Dragos Clipici

Parallel Random Coordinate Descent Method for Composite Minimization: Convergence Analysis and Error Bounds

In this paper we employ a parallel version of a randomized (block) coordinate descent method for minimizing the sum of a partially separable smooth convex function and a fully separable nonsmooth convex function. Under the assumption of Lipschitz continuity of the gradient of the smooth function, this method has a sublinear convergence rate. Linear convergence rate of the method is obtained for the newly introduced class of generalized error bound functions. We prove that the new class of generalized error bound functions encompasses both global/local error bound functions and smooth strongly convex functions. We also show that the theoretical estimates on the convergence rate depend on the number of blocks chosen randomly and a natural measure of separability of the smooth component of the objective function.

MPC for Power Systems Dispatch Based on Stochastic Optimization

Abstract In this paper we investigate the problem of optimal real-time power dispatch of an interconnection of conventional power generation plants, renewable resources and energy storage systems. The objective is to minimize imbalance costs and maximize profits whilst satisfying user demand. The managing company is able to trade energy on an electricity market. Energy prices, demand and renewable generation are considered stochastic processes. We show that under certain assumptions, the stochastic power dispatch problem over a finite horizon can be recast into a stochastic optimization formulation but with deterministic constraints. We carry out a systematic study of stochastic optimization methods to solve this problem. We also show that this problem can be approximated by a proper deterministic optimization problem using the sample average approximation method, which can then be solved by standard means.

Distributed model predictive control of leader-follower systems using an interior point method with efficient computations

This paper focuses on distributed model predictive control for large-scale systems comprised of interacting linear subsystems, where the necessary online computations can be distributed amongst them. A model predictive controller based on a distributed interior point method is derived, in which stabilizing control inputs can be computed distributively by every subsystem of the network. We introduce local terminal sets and costs, which together satisfy distributed invariance conditions for the whole system and guarantee stability of the closed-loop interconnected system. We show that the synthesis of both terminal sets and terminal cost functions can be done in a distributed framework.

On convergence of inexact projection gradient method for strongly convex minimization

Dual methods can handle easily complicated constraints in convex problems, but they have typically slow (sublinear) convergence rate in an average primal point, even when the original problem has smooth strongly convex objective function. Primal projected gradient-based methods achieve linear convergence for constrained, smooth and strongly convex optimization, but it is difficult to implement them, since they require exact projections onto the complicated primal feasible set. Therefore, in the present work we consider an inexact projection primal gradient algorithm for convex problems having strongly convex objective function and with Lipschitz continuous gradient. More precisely, we consider the Projected Gradient algorithm, where instead of an exact projection onto the complicated primal feasible set, an approximate projection, which is not necessarily feasible, is computed. We show that we can still achieve linear convergence for this scheme, provided that the approximate projection is computed with sufficient accuracy. Practical performance on quadratic programs coming from model predictive control applications shows encouraging results.

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.

A linear MPC algorithm for embedded systems with computational complexity guarantees

In this paper we propose a linear MPC scheme for embedded systems based on the dual fast gradient algorithm for solving the corresponding control problem. We establish computational complexity guarantees for the MPC scheme by appropriately deriving tight convergence estimates of order O(1/k2) for an average primal sequence generated by our proposed numerical optimization algorithm. We also show that these estimates rely heavily upon the dual optimal variables (Lagrange multipliers). Since convergence certification in embedded MPC is essential, we also derive tight bounds on the norm of these dual optimal variables. However, computing the norm for the Lagrange multipliers associated to multi-parametric optimization problems can quickly become intractable for high dimension and/or a large set of constraints. Therefore, we recast the problem in a mixed integer formulation by using auxiliary binary variables to characterize the complementarity conditions. We also show that this problem can be solved numerically efficiently.

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.