A Simple and Efficient Algorithm for Nonlinear Model Predictive Control

We present PANOC, a new algorithm for solving optimal control problems arising in nonlinear model predictive control (NMPC). A usual approach to this type of problems is sequential quadratic programming (SQP), which requires the solution of a quadratic program at every iteration and, consequently, inner iterative procedures. As a result, when the problem is ill-conditioned or the prediction horizon is large, each outer iteration becomes computationally very expensive. We propose a line-search algorithm that combines forward-backward iterations (FB) and Newton-type steps over the recently introduced forward-backward envelope (FBE), a continuous, real-valued, exact merit function for the original problem. The curvature information of Newton-type methods enables asymptotic superlinear rates under mild assumptions at the limit point, and the proposed algorithm is based on very simple operations: access to first-order information of the cost and dynamics and low-cost direct linear algebra. No inner iterative procedure nor Hessian evaluation is required, making our approach computationally simpler than SQP methods. The low-memory requirements and simple implementation make our method particularly suited for embedded NMPC applications

Projected Newton methods and optimization of multicommodity flows

Bibliography: p. 26-28."August 1981."Partial support provided by the National Science Foundation Grant ECS-79-20834 Defense Advanced Research Project Agency Grant ONR-N00014-75-C-1183by Dimitri P. Bertsekas and Eli M. Gafni

On the Local and Global Convergence of a Reduced Quasi-Newton Method

In optimization in R^n with m nonlinear equality constraints, we study the local convergence of reduced quasi-Newton methods, in which the updated matrix is of order n-m. In particular, we give necessary and sufficient conditions for q-superlinear convergence (in one step). We introduce a device to globalize the local algorithm which consists in determining a step on an arc in order to decrease an exact penalty function. We give conditions so that asymptotically the step will be equal to one

Some recent advances in projection-type methods for variational inequalities

AbstractProjection-type methods are a class of simple methods for solving variational inequalities, especially for complementarity problems. In this paper we review and summarize recent developments in this class of methods, and focus mainly on some new trends in projection-type methods

Forward-backward truncated Newton methods for convex composite optimization

This paper proposes two proximal Newton-CG methods for convex nonsmooth optimization problems in composite form. The algorithms are based on a a reformulation of the original nonsmooth problem as the unconstrained minimization of a continuously differentiable function, namely the forward-backward envelope (FBE). The first algorithm is based on a standard line search strategy, whereas the second one combines the global efficiency estimates of the corresponding first-order methods, while achieving fast asymptotic convergence rates. Furthermore, they are computationally attractive since each Newton iteration requires the approximate solution of a linear system of usually small dimension