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

Renbo Zhao

TL;DR

The Generalized Multiplicative Gradient 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 is developed and analyzed.

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.

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

TL;DR

The Generalized Multiplicative Gradient 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 is developed and analyzed.

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 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.
Paper Structure (28 sections, 29 theorems, 93 equations, 1 table, 1 algorithm)

This paper contains 28 sections, 29 theorems, 93 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Class
  3. Another Application of \ref{['eq:P']}
  4. Main Contributions
  5. Notations
  6. Background on Euclidean Jordan algebra
  7. Some Convex Analytic Properties of Legendre and LH Functions
  8. The GMG Method
  9. Additional Assumptions and Their Implications
  10. The GMG Method
  11. Convergence Rate Analysis
  12. Function Classes Satisfying Assumptions \ref{['assump:A']} and \ref{['assump:F']}
  13. First Scenario
  14. Background on hyperbolic polynomials and hyperbolicity cones
  15. Definitions of $f$ and $\mathsf{A}$
  16. ...and 13 more sections

Key Result

Lemma 3.1

Let $h:\mathbb{Y}\to \bar{\mathbb{R}}$ be a proper, closed and convex function. Then $h$ is Legendre if and only if $h^*$ is Legendre. Furthermore, $\nabla h:\mathsf{int}\,\mathsf{dom}\, h\to \mathsf{int}\,\mathsf{dom}\, h^*$ is a homeomorphism, whose inverse $(\nabla h)^{-1} = \nabla h^*$.

Theorems & Definitions (56)

  • Definition 1.1: Legendre function; Bauschke_00
  • Definition 1.2: Logarithmically homogeneous function
  • Lemma 3.1: Rock_70
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • proof
  • Proposition 4.1
  • ...and 46 more