Non-convex semi-infinite min-max optimization with noncompact sets. (English) Zbl 1373.90166
Summary: In this paper, first we study the non-convex sup-type functions with noncompact sets. Under quite mild conditions, the expressions of its derivative and subderivative along arbitrary direction are given. Furthermore, the structure of its subdifferential is characterized completely. Then, using these results, we establish first-order optimality conditions for semi-infinite min-max optimization problems. These results generalize and improve the corresponding results in the relevant literatures.
MSC:
| 90C34 | Semi-infinite programming |
| 90C46 | Optimality conditions and duality in mathematical programming |
| 65K10 | Numerical optimization and variational techniques |
Keywords:
semi-infinite optimization; sup-type function; subderivative; subdifferential; first-order optimality conditionsReferences:
| [1] | B. Betrò, An accelerated central cutting plane algorithm for linear semi-infinite programming,, Math. Program., 101, 479 (2004) · Zbl 1073.90053 · doi:10.1007/s10107-003-0492-5 |
| [2] | S. C. Fang, Solving quadratic semi-infinite programming problems by using relaxed cutting-plane scheme,, J. Comput. Appl. Math., 129, 89 (2001) · Zbl 0984.65062 · doi:10.1016/S0377-0427(00)00544-6 |
| [3] | D. Gale, A geometric duality theorem with economic applications,, Rev. Econ. Stud., 34, 19 (1967) · doi:10.2307/2296568 |
| [4] | M. A. Goberna, <em>Semi-Infinite Programming-Recent Advances</em>,, Kluwer (2001) · Zbl 0978.00025 · doi:10.1007/978-1-4757-3403-4 |
| [5] | F. Guerra-Vazquez, General semi-infinite programming: Symmetric Mangasarian-Fromovitz constraint qualification and the closure of the feasible set,, SIAM J. Optim., 20, 2487 (2010) · Zbl 1229.90244 · doi:10.1137/090775294 |
| [6] | A. Hantoute, A complete characterization of the subdifferential set of the supremum of an arbitrary family of convex functions,, J. Convex Anal., 15, 831 (2008) · Zbl 1165.52010 |
| [7] | A. Hantoute, Subdifferential calculus rules in convex analysis: A unifying approach via pointwise supremum functions,, SIAM J. Optim., 19, 863 (2008) · Zbl 1163.49014 · doi:10.1137/070700413 |
| [8] | R. Hettich, Semi-infinite programming: Theory, methods, and applications,, SIAM Rev., 35, 380 (1993) · Zbl 0784.90090 · doi:10.1137/1035089 |
| [9] | R. Hettich, Second order optimality conditions for generalized semi-infinite programming problems,, Optimization, 34, 195 (1995) · Zbl 0855.90129 · doi:10.1080/02331939508844106 |
| [10] | H. Th. Jongen, Generalized semi-infinite optimization: A first order optimality condition and examples,, Math. Program., 83, 145 (1998) · Zbl 0949.90090 · doi:10.1007/BF02680555 |
| [11] | N. Kanzi, Necessary optimality conditions for nonsmooth generalized semi-infinite programming problems,, Eur. J. Oper. Res., 205, 253 (2010) · Zbl 1188.90261 · doi:10.1016/j.ejor.2009.12.025 |
| [12] | N. Kanzi, Necessary optimality conditions for nonsmooth semi-infinite programming problems,, J. Global Optim., 49, 713 (2011) · Zbl 1254.90264 · doi:10.1007/s10898-010-9561-5 |
| [13] | N. Kanzi, Lagrange multiplier rules for non-differentiable DC generalized semi-infinite programming problems,, J. Global Optim., 56, 417 (2013) · Zbl 1300.90056 · doi:10.1007/s10898-011-9828-5 |
| [14] | C. Ling, A new smoothing Newton-type algorithm for semi-infinite programming,, J. Global Optim., 47, 133 (2010) · Zbl 1220.90144 · doi:10.1007/s10898-009-9462-7 |
