The Generalized Multiplicative Gradient Method for A Class of Convex Optimization Problems Over Symmetric Cones

TL;DR

This paper introduces the Generalized Multiplicative Gradient (GMG) method for convex optimization over symmetric cones, achieving an $O(1/k)$ convergence rate without requiring Lipschitz gradients, and demonstrates its efficiency across various applications.

Contribution

The paper develops the GMG method with a novel convergence analysis and compares its computational complexity favorably against existing methods in multiple applications.

Findings

GMG achieves $O(1/k)$ convergence rate.

GMG outperforms or matches other methods in computational complexity.

New theoretical bounds on curvature and inequalities in Euclidean Jordan Algebras.

Abstract

We develop and analyze the Generalized Multiplicative Gradient (GMG) method for solving a class of convex optimization problems over symmetric cones, where the objective function does not have Lipschitz gradient over the feasible region. This problem class includes several applications, such as positron emission tomography, D-optimal design, quantum state tomography and the dual problem of Nesterov's convex relaxation of the boolean quadratic problem. We show that the GMG method has a convergence rate of $O(1/k)$ in terms of the objective gap. Our analysis of the convergence rate is rather unconventional, and to that end, we establish several results that may be of independent interest, such as a curvature bound of the Legendre and logarithmically-homogeneous functions, and a Cauchy-Schwarz inequality in representative simple Euclidean Jordan Algebras. Finally, we compare the computational complexity of the GMG method with three other related first-order methods on several important applications, and we show that under certain mild assumptions, the GMG method achieves the best (or almost the best) computational complexities on all of the applications.