The Moment-Sos Hierarchy

Abstract

The Moment-SOS hierarchy initially introduced in optimization in 2000, is based on the theory of the K-moment problem and its dual counterpart, polynomials that are positive on K. It turns out that this methodology can be also applied to solve problems with positivity constraints "f (x) ≥ 0 for all x ∈ K" and/or linear constraints on Borel measures. Such problems can be viewed as specific instances of the "Generalized Problem of Moments" (GPM) whose list of important applications in various domains is endless. We describe this methodology and outline some of its applications in various domains.1991 Mathematics Subject Classification. 90C26 90C22 90C27 65K05 14P10 44A60.

Semi-algebraic Approximation Using Christoffel–Darboux Kernel

“…It turns out that it can also be used for effective computation as it fits perfectly the LP-based methodology described in [16] and the Moment-SOS (polynomial sums of squares) methodology described in e.g. [23,21,22]. In the latter methodology one may thus approximate the optimal solution µ of the measure-valued formulation by solving a hierarchy of semidefinite relaxations of the problem, whose size increases with d; see e.g.…”

“…More generally, moment information about the unknown function f in the format of Problem 1 is available when applying the Moment-SOS hierarchy [22] to solve Generalized Moment Problems where the involved Borel measures are Young measures. The necessary moment information is also given when considering empirical measures [31,25,32] if input data points lie on the graph of an unknown function f (e.g., as is the case in interpolation).…”

The Saddle Point Problem of Polynomials

“…This section reviews some basics in polynomial optimization. We refer to [9,33,34,36,37,57] for the books and surveys in this field.…”

Convex generalized Nash equilibrium problems and polynomial optimization

“…They are also useful for solving truncated moment problems [21,43] and tensor decompositions [44,45]. We refer to [31,32,34,35,37,42,50] for more references about polynomial optimization and moment problems.…”

“…We discuss how to solve the polynomial optimization problems in Algorithm 5.1, by using the Moment-SOS hierarchy of semidefinite relaxations [29,31,32,34,35]. We refer to the notation in Sects.…”