p_{1}\geq 0,\ldots,p_{m}\geq 0$. We consider Lasserre's relaxation hierarchy to solve the problem of minimizing a polynomial over…
The Lasserre Hierarchy is a set of semidefinite programs which yield increasingly tight bounds on optimal solutions to many NP-hard optimization problems. The hierarchy is parameterized by levels, with a higher level corresponding to a more…
We consider the problem of minimizing a continuous function f over a compact set K. We analyze a hierarchy of upper bounds proposed by Lasserre in [SIAM J. Optim. 21(3) (2011), pp. 864--885], obtained by searching for an optimal probability…
In this survey we consider polynomial optimization problems, asking to minimize a polynomial function over a compact semialgebraic set, defined by polynomial inequalities. This models a great variety of (in general, nonlinear nonconvex)…
In this note, we prove that for homogeneous polynomial optimization on the sphere, if the objective $f$ is generic in the input space, all feasible points satisfying the first order and second order necessary optimality conditions are local…
We address combinatorial problems that can be formulated as minimization of a partially separable function of discrete variables (energy minimization in graphical models, weighted constraint satisfaction, pseudo-Boolean optimization, 0-1…
Motivated by recent increased interest in optimization algorithms for non-convex optimization in application to training deep neural networks and other optimization problems in data analysis, we give an overview of recent theoretical…
This paper studies how to certify the convergence of Lasserre's hierarchy of semidefinite programming relaxations for solving multivariate polynomial optimization. We propose flat truncation as a general certificate for this purpose. Assume…
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…