We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the step-size parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem. Key words: Nonconvex programming; nonsmooth optimization; augmented Lagrangian; sharp Lagrangian; subgradient optimization.

A computational technique for unconstrained optimal control problems is presented. First an Euler discretization is carried out to obtain a finite-dimensional approximation of the continous-time (infinite-dimensional) problem. Then an inexact restoration (IR) method due to Birgin and Martínez is applied to the discretized problem to find an approximate solution. Convergence of the technique to a solution of the continuous-time problem is facilitated by the convergence of the IR method and the convergence of the discrete (approximate) solution as finer subdivisions are taken. It is shown that a special case of the IR method is equivalent to the projected Newton method for equality constrained quadratic optimization problems. The technique is numerically demonstrated by means of a scalar system and the van der Pol system, and comprehensive comparisons are made with the Newton and projected Newton methods.

Many practical optimal control problems include discrete decisions. These may be either time–independent parameters or time–dependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixed–integer linear programs, as well as in continuous optimal control problems, the combination of the two is yet an open field of research. We consider the question of lower bounds that can be obtained by a relaxation of the integer requirements. For general nonlinear mixed–integer programs such lower bounds typically suffer from a huge integer gap. We convexify (with respect to binary controls) and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem. Building on this theoretical result we present a novel algorithm to solve mixed–integer optimal control problems, with a focus on discrete–valued control functions. Our algorithm is based on the direct multiple shooting method, an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods. Its applicability is shown by a challenging application, the energy optimal control of a subway train with discrete gears and velocity limits.

The leapfrog algorithm, so-called because of its geometric nature, for solving a class of optimal control problems is proposed. Initially a feasible trajectory is given and subdivided into smaller pieces. In each subdivision, with the assumption that local optimal controls can easily be calculated, a piecewise-optimal trajectory is obtained. Then the junctions of these smaller pieces of optimal control trajectories are updated through a scheme of midpoint maps. Under some broad assumptions the sequence of trajectories is shown to converge to a trajectory that satisfies the Maximum Principle. The main advantages of the leapfrog algorithm are that (i) it does not need an initial guess for the costates, (ii) the piecewise-optimal trajectory generated in each iteration is feasible. These are illustrated through a numerical implementation of leapfrog on a problem involving the van der Pol system. Key words: Optimal control, two-point boundary-value problem, multiple shooting, geodesics, numerical methods, van der Pol system. 1