Preconditioned ADMM for a Class of Bilinear Programming Problems

Abstract

We design a novel preconditioned alternating direction method for solving a class of bilinear programming problems, where each subproblem is solved by adding a positive-definite regularization term with a proximal parameter. By the aid of the variational inequality, the global convergence of the proposed method is analyzed and a worst-case O(1/ ) convergence rate in an ergodic sense is established. Several preliminary numerical examples, including the Markowitz portfolio optimization problem, are also tested to verify the performance of the proposed method.

Partial Exactness for the Penalty Function of Biconvex Programming

“…It is ubiquitous nowadays in fields such as control [3][4][5], machine learning [6,7], signal and information processing [8,9], communication [10,11], and also NP-hard problems [12]. The existing research on multi-convex programming mainly solves some very special models [12][13][14][15][16][17][18][19]. These studies all give specific methods for each special model.…”

“…One is the simplex algorithm based on sub-problems, and the other is the alternating direction method. For example, Liang and Bai [15] and Hajinezhad and Shi [16] both proposed the alternating direction method of multipliers (ADMM) algorithm for two special bilinear programming problems, where the extended Lagrangian penalty function uses a square penalty. Furthermore, Charkhgard et al [17] presented a multi-linear programming algorithm using the linear programming algorithm.…”