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.

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We propose and analyze an inexact version of the modified subgradient (MSG) algorithm, which we call the IMSG algorithm, for nonsmooth and nonconvex optimization over a compact set. We prove that under an approximate, i.e. inexact, minimization of the sharp augmented Lagrangian, the main convergence properties of the MSG algorithm are preserved for the IMSG algorithm. Inexact minimization may allow to solve problems with less computational effort. We illustrate this through test problems, including an optimal bang–bang control problem, under several different inexactness schemes.

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

Abstract—An on-off iterative adaptive controller has been developed that is applicable to servo systems performing repeated motions under extremely strict power constraints. The motivation for this approach is the control of piezoelectric actuators in autonomous micro-robots, where power consumption in analog circuitry and/or for position sensing may be much larger than that of the actuators themselves. The control algorithm optimizes the switching instances between ‘on’ and ‘off ’ inputs to the actuator using a stochastic approximation of the gradient of an objective function, namely that the system reach a specified output value at a specified time. This allows rapid convergence of system output to the desired value using just a single sensor measurement per iteration and discrete voltage inputs. M I.

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