Linear Optimization with Cones of Moments and Nonnegative Polynomials

Let A be a finite subset of N^n and R[x]_A be the space of real polynomials whose monomial powers are from A. Let K be a compact basic semialgebraic set of R^n such that R[x]_A contains a polynomial that is positive on K. Denote by P_A(K) the cone of polynomials in R[x]_A that are nonnegative on K. The dual cone of P_A(K) is R_A(K), the set of all A-truncated moment sequences in R^A that admit representing measures supported in K. Our main results are: i) We study the properties of P_A(K) and R_A(K) (like interiors, closeness, duality, memberships), and construct a convergent hierarchy of semidefinite relaxations for each of them. ii) We propose a semidefinite algorithm for solving linear optimization problems with the cones P_A(K) and R_A(K), and prove its asymptotic and finite convergence; a stopping criterion is also given. iii) We show how to check whether P_A(K) and R_A(K) intersect affine subspaces; if they do, we show to get get a point in the intersections; if they do not, we prove certificates for the non-intersecting

A Semidefinite Hierarchy for Containment of Spectrahedra

A spectrahedron is the positivity region of a linear matrix pencil and thus the feasible set of a semidefinite program. We propose and study a hierarchy of sufficient semidefinite conditions to certify the containment of a spectrahedron in another one. This approach comes from applying a moment relaxation to a suitable polynomial optimization formulation. The hierarchical criterion is stronger than a solitary semidefinite criterion discussed earlier by Helton, Klep, and McCullough as well as by the authors. Moreover, several exactness results for the solitary criterion can be brought forward to the hierarchical approach. The hierarchy also applies to the (equivalent) question of checking whether a map between matrix (sub-)spaces is positive. In this context, the solitary criterion checks whether the map is completely positive, and thus our results provide a hierarchy between positivity and complete positivity.Comment: 24 pages, 2 figures; minor corrections; to appear in SIAM J. Opti

Interiors of completely positive cones

A symmetric matrix $A$ is completely positive (CP) if there exists an entrywise nonnegative matrix $B$ such that $A = BB^T$. We characterize the interior of the CP cone. A semidefinite algorithm is proposed for checking interiors of the CP cone, and its properties are studied. A CP-decomposition of a matrix in Dickinson's form can be obtained if it is an interior of the CP cone. Some computational experiments are also presented

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic programming. The recent advances in various topics of modern optimization have also been revamping the area of machine learning. Motivated by the crucial role of optimization theory in the design, analysis, control and operation of real-world systems, this tutorial paper offers a detailed overview of some major advances in this area, namely conic optimization and its emerging applications. First, we discuss the importance of conic optimization in different areas. Then, we explain seminal results on the design of hierarchies of convex relaxations for a wide range of nonconvex problems. Finally, we study different numerical algorithms for large-scale conic optimization problems.Comment: 18 page