Fast primal-dual algorithms via dynamics for linearly constrained convex optimization problems

By time discretization of a primal-dual dynamical system, we propose an inexact primal-dual algorithm, linked to the Nesterov's acceleration scheme, for the linear equality constrained convex optimization problem. We also consider an inexact linearized primal-dual algorithm for the composite problem with linear constrains. Under suitable conditions, we show that these algorithms enjoy fast convergence properties. Finally, we study the convergence properties of the primal-dual dynamical system to better understand the accelerated schemes of the proposed algorithms. We also report numerical experiments to demonstrate the effectiveness of the proposed algorithms

Fast Primal-Dual Gradient Method for Strongly Convex Minimization Problems with Linear Constraints

In this paper we consider a class of optimization problems with a strongly convex objective function and the feasible set given by an intersection of a simple convex set with a set given by a number of linear equality and inequality constraints. A number of optimization problems in applications can be stated in this form, examples being the entropy-linear programming, the ridge regression, the elastic net, the regularized optimal transport, etc. We extend the Fast Gradient Method applied to the dual problem in order to make it primal-dual so that it allows not only to solve the dual problem, but also to construct nearly optimal and nearly feasible solution of the primal problem. We also prove a theorem about the convergence rate for the proposed algorithm in terms of the objective function and the linear constraints infeasibility.Comment: Submitted for DOOR 201

Fast convergence of primal-dual dynamics and algorithms with time scaling for linear equality constrained convex optimization problems

We propose a primal-dual dynamic with time scaling for a linear equality constrained convex optimization problem, which consists of a second-order ODE for the primal variable and a first-order ODE for the dual variable. Without assuming strong convexity, we prove its fast convergence property and show that the obtained fast convergence property is preserved under a small perturbation. We also develop an inexact primal-dual algorithm derived by a time discretization, and derive the fast convergence property matching that of the underlying dynamic. Finally, we give numerical experiments to illustrate the validity of the proposed algorithm

Primal and dual multi-objective linear programming algorithms for linear multiplicative programmes

Multiplicative programming problems (MPPs) are global optimization problems known to be NP-hard. In this paper, we employ algorithms developed to compute the entire set of nondominated points of multi-objective linear programmes (MOLPs) to solve linear MPPs. First, we improve our own objective space cut and bound algorithm for convex MPPs in the special case of linear MPPs by only solving one linear programme in each iteration, instead of two as the previous version indicates. We call this algorithm, which is based on Benson’s outer approximation algorithm for MOLPs, the primal objective space algorithm. Then, based on the dual variant of Benson’s algorithm, we propose a dual objective space algorithm for solving linear MPPs. The dual algorithm also requires solving only one linear programme in each iteration. We prove the correctness of the dual algorithm and use computational experiments comparing our algorithms to a recent global optimization algorithm for linear MPPs from the literature as well as two general global optimization solvers to demonstrate the superiority of the new algorithms in terms of computation time. Thus, we demonstrate that the use of multi-objective optimization techniques can be beneficial to solve difficult single objective global optimization problems