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Description: |
In this talk we will focus on the numerical optimization of extended-value, nonsmooth functions, without using derivatives. We will start by reviewing the class of directional Direct Search Methods, paying particular attention to the situation where the objective function is discontinuous at a limit point of the sequence of iterates generated by the algorithms. Using Rockafeller generalized derivatives, the known convergence results obtained under Lipschitz continuity can be generalized to the discontinuous case.
Direct Search is a robust but inefficient technique. We show how the use of quadratic interpolation in the search step of the algorithms and an appropriate ordering strategy in their poll step can significantly improve the efficiency.
Finally, we generalize Direct Search to Multiobjective Optimization by introducing a new algorithm: Direct MultiSearch. Under common assumptions, it can be proved that at least one limit point of the sequence of iterates lies in a critical form of the Pareto frontier. Computational results will be reported, showing that Direct MultiSearch, even in its simplest form, has an impressive capability of generating the whole Pareto frontier.
This is joint work with José F. Aguilar Madeira, Humberto Rocha, A. Ismael F. Vaz, and Luís Nunes Vicente.
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