A Tight SDP Relaxation for the Cubic-Quartic Regularization Problem

• 成果类型: Article; Early Access
• 署名作者: Zhou, Jinling; Liu, Xin; Nie, Jiawang; Tang, Xindong
• 署名单位: Xiangtan University; Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS; Chinese Academy of Sciences; University of Chinese Academy of Sciences, CAS; University of California System; University of California San Diego; Hong Kong Baptist University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-026-02413-6
• 发表日期: 2026-08-21
• 关键词: regularization; polynomial; semidefinite program; Tight relaxation; CASE EVALUATION COMPLEXITY; optimization; newtons; squares

摘要

This paper studies how to compute global minimizers of the cubic-quartic regularization (CQR) problem where f0 is a constant, g is an n-dimensional vector, H is an n-by-n symmetric matrix, and & Vert;s & Vert; denotes the Euclidean norm of s. The parameter sigma is nonnegative while beta can have any sign. The CQR problem arises as a critical subproblem for getting efficient regularization methods for solving unconstrained nonlinear optimization. Its properties are recently well studied by Cartis and Zhu [cubic-quartic regularization models for solving polynomial subproblems in third-order tensor methods, Math. Program, 2025]. We propose a structured semidefinite programming (SDP) relaxation method for solving the CQR problem globally. The SDP relaxation has only three symmetric positive semidefinite matrix variables of sizes (n+1) -by- (n+1) , 3-by-3 and 2-by-2 respectively. We show that our SDP relaxation is tight if and only if & Vert;s & lowast;& Vert;(beta+3 sigma & Vert;s & lowast;& Vert;)>= 0 holds for a global minimizer s & lowast; . When s & lowast;not equal 0 , this aligns with the sufficient global optimality condition beta+3 sigma & Vert;s & lowast;& Vert;>= 0 given by Cartis and Zhu. In particular, if either beta >= 0 or H has a nonpositive eigenvalue, then the SDP relaxation is shown to be tight. Second, we show that all nonzero global minimizers have the same Euclidean norm for the tight case. Third, we give an algorithm to detect tightness and to obtain the set of all global minimizers. Numerical experiments demonstrate that our SDP relaxation method is both effective and computationally efficient. This paper gives a polynomial time algorithm for solving the CQR problem globally, under the sufficient global optimality condition.