Abstract
An algorithm for univariate optimization using a linear lower bounding function is extended to a nonsmooth case by using the generalized gradient instead of the derivative. A convergence theorem is proved under the condition of semismoothness. This approach gives a globally superlinear convergence of algorithm, which is a generalized Newton-type method.
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References
Bazaraa M.S., Sherali H.D., Shetty C.M., Nonlinear programming, theory and algorithms, John Wiley & Sons Inc., New York, 1993
Bromberg M., Chang T.S., Global optimization using linear lower bounds: one dimensional case, In: Proceedings of the 29th Conference on Decision and Control, Honolulu, Hawaii, 1990
Chen X., Superlinear convergence of smoothing quasi-Newton methods for nonsmooth equations, J. Comput. Appl. Math., 1997, 80, 105–126
Clarke F.H., Optimization and nonsmooth analysis, John Wiley & Sons Inc., New York, 1983
Conn A.R., Gould N.I.M., Toint P.L., Trust-region methods, SIAM, Philadelphia, 2000
Dennis Jr. J.E., Moré J.J., A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comp., 1974, 28, 549–560
Famularo D., Sergeyev Ya.D., Pugliese P., Test problems for Lipschitz univariate global optimization with multiextremal constraints, In: Dzemyda G., Saltenis V., Žilinskas A. (Eds.), Stochastic and global optimization, Kluwer Academic Publishers, Dordrecht, 2002
Hansen P., Jaumard B., Lu S.H., Global optimization of univariate Lipschitz functions I: survey and properties, Math. Program., 1992, 55, 251–273
Harker P.T., Xiao B., Newton’s method for the nonlinear complementarity problem: a B-differentiable equation approach, Math. Program., 1990, 48, 339–357
Kahya E., A class of exponential quadratically convergent iterative formulae for unconstrained optimization., Appl. Math. Comput., 2007, 186, 1010–1017
Mifflin R., Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 1977, 15, 959–972
Pang J.S., Newton’s method for B-differentiable equations, Math. Oper. Res., 1990, 15, 311–341
Potra F.A., Qi L., Sun D., Secant methods for semismooth equations, Numer. Math., 1998, 80, 305–324
Qi L., Sun J., A nonsmooth version of Newton’s method, Math. Program., 1993, 58, 353–367
Sergeyev Ya.D., Daponte P., Grimaldi D., Molinaro A., Two methods for solving optimization problems arising in electronic measurements and electrical engineering, SIAM J. Optim., 1999, 10, 1–21
Shapiro A., On concepts of directional differentiability, J. Optim. Theory Appl., 1990, 66, 477–487
Smietanski M.J., A new versions of approximate Newton method for solving nonsmooth equations, Ph.D. thesis, University of Lódz, Poland, 1999 (in Polish)
Tseng C.L., A Newton-type univariate optimization algorithm for locating the nearest extremum, Eur. J. Oper. Res., 1998, 105, 236–246
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Smietanski, M.J. A nonsmooth version of the univariate optimization algorithm for locating the nearest extremum (locating extremum in nonsmooth univariate optimization). centr.eur.j.math. 6, 469–481 (2008). https://doi.org/10.2478/s11533-008-0039-3
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DOI: https://doi.org/10.2478/s11533-008-0039-3