Double Smoothing Technique for Large-Scale Linearly Constrained Convex Optimization

Abstract

In this paper, we propose an efficient approach for solving a class of large-scale convex optimization problems. The problem we consider is the minimization of a convex function over a simple (possibly infinite-dimensional) convex set, under the additional constraint Au ∈ T , where A is a linear operator and T is a convex set whose dimension is small compared to the dimension of the feasible region. In our approach, we dualize the linear constraints, solve the resulting dual problem with a purely dual gradient-type method and show how to reconstruct an approximate primal solution. Because the linear constraints have been dualized, the dual objective function typically becomes separable, and therefore easy to compute. In order to accelerate our scheme, we introduce a novel double smoothing technique that involves regularization of the dual problem to allow the use of a fast gradient method. As a result, we obtain a method with complexity O(1 ln 1) gradient iterations, where is the desired accuracy for the primal-dual solution. Our approach covers, in particular, optimal control problems with a trajectory governed by a system of linear differential equations, where the additional constraints can, for example, force the trajectory to visit some convex sets at certain moments in time.

Double smoothing for time-varying distributed multiuser optimization

“…This last property is actually very important when (as in our case) at each discrete time the regularized problem is changing due to its time-varying nature. For further details we refer to the original works on (time-invariant) regularization and double smoothing techniques [2], [17].…”

Robust Principal Component Analysis Based on Low-Rank and Block-Sparse Matrix Decomposition

“…Secondly, we compare to alternatives in the literature; the basic ingredients involved in TFOCS are well-known in the optimization community, and there are many variants and applications. The TFOCS algorithm also motivated [45], which promotes an alternative approach that smooths both the primal and the dual. The TFOCS algorithm also motivated [45], which promotes an alternative approach that smooths both the primal and the dual.…”

“…Another option is the double-smoothing technique proposed by [45]. This is the approach most similar with our own.…”

A dynamic game behavior: Demand side management based on utility maximization with renewable energy and storage integration

“…where is a proximity function [11]of a given nonempty, closed and convex set . It is continuous, strongly convex with a convexity parameter 0 and .…”