In the framework of the internship, and the beginning of doctoral thesis, of A. Hermant, we have studied the junction conditions for optimal control problems with only one control and one state constraint, of arbitrary order q. For these problems there exists an ``alternative formulation'' to the classical Pontryaguin principle, whose Hamiltonian involves the q-times derivative of the state constraint. We have clarified the question of equivalence between the classical and alternative formulations, by stating a complete set of ``additional conditions''. We have showed that, under weak conditions, the Jacobian of the shooting algorithm is invertible.

An INRIA report will be published at the beginning of 2006.

in collaboration with N. Bérend (Onera) and Ch. Talbot (Cnes).

We report here the work done in the framework of the thesis of J. Laurent-Varin (defendedin November 2005), whose subject is the numerical methods for multiarc problems, i.e., when several connected trajectories are to be optimized simultaneously (separations of stages of launchers, orbital rendez-vous). The work done previously was an implementation of a interior-point algorithm, combined with a refinement procedure, for the single arc case. We reported in 2004 a theoretical study of multiarc problems. The main effort in 2005 was the implementation of the method. We use an elimination procedure of variables associated with arcs, and obtain a reduced system whose unknowns are the variables associated with nodes.

The method has been applied to a model deriving from the FSSC16 concept of two-stages launcher. After separation, the re-usable booster comes back to Earth using an optimized path. Our algorithm was able to optimize this model.

In [ CRAS-2005 ], we present an approximation scheme of the type of Markov chain approximation for the resolution of second order HJB equations. Our scheme is a generalization of the approximations suggested by [ RM-CF ]. It is based on an approximation of the characteristics and an interpolation on a grid which is not necessarily regular. We prove that when the diffusion step and the interpolation weights satisfy a 'balance conditions', the scheme is consistent (which guarantees its convergence).

We have continued the analysis and implementation of splitting type algorithms for second order HJB equations. We use the Strang decomposition for obtaining second-order approximations for problems with smooth solutions. Extensive numerical results for schemes, combined or not with the Strang decomposition, confirm the predictins of the theory.

We have established error estimates for stochastic impulse control problems. We use the techniques of Barles and Jakobsen, and the analysis of cascade problems. The error estimate combines an error estimate for an arbitrary solution of the cascade problem, with the estimate due to Ishii of the distance between that solution and the one of the impulse problem. The results were presented in INRIA report RR 5441.

-algorithms and theory

For the -superlinear bundle algorithm for convex minimization described in , see Report 2004, we consider a line search along the lines of . We are currently developing a battery of test problems to validate the results.

This work is a follow-up of our more theoretical research on variational analysis, see and and .

In we propose an algorithm for solving nonsmooth convex constrained optimization problems, which combines the ideas of the proximal bundle methods with the filter strategy for evaluating candidate points. The resulting algorithm inherits some attractive features from both approaches. On the one hand, it allows an effective control of the size of quadratic programming subproblems via the compression and aggregation techniques of the proximal bundle methods. On the other hand, the filter criterion for accepting a candidate point as the new iterate is expected to be easier to satisfy than the usual descent condition in bundle methods. Preliminary computational results, including comparisons with the (nonfilter) infeasible bundle method in , see Report 2004, are also reported.

The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with the critical points of the original function. This suggests that the many uses of proximal points, and their corresponding proximal envelopes (Moreau Envelopes), will have a natural extension from Convex Optimization to Nonconvex Optimization. For example, the inexact proximal point methods for convex optimization might be redesigned to work for nonconvex functions. In order to begin the practical implementation of proximal points in a nonconvex setting, a first crucial step would be to design efficient methods of approximating nonconvex proximal points. This would provide a solid foundation on which future design and analysis for nonconvex proximal point methods could flourish.

In we present a methodology based on the computation of proximal points of piecewise affine models of the nonconvex function. These models can be built with only the knowledge obtained from a black box providing, for each point, the function value and one subgradient. Convergence of the method is proven for the class of nonconvex functions that are prox-bounded and lower- and encouraging preliminary numerical testing is reported.