A Tight SDP Relaxation for the Cubic-Quartic Regularization Problem

TL;DR

This paper introduces a practical and efficient semidefinite programming relaxation method to compute global minimizers of the cubic-quartic regularization problem, a key subproblem in nonlinear optimization.

Contribution

It proposes a tight SDP relaxation approach for the CQR problem, providing conditions for tightness, and develops an algorithm to find all global minimizers.

Findings

SDP relaxation is tight under certain conditions.

All nonzero global minimizers share the same norm.

Numerical experiments confirm effectiveness and efficiency.

Abstract

This paper studies how to compute global minimizers of the cubic-quartic regularization (CQR) problem \[ \min_{s \in \mathbb{R}^n} \quad f_0+g^Ts+\frac{1}{2}s^THs+\frac{\beta}{6} \| s \|^3+\frac{\sigma}{4} \| s \|^4, \] where $f_0$ is a constant, $g$ is an $n$-dimensional vector, $H$ is a $n$-by-$n$ symmetric matrix, and $\| s \|$ denotes the Euclidean norm of $s$. The parameter $\sigma \ge 0$ 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]. However, a practical method for computing global minimizers of the CQR problem still remains elusive. To this end, we propose a semidefinite programming (SDP) relaxation method for solving the CQR problem globally. First, we show that our SDP relaxation is tight if and only if $\| s^* \| ( \beta + 3 \sigma \| s^* \|) \ge 0$ holds for a global minimizer $s^*$. In particular, if either $\beta \ge 0$ 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 length 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, providing the first practical method for globally solving the CQR problem.