Moment and polynomial optimization has received high attention in recent decades. It has
beautiful theory and efficient methods, as well as broad applications for various …

Let S={x∈ Rn∣ g1 (x)≥ 0,…, gm (x)≥ 0} be a basic closed semialgebraic set defined by
real polynomials gi. Putinar's Positivstellensatz says that, under a certain condition stronger …

It is undeniable that geometric ideas have been very important to the foundations of modern
discrete optimization. The influence that geometric algorithms have in optimization was …

This paper studies the so-called biquadratic optimization over unit spheres
x∈R^n,y∈R^m1≦i,k≦n,\,1≦j,l≦mb_ijklx_iy_jx_ky_l, subject to ‖x‖=1, ‖y‖=1. We …

This paper considers polynomial optimization with unbounded sets. We give a
homogenization formulation and propose a hierarchy of Moment-SOS relaxations to solve it …

Given polynomials f (x), gi (x), hj (x), we study how to minimize f (x) on the set S=\left {x ∈ R^
n:\, h_1 (x)= ⋯= h_ m_1 (x)= 0,\g_1 (x) ≧ 0, ..., g_ m_2 (x) ≧ 0\right\}. Let f min be the …

Let f,f_1,...,f_s be n-variate polynomials with rational coefficients of maximum degree D and
let V be the set of common complex solutions of F=(f_1,...,f_s). We give an algorithm which …

This paper proposes tight semidefinite relaxations for polynomial optimization. The
optimality conditions are investigated. We show that generally Lagrange multipliers can be …

In this paper, we consider polynomial optimization with correlative sparsity. We construct
correlatively sparse Lagrange multiplier expressions (CS-LMEs) and propose CS-LME …

In a first contribution, we revisit two certificates of positivity on (possibly non-compact) basic
semialgebraic sets due to Putinar and Vasilescu (CR Acad Sci Ser I Math 328 (6): 495–499 …