Abstract
We propose two restricted memory level bundle-like algorithms for minimizing a convex function over a convex set. If the memory is restricted to one linearization of the objective function, then both algorithms are variations of the projected subgradient method. The first algorithm, proposed in Hilbert space, is a conceptual one. It is shown to be strongly convergent to the solution that lies closest to the initial iterate. Furthermore, the entire sequence of iterates generated by the algorithm is contained in a ball with diameter equal to the distance between the initial point and the solution set. The second algorithm is an implementable version. It mimics as much as possible the conceptual one in order to resemble convergence properties. The implementable algorithm is validated by numerical results on several two-stage stochastic linear programs.
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References
Alber, Y.I., Iusem, A.N.: Extension of subgradient techniques for nonsmooth optimization in banach spaces. Set Valued Anal. 9, 315–335 (2001). doi:10.1023/A:1012665832688
Alber, Y.I., Iusem, A.N., Solodov M.V.: On the projected subgradient method for nonsmooth convex optimization in a Hilbert space. Math. Program. 81, 23–35 (1998). doi:10.1007/BF01584842
Bazaraa, M.S., Sherali, H.D.: On the choice of step size in subgradient optimization. Eur. J. Oper. Res. 7, 380–388 (1981)
Bello Cruz, J.Y., Iusem, A.N.: A strongly convergent method for nonsmooth convex minimization in hilbert spaces. Numer. Funct. Anal. Optim. 32, 1009–1018 (2011)
Ben-Tal, A., Nemirovski, A.: Non-euclidean restricted memory level method for large-scale convex optimization. Math. Program. 102, 407–456 (2005)
Brannlund, U., Kiwiel, K.C., Lindberg, P.O.: A descent proximal level bundle method for convex nondifferentiable optimization. Oper. Res. Lett. 17, 121–126 (1995)
Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)
Fábián, C.: Bundle-type methods for inexact data. Central Eur. J. Oper. Res. 8, 35–55 (special issue, T. Csendes and T. Rapcsák, eds.) (2000)
Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. no. 305-306 in Grund. der math. Wiss, Springer (two volumes) (1993)
Kim, S., Ahn, H., Cho, S.-C.: Variable target value subgradient method. Math. Program. 49, 359–369 (1990). doi:10.1007/BF01588797
Kiwiel, K.C.: Proximal level bubdle methods for convex nondiferentiable optimization, saddle-point problems and variational inequalities. Math. Program. 69, 89–109 (1995)
Lan, G.: Bundle-type methods uniformly optimal for smooth and nonsmooth convex optimization. Technical report 2796, University of Florida, Departament of Industrial and Systems Engineering, November 2010. Optimization Online
Lemaréchal, C., Nemirovskii, A., Nesterov, Y.: New variants of bundle methods. Math. Program. 69, 111–147 (1995)
Mak, W.-K., Morton, D., Wood, R.: Monte Carlo bounding techniques for determining solution quality in stochastic programs. Oper. Res. Lett. 24, 47–56 (1999) cited By (since 1996)140
Nesterov, Y.: Subgradient methods for huge-scale optimization problems. Math. Program., 1–23 (2013). doi:10.1007/s10107-013-0686-4
Norkin, V., Pflug, G., Ruszczyński, A.: A branch and bound method for stochastic global optimization. Math. Program. Ser B 83, 425–450 (1998) cited By (since 1996)89
Oliveira, W., Sagastizábal, C.: Level bundle methods for oracles with on-demand accuracy. Preprint, Instituto Nacional de Matemática Pura e Aplicada. http://www.optimization-online.org/DB_HTML/2012/03/3390.html (2012)
Oliveira, W., Sagastizábal, C., Scheimberg, S.: Inexact bundle methods for two-stage stochastic programming. SIAM J. Optim. 21, 517–544 (2011)
Polyak, B.: A general method for solving extremal problems. Soviet Mathematics Doklady 8, 593–597 (1967)
Polyak, B.: Minimization of unsmooth functionals. USSR Comput. Math. Math. Phys. 9, 14–29 (1969)
Richtárik, P.: Approximate level method for nonsmooth convex minimization. J. Optim. Theory Appl. 152, 334–350 (2012) cited By (since 1996)1
Shapiro, A.: Monte Carlo sampling methods. Handbooks Oper. Res. Manag. Sci. 10, 353–425 (2003) cited By (since 1996)78
Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming, Modeling and Theory. MPS-SIAM Series on Optimization, SIAM—Society for Industrial and Applied Mathematics and Mathematical Programming Society, Philadelphia (2009)
Shor, N.Z.: Minimization methods for non-differentiable functions. ZAMM J. Appl. Math. Mech. 66, 575–575 (1986)
Zverovich, V., Fábián, C., Ellison, E., Mitra, G.: A computational study of a solver system for processing two-stage stochastic lps with enhanced benders decomposition. Math. Program. Comput. 4, 211–238 (2012)
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The authors are grateful to the reviewers for their insightful comments and remarks. Research for this paper was partially supported by PROCAD-nf—UFG/UnB/IMPA research and PRONEX—CNPq-FAPERJ—Optimization research, and by project CAPES-MES-CUBA 226/2012 “Modelos de Otimização e Aplicações ”. The first author thanks CNPq and the Institute for Pure and Applied Mathematics (IMPA), in Rio de Janeiro, Brazil, where he was a visitor while working on this paper.
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Bello Cruz, J.Y., de Oliveira, W. Level bundle-like algorithms for convex optimization. J Glob Optim 59, 787–809 (2014). https://doi.org/10.1007/s10898-013-0096-4
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DOI: https://doi.org/10.1007/s10898-013-0096-4