# Project : tropics

## Section: New Results

Keywords : optimization, Sequential Quadratic Programming, One-shot.

### SQP-One-Shot optimization

Participants : Francois Courty, Alain Dervieux.

The class of methods that applies best to optimal control with a state equation is the Sequential Quadratic Programming. We refer for example to the monography of Nocedal and Wright [43].

SQP methods are sophisticated methods combining a lot of useful heuristics. They enjoy robustness properties due for example to Trust Region heuristics relying on powerful theory (Wolfe criteria for gradient convergence), and due to quasi Newton formulas such as BFGS. However, they are not well adapted to large scale systems such as those handled in Optimal Control loops with adjoints. Indeed, the standard SQP methods involve at each main iteration to solve several linearized state systems. This difficult point has been identified by many researchers in optimal shape design and the result is that SQP methods have been not always applied, but instead, either less modern but less complex algorithms like gradient algorithm were applied [40], or algorithms for the simultaneous solution of the KKT optimality equations were proposed [46]. The latter class of algorithm is in fact an important key for large scale optimization. However, existing one-shot algorithm are deprived of the many robustness heuristics that are involved in SQP modern algorithms. We have derived a family of one-shot-SQP algorithm devoted to the robust application of the one-shot principle:

they solve progressively the three equations of optimality, yielding a good complexity for obtaining the final result,

they involve some important features of SQP allowing for a quasi-black box resolution of a new problem.

Results have been described and published in [18][19][24].