Polynomial Convergence of Infeasible-Interior-Point Methods Over Symmetric Cones

Polynomial Convergence of Infeasible-Interior-Point Methods Over Symmetric Cone

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

Stat Optim Inf Comput

In this paper, an improved Interior-Point Method (IPM) for solving symmetric optimization problems is presented. Symmetric optimization (SO) problems are linear optimization problems over symmetric cones. In particular, the method can be efficiently applied to an important instance of SO, a Controlled Tabular Adjustment (CTA) problem which is a method used for Statistical Disclosure Limitation (SDL) of tabular data. The presented method is a full Nesterov-Todd step infeasible IPM for SO. The algorithm converges to |-approximate solution from any starting point whether feasible or infeasible. Each iteration consists of the feasibility step and several centering steps, however, the iterates are obtained in the wider neighborhood of the central path in comparison to the similar algorithms of this type which is the main improvement of the method. However, the currently best known iteration bound known for infeasible short-step methods is still achieved.CC999999/ImCDC/Intramural CDC HHSUnited States/2022-01-01T00:00:00Z34141814PMC820532010747vault:3716

Decomposition-Based Method for Sparse Semidefinite Relaxations of Polynomial Optimization Problems

We consider polynomial optimization problems pervaded by a sparsity pattern. It has been shown in [1, 2] that the optimal solution of a polynomial programming problem with structured sparsity can be computed by solving a series of semidefinite relaxations that possess the same kind of sparsity. We aim at solving the former relaxations with a decompositionbased method, which partitions the relaxations according to their sparsity pattern. The decomposition-based method that we propose is an extension to semidefinite programming of the Benders decomposition for linear programs [3] .Polynomial optimization, Semidefinite programming, Sparse SDP relaxations, Benders decomposition