Joseph Frédéric Bonnans

A family of variable metric proximal methods

We consider conceptual optimization methods combining two ideas: the Moreau—Yosida regularization in convex analysis, and quasi-Newton approximations of smooth functions. We outline several approaches based on this combination, and establish their global convergence. Then we study theoretically the local convergence properties of one of these approaches, which uses quasi-Newton updates of the objective function itself. Also, we obtain a globally and superlinearly convergent BFGS proximal method. At each step of our study, we single out the assumptions that are useful to derive the result concerned.

Second-order Sufficiency and Quadratic Growth for Nonisolated Minima

For standard nonlinear programming problems, the weak second-order sufficient condition is equivalent to the quadratic growth condition as far as the set of minima consists of isolated points and some qualification hypothesis holds. This kind of condition is instrumental in the study of numerical algorithms and sensitivity analysis. The arm of the paper is to study the relations between various types of sufficient conditions and quadratic growth in cases when the set of minima may have nonisolated points.

Pseudopower expansion of solutions of generalized equations and constrained optimization problems

We show that the solution of a strongly regular generalized equation subject to a scalar perturbation expands in pseudopower series in terms of the perturbation parameter, i.e., the expansion of orderk is the solution of generalized equations expanded to orderk and thus depends itself on the perturbation parameter. In the polyhedral case, this expansion reduces to a usual Taylor expansion. These results are applied to the problem of regular perturbation in constrained optimization. We show that, if the strong regularity condition is satisfied, the property of quadratic growth holds and, at least locally, the solutions of the optimization problem and of the associated optimality system coincide. If, in addition the number of inequality constraints is finite, the solution and the Lagrange multiplier can be expanded in Taylor series. If the data are analytic, the solution and the multiplier are analytic functions of the perturbation parameter.

Maximum Principles in the Optimal Control of Semilinear Elliptic Systems

Abstract We obtain Pontryagins maximum principle for local solutions of an optimal control problem with a monotone semilinear elliptic system and an integral cost. We also use Ekelands principle in order to derive some optimality conditions for approximate solutions.

The shooting approach to optimal control problems

We give an overview of the shooting technique for solving deterministic optimal control problems. This approach allows to reduce locally these problems to a finite dimensional equation. We first recall the basic idea, in the case of unconstrained or control constrained problems, and show the link with second-order optimality conditions and the analysis or discretization errors. Then we focus on two cases that are now better undestood: state constrained problems, and affine control systems. We end by discussing extensions to the optimal control of a parabolic equation.

Analytic Singular Perturbations of Elliptic Systems

Abstract We study a singular perturbation problem for a system defined under a variational form. We show the analytic dependence of the solution of the equation with respect to a small, nonnull parameter e, and make explicit the terms of the power series. This result improves a theorem of Chap. I of J. L. Lions (“Perturbations singulieres dans les problemes aux limites et en controle optimal,” Springer-Verlag, Berlin 1973) in which the variational forms are supposed to be symmetric and no analycity result is given. We give an application to the study of a stationary thermical system with a small convection coefficient.

Application of a New Class of Augmented Lagrangians to the Control of Distributed Parameter Systems

Abstract This paper describes a new method to solve optimal control problems of distributed parameter systems. The study is restricted to problems with a convex criterion and a state equation affine with respect to the pair (state, control). We show the equivalence between the initial problem and the minimisation with respect to the control, the state and also the costate of some augmented lagrangian, obtained by addition of the lagrangian and of two terms penalizing the state and costate equations. An example is treated, and numerical results are given.