Valentin Nedelcu

Computational Complexity of Inexact Gradient Augmented Lagrangian Methods: Application to Constrained MPC

We study the computational complexity certification of inexact gradient augmented Lagrangian methods for solving convex optimization problems with complicated constraints. We solve the augmented Lagrangian dual problem that arises from the relaxation of complicating constraints with gradient and fast gradient methods based on inexact first order information. Moreover, since the exact solution of the augmented Lagrangian primal problem is hard to compute in practice, we solve this problem up to some given inner accuracy. We derive relations between the inner and the outer accuracy of the primal and dual problems and we give a full convergence rate analysis for both gradient and fast gradient algorithms. We provide estimates on the primal and dual suboptimality and on primal feasibility violation of the generated approximate primal and dual solutions. Our analysis relies on the Lipschitz property of the dual function and on inexact dual gradients. We also discuss implementation aspects of the proposed algor...

Rate Analysis of Inexact Dual First-Order Methods Application to Dual Decomposition

We propose and analyze two dual methods based on inexact gradient information and averaging that generate approximate primal solutions for smooth convex 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 for which we prove sublinear rate of convergence. In particular, we provide a complete rate analysis and 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 linearly convergent parallel coordinate descent algorithm. Our analysis relies on the Lipschitz property of the dual function and inexact dual gradients. Further, we combine these methods with dual decomposition and constraint tightening and apply this framework to linear model predictive control obtaining a suboptimal and feasible control scheme.

On linear convergence of a distributed dual gradient algorithm for linearly constrained separable convex problems

In this paper we propose a fully distributed dual gradient algorithm for minimizing linearly constrained separable convex problems and analyze its rate of convergence. In particular, we prove that under the assumption of strong convexity and Lipschitz continuity of the gradient of the primal objective function we have a global error bound type property for the dual problem. Using this error bound property we devise a fully distributed dual gradient scheme, i.e. a gradient scheme based on a weighted step size, for which we derive global linear rate of convergence for both dual and primal suboptimality and for primal feasibility violation. Numerical simulations are also provided to confirm our theory.

Iteration complexity of an inexact augmented Lagrangian method for constrained MPC

In this paper we discuss the iteration complexity certification for solving constrained MPC problems for linear systems using an inexact augmented Lagrangian approach. We solve the augmented dual problem that arises from Lagrange relaxation of the linear constraints coming from the dynamics with the dual gradient method. Since the exact solution of the primal augmented Lagrange problem is usually impossible to compute in practical situations, we consider an inexact version of the dual gradient method. We discuss the relation between the inner and the outer accuracy of the primal and dual problem and we derive lower bounds on both primal and dual gap but also primal feasibility. We prove that we can obtain an o accurate solution in terms of the primal optimality and feasibility in at most O(1/ϵ) iterations, provided that the inner problems are solved with accuracy ε2.

Linear model predictive control based on approximate optimal control inputs and constraint tightening

In this paper we propose a model predictive control scheme for discrete-time linear time-invariant systems based on inexact numerical optimization algorithms. We assume that the solution of the associated quadratic program produced by some numerical algorithm is possibly neither optimal nor feasible, but the algorithm is able to provide estimates on primal suboptimality and primal feasibility violation. By tightening the complicating constraints we can ensure the primal feasibility of the approximate solutions generated by the algorithm. Finally, we derive a control strategy that has the following properties: the constraints on the states and inputs are satisfied, asymptotic stability of the closed-loop system is guaranteed, and the number of iterations needed for a desired level of suboptimality can be determined.

Complexity of an Inexact Augmented Lagrangian Method: Application to Constrained MPC

We propose in this paper an inexact dual gradient algorithm based on augmented Lagrangian theory and inexact information for the values of dual function and its gradient. We study the computational complexity certification of the proposed method and we provide estimates on primal and dual suboptimality and also on primal infeasibility. We also discuss implementation aspects of the proposed algorithm on constrained model predictive control problems for embedded linear systems and provide numerical tests to certify the efficiency of the method.

Optimal voltage control for loss minimization based on sequential convex programming

This paper focuses on the active power loss minimization by optimal voltage control in a power system using a new optimization algorithm. The cost function is assumed to be convex. The algorithm we propose to address the numerical solution of this problem is based on the exploitation of the convex problem structure using a sequential convex programming framework that linearizes the nonlinear power balance constraints at each iteration. The convex subproblem is then solved using a dual fast gradient method. We provide mathematical guarantees for the linear convergence of the algorithm towards a local solution. This approach allows an optimal voltage for each bus, while achieving the (local) economical optimum of the whole power grid. The newly developed algorithm can be run over large electricity networks, as we show on several numerical simulations using the classical IEEE bus test cases.

Feasible distributed MPC scheme for network systems based on an inexact dual gradient method

In this paper we propose an inexact dual gradient method for solving large-scale smooth convex optimization problems. For the proposed algorithm we provide estimates on primal and dual suboptimality and primal infeasibility. We solve the inner problems by means of a parallel coordinate descent method with linear convergence rate. We adapt our method using constraint tightening and obtain a distributed MPC strategy for network systems which guarantees feasibility.

A primal-dual consensus algorithm for convex problems from network control and estimation*

Abstract In this paper we propose a distributed algorithm for solving separable convex optimization problems on graphs arising for example from estimation and control in networks. We derive a primal-dual decomposition algorithm for solving this type of separable optimization problems in a distributed fashion given restrictions on the communication topology. The proposed algorithm is based on consensus principles in combination with local subgradient updates for the primal-dual variables. The main novelty of our algorithm consists in the way the weights in the consensus process are updated using arguments from optimization theory, while in most of the existing distributed algorithms based on consensus principles the weights have to be tuned.