Algorithms for the continuous nonlinear resource allocation problem---new implementations and numerical studies

Patriksson (2008) provided a then up-to-date survey on the continuous,separable, differentiable and convex resource allocation problem with a single resource constraint. Since the publication of that paper the interest in the problem has grown: several new applications have arisen where the problem at hand constitutes a subproblem, and several new algorithms have been developed for its efficient solution. This paper therefore serves three purposes. First, it provides an up-to-date extension of the survey of the literature of the field, complementing the survey in Patriksson (2008) with more then 20 books and articles. Second, it contributes improvements of some of these algorithms, in particular with an improvement of the pegging (that is, variable fixing) process in the relaxation algorithm, and an improved means to evaluate subsolutions. Third, it numerically evaluates several relaxation (primal) and breakpoint (dual) algorithms, incorporating a variety of pegging strategies, as well as a quasi-Newton method. Our conclusion is that our modification of the relaxation algorithm performs the best. At least for problem sizes up to 30 million variables the practical time complexity for the breakpoint and relaxation algorithms is linear

Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems

Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. Deriving efficient strategies which jointly brings into play the primal and the dual problems is however a more recent idea which has generated many important new contributions in the last years. These novel developments are grounded on recent advances in convex analysis, discrete optimization, parallel processing, and non-smooth optimization with emphasis on sparsity issues. In this paper, we aim at presenting the principles of primal-dual approaches, while giving an overview of numerical methods which have been proposed in different contexts. We show the benefits which can be drawn from primal-dual algorithms both for solving large-scale convex optimization problems and discrete ones, and we provide various application examples to illustrate their usefulness