Abstract
Recently a new derivative-free algorithm has been proposed for the solution of linearly constrained finite minimax problems. This derivative-free algorithm is based on a smoothing technique that allows one to take into account the non-smoothness of the max function. In this paper, we investigate, both from a theoretical and computational point of view, the behavior of the minmax algorithm when used to solve systems of nonlinear inequalities when derivatives are unavailable. In particular, we show an interesting property of the algorithm, namely, under some mild conditions regarding the regularity of the functions defining the system, it is possible to prove that the algorithm locates a solution of the problem after a finite number of iterations. Furthermore, under a weaker regularity condition, it is possible to show that an accumulation point of the sequence generated by the algorithm exists which is a solution of the system. Moreover, we carried out numerical experimentation and comparison of the method against a standard pattern search minimization method. The obtained results confirm that the good theoretical properties of the method correspond to interesting numerical performance. Moreover, the algorithm compares favorably with a standard derivative-free method, and this seems to indicate that extending the smoothing technique to pattern search algorithms can be beneficial.
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References
Bertsekas, D.P.: Constrained Optimization and Lagrange Multipliers Methods. Academic Press, New York (1982)
Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program 91, 201–213 (2002)
Gould, N.I.M., Orban, D., Toint, Ph.L.: CUTEr and SifDec: a constrained and unconstrained testing environment, revisited. ACM T. Math. Softw. 29, 373–394 (2003)
Han, S.P., Mangasarian, O.L.: Exact penalty functions in nonlinear programming. Math. Program 17, 251–269 (1979)
Kolda, T.G., Lewis, R.M., Torczon, V.: Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods. SIAM Rev. 45, 385–482 (2003)
Liuzzi, G., Lucidi, S.: A derivative-free algorithm for systems of nonlinear inequalities. TR 17/05, Dept. of Computer and Systems Science, “Sapienza” University of Rome (2006)
Liuzzi, G., Lucidi, S., Sciandrone, M.: A derivative-free algorithm for linearly constrained finite minimax problems. SIAM J. Optim. 16, 1054–1075 (2006)
Lucidi, S., Sciandrone, M., Tseng, P.: Objective-derivative-free methods for constrained optimization. Math. Program Ser. A 92, 37–59 (2002)
May, J.H.: Linearly constrained nonlinear programming: a solution method that does not require analytic derivatives. Ph.D. Thesis. Yale University (1974)
Torczon, V.: http://www.cs.wm.edu/~va/software/
Lewis, R.M., Torczon, V.: Pattern search methods for linearly constrained minimization. SIAM J. Optim. 10, 917–941 (2000)
Xu, S.: Smoothing method for minimax problems. Comput. Optim. Appl. 20, 267–279 (2001)
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Liuzzi, G., Lucidi, S. A derivative-free algorithm for systems of nonlinear inequalities. Optim Lett 2, 521–534 (2008). https://doi.org/10.1007/s11590-008-0078-5
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DOI: https://doi.org/10.1007/s11590-008-0078-5