Derivative-Free Optimization: Future Challenges and New Applications

Research grant PTDC/MAT/098214/2008 funded by FCT.

January 2010 - December 2012

Optimization without derivatives finds numerous and increasing applications in the industry and in computational sciences. In part this is because of the growing sophistication of computer hardware and mathematical algorithms and software, which allows expensive simulations and opens new possibilities for optimization. On the other hand, we deal more frequently with binary codes (for which the source is unavailable or owned) and legacy codes (written in the past and no longer maintained). Thus, in many circumstances, the alternatives of Derivative-Free Optimization (DFO) cannot be applied: (i) derivatives are unavailable (e.g., absence of adjoint codes); (ii) the application of automatic differentiation is too complex or impossible; (iii) even when derivatives are available, the contamination by noise and the need to search for global minimizers make them useless.

The proposed work is organized around three main tasks:
(i) Model-based methods (uncertainty, parallelization, constraints, multilevel DFO);
(ii) Direct-search methods (global and multiobjective optimization & other issues);
(iii) DFO in practice
(the DFO applications of interest to us lie in computational systems biology,
data mining and information science, and computational engineering for
space technology).

**Papers:**

- Le Thi Hoai An, A. I. F. Vaz, and L. N. Vicente, Optimizing radial basis functions by D.C. programming and its use in direct search for global derivative-free optimization, TOP, 20 (2012) 190-214.
- A. S. Bandeira, K. Scheinberg, and L. N. Vicente, Computation of sparse low degree interpolating polynomials and their application to derivative-free optimization, Mathematical Programming, 134 (2012) 223-257.
- A. S. Bandeira, K. Scheinberg, and L. N. Vicente, On partially sparse recovery, preprint 11-13, Dept. Mathematics, Univ. Coimbra.
- R. P. Brito and L. N. Vicente, Efficient cardinality/mean-variance portfolios, in System Modeling and Optimization, Springer series IFIP Advances in Information and Communication Technology, edited by C. Pötzsche, C. Heuberger, B. Kaltenbacher, and F. Rendl, 2014.
- A. R. Conn and L. N. Vicente, Bilevel derivative-free optimization and its application to robust optimization, Optimization Methods and Software, 27 (2012) 561-577.
- A. L. Custódio, J. F. A. Madeira, A. I. F. Vaz, and L. N. Vicente, Direct multisearch for multiobjective optimization, SIAM Journal on Optimization, 21 (2011) 1109-1140.
- A. L. Custódio, H. Rocha, and L. N. Vicente, Incorporating minimum Frobenius norm models in direct search, Computational Optimization and Applications, 46 (2010) 265-278.
- Y. Diouane, S. Gratton, and L. N. Vicente, Globally convergent evolution strategies, to appear in Mathematical Programming.
- R. Garmanjani and L. N. Vicente, Smoothing and worst-case complexity for direct-search methods in nonsmooth optimization, IMA Journal of Numerical Analysis, 33 (2013) 1008-1028.
- S. Gratton and L. N. Vicente, A surrogate management framework using rigorous trust-region steps, Optimization Methods and Software, 29 (2014) 10-23.
- J. M. Fernandes, A. I. F. Vaz, and L. N. Vicente, Modeling binary stars: age, helium abundance, and convection parameters, Monthly Notices of the Royal Astronomical Society, 425 (2012) 3104-3111.
- L. N. Vicente, Worst case complexity of direct search, EURO Journal on Computational Optimization, 1 (2013) 143-153.
- L. N. Vicente and A. L. Custódio, Analysis of direct searches for discontinuous functions, Mathematical Programming, 133 (2012) 299-325.

- Direct multisearch for multiobjective optimization: dms.
- Global derivative-free optimization: PSwarm.
- Direct search: sid-psm.