An inexact projected gradient method with rounding and lifting by nonlinear programming for solving rank-one semidefinite relaxation of polynomial optimization

We consider solving high-order and tight semidefinite programming (SDP) relaxations of nonconvex polynomial optimization problems (POPs) that often admit degenerate rank-one optimal solutions. Instead of solving the SDP alone, we propose a new algorithmic framework that blends local search using the...

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Bibliographic Details
Published in:Mathematical programming Vol. 201; no. 1-2; pp. 409 - 472
Main Authors: Yang, Heng, Liang, Ling, Carlone, Luca, Toh, Kim-Chuan
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2023
Springer
Subjects:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Methods
Numerical Analysis
Theoretical
Semidefinite programming
Degeneracy
Polynomial optimization
90C22
90C55
90C23
90C06
Nonlinear programming
Inexact projected gradient method
Rank-one solutions
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We consider solving high-order and tight semidefinite programming (SDP) relaxations of nonconvex polynomial optimization problems (POPs) that often admit degenerate rank-one optimal solutions. Instead of solving the SDP alone, we propose a new algorithmic framework that blends local search using the nonconvex POP into global descent using the convex SDP. In particular, we first design a globally convergent inexact projected gradient method (iPGM) for solving the SDP that serves as the backbone of our framework. We then accelerate iPGM by taking long, but safeguarded , rank-one steps generated by fast nonlinear programming algorithms. We prove that the new framework is still globally convergent for solving the SDP. To solve the iPGM subproblem of projecting a given point onto the feasible set of the SDP, we design a two-phase algorithm with phase one using a symmetric Gauss–Seidel based accelerated proximal gradient method (sGS-APG) to generate a good initial point, and phase two using a modified limited-memory BFGS (L-BFGS) method to obtain an accurate solution. We analyze the convergence for both phases and establish a novel global convergence result for the modified L-BFGS that does not require the objective function to be twice continuously differentiable. We conduct numerical experiments for solving second-order SDP relaxations arising from a diverse set of POPs. Our framework demonstrates state-of-the-art efficiency, scalability, and robustness in solving degenerate SDPs to high accuracy, even in the presence of millions of equality constraints.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-022-01912-6