A Stable and Efficient Method for Solving a Convex Quadratic Program with Application to Optimal Control

Abstract

Abstract.A method is proposed for reducing the cost of computing search directions in an interior point method for a quadratic program. The KKT system is partitioned and modified, based on the ratios of the slack variables and dual variables associated with the inequality constraints, to produce a smaller, approximate linear system. Analytical and numerical results are included that suggest the distribution of eigenvalues of the new, approximate system matrix is improved, which makes it more amenable to being solved with an iterative linear solver. For this purpose, new preconditioners are also presented to allow iterative methods, such as MINRES, to be used. Numerical results indicate that the computational complexity of the proposed method scales well when applied to a finite horizon discrete-time optimal control problem with linear dynamics, quadratic cost, and linear inequality constraints, which arises in model predictive control applications.Key words. interior point methods, quadratic programming, ill-conditioning, large-scale problems, inexact methods, iterative methods, predictive control AMS subject classifications. 90C51, 90C20, 90C06, 15A12, 49M15, 49N05 DOI. 10.1137/11082960X1. Introduction. Interior point methods (IPMs) have proved to be an efficient way of solving linear, quadratic, and nonlinear programming problems. Quadratic programs (QPs) arise in many applications, such as least-squares regression with linear constraints, robust data fitting, support vector machines, and predictive control problems [5,14,15,21]. They also appear in solving a nonlinear programming problem with sequential quadratic programming, in which a series of QPs is solved [24, Chap. 18].Many primal-dual IPMs for solving a QP involve finding a solution to the KarushKuhn-Tucker (KKT) conditions by a Newton-type method [30], in which a linear system of equations is formed and solved at each IPM iteration. Solving this linear system is the main contributor to the total computational cost of an IPM.The linear system that arises in this process is often sparse and large. Its size is usually reduced by block elimination, which provides alternative linear systems, as reviewed in section 2. However, the reduction in size may adversely affect the sparsity of the system matrix. In section 3 we present a new method that replaces the linear system with a smaller, approximate linear system, in which sparsity has been preserved. An upper bound on the error introduced, which decreases with each IPM iteration, is also presented. A nice property of this approximate linear system is that

Fast model predictive control for linear periodic systems with state and control constraints

“…This example is taken from . Consider a system of four equal masses connected by springs and to walls at the ends, see Figure .…”

Nonlinear predictive control on a heterogeneous computing platform

“…NLP (5) incorporates both integration and optimization, which potentially opens the possibility of accelerating both subproblems. Primal-dual interior-point algorithms have proven their efficiency for the numerical solution of optimal control problems [28]. Moreover, with interior-point algorithms, the structure of the KKT matrix associated with the OCP remains fixed (unlike with active set methods), which is desirable for hardware implementations [16].…”

Computer architectures to close the loop in real-time optimization

“…For example, with inexact Newton methods the resulting linear equations are only solved approximately using iterative solvers, such as the conjugate gradient method. This might result in more Newton steps, but since each iteration is more efficient than for an exact Newton method with Cholesky factorizations, the overall time taken by an inexact method might be less than that for an exact method [12].…”

“…The decision variables (x s and x h ) for those problems can be both continuous (discretization frequency, termination tolerance, chip clock frequency) and discrete (parallelization level, pipeline depth, model order), which makes solving the optimization problem non-trivial. Furthermore, derivative information is not always available for all objectives functions of (12) or this information might be unreliable. Evaluation of an objective function itself can be a time consuming task in some cases.…”