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Keywords:
nonsmooth optimization; nonsmooth equations; minimax problems; Newton methods; KKT systems; quasidifferential calculus
nonsmooth optimization; nonsmooth equations; minimax problems; Newton methods; KKT systems; quasidifferential calculus
Summary:
An equivalent model of nonsmooth equations for a constrained minimax problem is derived by using a KKT optimality condition. The Newton method is applied to solving this system of nonsmooth equations. To perform the Newton method, the computation of an element of the $b$-differential for the corresponding function is developed.
References:
[1] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski: Nonsmooth Analysis and Control Theory. Springer-Verlag, New York, 1998. MR 1488695
[2] V. F. Demyanov: On a relation between the Clarke subdifferential and the quasi-differential. Vestn. Leningr. Univ., Math. 13 (1981), 183–189. Zbl 0473.49008
[3] V. F. Demyanov, A. M. Rubinov: Constructive Nonsmooth Analysis. Peter Lang, Frankfurt am Main, 1995. MR 1325923
[4] F. Facchinei, A. Fischer, C. Kanzow, and J. Peng: A simple constrained optimization reformulation of KKT systems arising from variational inequalities. Appl. Math. Optimization 40 (1999), 19–37. DOI 10.1007/s002459900114 | MR 1685651
[5] Y. Gao: Demyanov difference of two sets and optimality conditions of Lagrange multipliers type for constrained the quasidifferential optimization. J. Optimization Theory Appl. 104 (2000), 377–394. DOI 10.1023/A:1004613814084 | MR 1752323
[6] Y. Gao: Newton methods for solving nonsmooth equations via a new subdifferential. Math. Methods Oper. Res. 54 (2001), 239–257. DOI 10.1007/s001860100150 | MR 1873344 | Zbl 1031.90069
[7] J. B. Hiriart-Urruty, C. Lemaréchal: Convex Analysis and Minimization. Springer-Verlag, Berlin, 1993.
[8] D. Li, N. Yamashita, and M. Fukushima: Nonsmooth equations based BFGS method for solving KKT system in mathematical programming. J. Optimization Theory Appl. 109 (2001), 123–167. DOI 10.1023/A:1017565922109 | MR 1833427
[9] J. V. Outrata: Minimization of nonsmooth nonregular functions: Applications to discrete-time optimal control problems. Probl. Control Inf. Theory 13 (1984), 413–424. MR 0782437
[10] J. S. Pang, D. Ralph: Piecewise smoothness, local invertibility, and parametric analysis of normal maps. Math. Oper. Res. 21 (1996), 401–426. DOI 10.1287/moor.21.2.401 | MR 1397221
[11] F. A. Potra, L. Qi, and D. Sun: Secant methods for semismooth equations. Numer. Math. 80 (1998), 305–324. DOI 10.1007/s002110050369 | MR 1645041
[12] L. Qi: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18 (1993), 227–244. DOI 10.1287/moor.18.1.227 | MR 1250115 | Zbl 0776.65037
[13] L. Qi, H. Jiang: Semismooth Karush-Kuhn-Tucker equations and convergence analysis of Newton and quasi-Newton methods for solving these equations. Math. Oper. Res. 22 (1997), 301–325. DOI 10.1287/moor.22.2.301 | MR 1450794
[14] L. Qi, J. Sun: A nonsmooth version of Newton’s method. Mathematical Program., Ser. A 58 (1993), 353–367. DOI 10.1007/BF01581275 | MR 1216791
[15] D. Sun, J. Han: Newton and quasi-Newton methods for a class of nonsmooth equations and related problems. SIAM J. Optim. 7 (1997), 463–480. DOI 10.1137/S1052623494274970 | MR 1443629
[16] Y. H. Yu, L. Gao: Nonmonotone line search algorithm for constrained minimax problems. J. Optimization Theory Appl. 115 (2002), 419–446. DOI 10.1023/A:1020896407415 | MR 1950702
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