Abstract
In this paper, a hybrid algorithm for solving finite minimax problem is presented. In the algorithm, we combine the trust-region methods with the line-search methods and curve-search methods. By means of this hybrid technique, the algorithm, according to the specific situation at each iteration, can adaptively performs the trust-region step, line-search step or curve-search step, so as to avoid possibly solving the trust-region subproblems many times, and make better use of the advantages of different methods. Moreover, we use second-order correction step to circumvent the difficulties of the Maratos effect occurred in the nonsmooth optimization. Under mild conditions, we prove that the new algorithm is of global convergence and locally superlinear convergence. The preliminary experiments show that the new algorithm performs efficiently.
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References
Charalambous, C., & Conn, A. R. (1978). An efficient method to solve the minimax problem directly. SIAM Journal on Numerical Analysis, 15, 162–187.
Conn, A. R., Gould, N. I., & Toint, P. L. (2000). Trust-region methods. Philadelphia: SIAM.
Dem’yanov, V. F., & Malozemov, V. N. (1974). Introduction to minimax. New York: Wiley.
Di Pillo, G., Grippo, L., & Lucidi, S. (1993). A smooth method for the finite minimax problem. Mathematical Programming, 60, 187–214.
Du, D. Z. & Pardalos, P. M. (Eds.) (1995). Minimax and applications. Dordrecht: Kluwer.
Fletcher, R. (1982). Second order correction for nondifferentiable optimization problems. In G. A. Watson (Ed.), Numerical analysis (pp. 85–114). Berlin: Springer.
Fletcher, R. (1987). Practical methods of optimization (2nd edn.) New York: Wiley.
Gaudioso, M., & Monaco, M. F. (1982). A boundle type approach to the unconstrained minimization of convex nonsmooth functions. Mathematical Programming, 23, 216–226.
Gigola, C., & Gomez, S. (1990). A regularization method for solving the finite convex min–max problem. SIAM Journal on Numerical Analysis, 27, 1621–1634.
Hager, W. W. (1999). Stabilized sequential quadratic programming. Computational Optimization and Applications, 12, 253–273.
Han, S. P. (1981). Variable metric methods for minimizing a class of nondifferentiable functions. Mathematical Programming, 20, 1–13.
Jian, J. B. (2006). New sequential quadratically constrained quadratic programming method of feasible directions and its convergence rate. Journal of Optimization Theorey and Application, 129, 109–130.
Jian, J.-B., Quan, R., & Hu, Q. J. (2007). A new superlinearly convergent SQP algorithm for nonlinear minimax problem. Acta Mathematicae Applicatae Sinica, 23, 395–410. English series.
Li, X.-S., & Fang, S.-C. (1997). On the entropic regularization method for solving min-max problems with applications. Mathematical Methods of Operations Research, 46, 119–130.
Luksan, L. (1986). A compact variable metric algorithm for nonlinear minimax approximation. Computing, 36, 355–373.
Michael Gertz, E. (2004). A quasi-Newton trust-region method. Mathematical Programming, 100, 447–470.
Murray, W., & Overton, M. L. (1980). A projected Lagrangian algorithm for nonlinear minimax optimization. SIAM Journal on Scientific and Statistic Computing, 1, 345–370.
Nocedal, J., & Wright, S. J. (2006). Numerical optimization. Beijing: Science Press.
Nocedal, J., & Yuan, Y. (1998). Combining trust region and line search techniques. In Y. Yuan (Ed.), Advances in nonlinear programming (pp. 153–175). Dordrecht: Kluwer.
Overton, M. L. (1982). Algorithms for nonlinear l 1 and l ∞ fitting. In M. J. D. Powell (Ed.), Nonlinear optimization 1981 (pp. 213–221). London: Academic Press.
Ploak, E. (1989). Basics of minimax algorithms. In F. H. Clarke, V. F. Demyanov, & F. Giannessi (Eds.), Nonsmooth optimization and related topics (pp. 343–369). New York: Plenum.
Polak, E., Mayne, D. H., & Higgins, J. E. (1991). Superlinearly convergent algorithm for min-max problems. Journal of Optimization Theorey and Application, 69, 407–439.
Powell, M. J. D. (1984). On the global convergence of trust region algorithm for unconstrained minimization. Mathematical Programming, 29, 297–303.
Shen, P. P., & Wang, Y. J. (2005). A new pruning test for finding all global minimizers of nonsmooth functions. Applied Mathematics and Computation, 168, 739–755.
Sun, W. (2004). Nonmonotone trust region method for solving optimization problems. Applied Mathematics and Computation, 156, 159–174.
Vardi, A. (1992). New minimax algorithm. Journal of Optimization Theorey and Application, 75, 613–634.
Wang, F. S., Zhang, K. C., & Tan, X. L. (2006). A fractional programming algorithm based on conic quasi-Newton trust region method for unconstrained minimization. Applied Mathematics and Computation, 181, 1061–1067.
Watson, G. A. (1979). The minimax solution of an overdetermined system of nonlinear equations. Journal of the Institute for Mathematics and Its Applications, 23, 167–180.
Xu, S. (2001). Smoothing method for minimax problems. Computational Optimization and Applications, 20, 267–279.
Yi, X. (2002). The sequential quadratic programming method for solving minimax problem. Journal of Systems Science and Mathematical Sciences, 22, 355–364.
Yi, X. (2004). A Newton like algorithm for solving minimax problem. Journal on Numerical Methods and Computer Applications, 2, 108–115.
Yuan, Y. (1985). On the superlinear convergence of a trust-region algorithm for nonsmooth optimization. Mathematical Programming, 31, 269–285.
Yuan, Y. (1995). On the convergence of a new trust region algorithm. Numerische Mathematik, 70, 515–539.
Zhou, J. L., & Tits, A. L. (1993). Nonmonotone line search for minimax problems. Journal of Optimization Theorey and Application, 76, 455–476.
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Wang, F., Zhang, K. A hybrid algorithm for nonlinear minimax problems. Ann Oper Res 164, 167–191 (2008). https://doi.org/10.1007/s10479-008-0401-7
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DOI: https://doi.org/10.1007/s10479-008-0401-7