Abstract
Minimax problems can be approached by reformulating them into smooth problems with constraints or by dealing with the non-smooth objective directly. We focus on verified enclosures of all globally optimal points of such problems. In smooth problems in branch and bound algorithms, interval Newton methods can be used to verify existence and uniqueness of solutions, to be used in eliminating regions containing such solutions, and point Newton methods can be used to obtain approximate solutions for good upper bounds on the global optimum. We analyze smooth reformulation approaches, show weaknesses in them, and compare reformulation to solving the non-smooth problem directly. In addition to analysis and illustrative problems, we exhibit the results of numerical computations on various test problems.
Similar content being viewed by others
References
Hall, J.: Homepage of C-LP, https://projects.coin-or.org/Clp (2002)
Hansen E.R.: Global Optimization Using Interval Analysis. Marcel Dekker, Inc., New York (1992)
Hu, C.: Optimal preconditioners for the interval Newton method. Ph.D. thesis, University of Southwestern Louisiana (1990)
Jaulin L.: Reliable minimax parameter estimation. Reliab. Comput. 7(3), 231–246 (2001)
Kearfott B.: A proof of convergence and an error bound for the method of bisection in \({\mathbf {R}^n}\) . Math. Comput. 32(144), 1147–1153 (1978)
Kearfott R.B.: Discussion and empirical comparisons of linear relaxations and alternate techniques in validated deterministic global optimization. Optim. Methods Softw. 21, 715–731 (2006)
Kearfott R.B.: Rigorous Global Search: Continuous Problems. Kluwer, Dordrecht, Netherlands (1996)
Kearfott R.B., Hongthong S.: Validated linear relaxations and preprocessing: some experiments. SIAM J. Optim. 16(2), 418–433 (2005)
Kearfott R.B.: GlobSol user guide. Optim. Methods Softw. 24(4–5), 687–708 (2009)
Kearfott R.B.: Interval computations, rigour and non-rigour in deterministic continuous global optimization. Optim. Methods Softw. 26(2), 259–279 (2011)
Kearfott R.B., Du k.: The cluster problem in multivariate global optimization. J. Glob. Optim. 5, 253–265 (1994)
Kearfott R.B., Muñoz Humberto: Slope interval, generalized gradient, semigradient, and slant derivative. Reliab. Comput. 10(3), 163–193 (2004)
Lemaréchal, C.: Nondifferentiable optimization. In: Powell, M.J.D. (ed.): Proceedings of the NATO Advanced Research Institute on ‘Nonlinear Optimization’, Trinity Hall, Cambridge, 13–24 July 1981
Lukšan, L., Vlček, J.: Test problems for nonsmooth unconstrained and linearly constrained optimization. Technical report 798, Institute of Computer Science (2000)
Moré, J.J., Garbow, B.S., Hillstrom, K.E.: User guide for MINPACK-1. Technical report ANL-80-74, Argonne National Laboratories (1980)
Neumaier A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)
Shen Z., Neumaier A., Eiermann M.C.: Solving minimax problems by interval methods. BIT 30, 742–751 (1990)
Wächter, A.: Homepage of IPOPT, https://projects.coin-or.org/Ipopt(2002)
Watson G.A.: The minimax solution of an overdetermined system of non-linear equations. J. Inst. Math. Appl. 23(2), 167–180 (1979)
Watson G.A.: Approximation in normed linear spaces (a historical survey of numerical methods). J. Comput. Appl. Math. 121, 1–36 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kearfott, R.B., Muniswamy, S., Wang, Y. et al. On smooth reformulations and direct non-smooth computations for minimax problems. J Glob Optim 57, 1091–1111 (2013). https://doi.org/10.1007/s10898-012-0014-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-012-0014-1