Rate analysis of inexact dual first order methods: Application to distributed MPC for network systems

In this paper we propose and analyze two dual methods based on inexact gradient information and averaging that generate approximate primal solutions for smooth convex optimization problems. The complicating constraints are moved into the cost using the Lagrange multipliers. The dual problem is solved by inexact first order methods based on approximate gradients and we prove sublinear rate of convergence for these methods. In particular, we provide, for the first time, estimates on the primal feasibility violation and primal and dual suboptimality of the generated approximate primal and dual solutions. Moreover, we solve approximately the inner problems with a parallel coordinate descent algorithm and we show that it has linear convergence rate. In our analysis we rely on the Lipschitz property of the dual function and inexact dual gradients. Further, we apply these methods to distributed model predictive control for network systems. By tightening the complicating constraints we are also able to ensure the primal feasibility of the approximate solutions generated by the proposed algorithms. We obtain a distributed control strategy that has the following features: state and input constraints are satisfied, stability of the plant is guaranteed, whilst the number of iterations for the suboptimal solution can be precisely determined.Comment: 26 pages, 2 figure

An optimal first-order primal-dual gap reduction framework for constrained convex optimization

We introduce an analysis framework for constructing optimal first-order primal-dual methods for the prototypical constrained convex optimization template. While this class of methods offers scalability advantages in obtaining numerical solutions, they have the disadvantage of producing sequences that are only approximately feasible to the problem constraints. As a result, it is theoretically challenging to compare the efficiency of different methods. To this end, we rigorously prove in the worst-case that the convergence of primal objective residual in first-order primal-dual algorithms must compete with their constraint feasibil- ity convergence, and mathematically summarize this fundamental trade-off. We then provide a heuristic-free analysis recipe for constructing optimal first-order primal-dual algorithms that can obtain a desirable trade-off between the primal objective residual and feasibility gap and whose iteration convergence rates cannot be improved. Our technique obtains a smoothed estimate of the primal-dual gap and drives the smoothness parameters to zero while simultaneously minimizing the smoothed gap using problem first-order oracles

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

Getting Feasible Variable Estimates From Infeasible Ones: MRF Local Polytope Study

This paper proposes a method for construction of approximate feasible primal solutions from dual ones for large-scale optimization problems possessing certain separability properties. Whereas infeasible primal estimates can typically be produced from (sub-)gradients of the dual function, it is often not easy to project them to the primal feasible set, since the projection itself has a complexity comparable to the complexity of the initial problem. We propose an alternative efficient method to obtain feasibility and show that its properties influencing the convergence to the optimum are similar to the properties of the Euclidean projection. We apply our method to the local polytope relaxation of inference problems for Markov Random Fields and demonstrate its superiority over existing methods.Comment: 20 page, 4 figure