| [15] | Q. Liu, On the convergence of a smoothed penalty algorithm for semi-infinite programming,, Math. Methods Oper. Res., 78, 203 (2013) · Zbl 1291.90271 · doi:10.1007/s00186-013-0440-y |
| [16] | M. López, Semi-infinite programming,, Eur. J. Oper. Res., 180, 491 (2007) · Zbl 1124.90042 · doi:10.1016/j.ejor.2006.08.045 |
| [17] | S. K. Mishra, Nonsmooth semi-infinite programming problem using limiting subdifferentials,, J. Global Optim., 53, 285 (2012) · Zbl 1274.90442 · doi:10.1007/s10898-011-9690-5 |
| [18] | Q. Ni, A truncated projected Newton-type algorithm for large-scale semi-infinite programming,, SIAM J. Optim., 16, 1137 (2006) · Zbl 1131.90064 · doi:10.1137/040619867 |
| [19] | E. Polak, <em>Optimization: Algorithms and Consistent Approximations</em>,, Springer-Verlag (1997) · Zbl 0899.90148 · doi:10.1007/978-1-4612-0663-7 |
| [20] | L. Q. Qi, Semismooth Newton methods for solving semi-infinite programming problems,, J. Global Optim., 27, 215 (2003) · Zbl 1033.90137 · doi:10.1023/A:1024814401713 |
| [21] | R. T. Rockafellar, <em>Convex Analysis</em>,, Princeton University Press (1970) · Zbl 0193.18401 |
| [22] | R. T. Rockafellar, <em>Variational Analysis</em>,, Springer (1998) · Zbl 0888.49001 · doi:10.1007/978-3-642-02431-3 |
| [23] | J. J. Rückmann, First-order optimality conditions in generalized semi-infinite programming,, J. Optim. Theory and Appl., 101, 677 (1999) · Zbl 0956.90055 · doi:10.1023/A:1021746305759 |
| [24] | A. Shapiro, On duality theory of convex semi-infinite programming,, Optimization, 54, 535 (2005) · Zbl 1147.90403 · doi:10.1080/02331930500342823 |
| [25] | A. Shapiro, Semi-infinite programming, duality, discretization and optimality condition,, Optimization, 58, 133 (2009) · Zbl 1158.90410 · doi:10.1080/02331930902730070 |
| [26] | V. N. Solov’ev, The subdifferential and the directional derivatives of the maximum of a family of convex functions,, Izv. Math., 62, 807 (1998) · Zbl 0921.49009 · doi:10.1070/im1998v062n04ABEH000192 |
| [27] | V. N. Solov’ev, The subdifferential and the directional derivatives of the maximum of a family of convex functions. II,, Izv. Math., 65, 99 (2001) · Zbl 1017.49020 · doi:10.1070/im2001v065n01ABEH000323 |
| [28] | O. Stein, On optimality conditions for generalized semi-infinite programming problems,, J. Optim. Theory Appl., 104, 443 (2000) · Zbl 0964.90047 · doi:10.1023/A:1004622015901 |
| [29] | O. Stein, First order optimality conditions for degenerate index sets in generalized semi-infinite programming,, Math. Oper. Res., 26, 565 (2001) · Zbl 1073.90562 · doi:10.1287/moor.26.3.565.10583 |
| [30] | O. Stein, Feasible method for generalized semi-infinite programming,, J. Optim. Theory Appl., 146, 419 (2010) · Zbl 1202.90256 · doi:10.1007/s10957-010-9674-5 |
| [31] | Y. Tanaka, A globally convergent SQP method for semi-infinite nonlinear optimization,, J. Comput. Appl. Math., 23, 141 (1988) · Zbl 0685.90080 · doi:10.1016/0377-0427(88)90276-2 |
| [32] | X. J. Tong, A semi-infinite programming algorithm for solving optimal power flow with transient stability constraints,, J. Comput. Appl. Math., 217, 432 (2008) · Zbl 1160.65029 · doi:10.1016/j.cam.2007.02.026 |
| [33] | A. I. F. Vaz, Air pollution control with semi-infinite programming,, Appl. Math. Modelling, 33, 1957 (2009) · Zbl 1205.90319 · doi:10.1016/j.apm.2008.05.008 |
| [34] | F. G. Vázquez, Extensions of the Kuhn-Tucker constraint qualification to generalized semi-infinite programming,, SIAM J. Optim., 15, 926 (2005) · Zbl 1114.90141 · doi:10.1137/S1052623403431500 |
| [35] | C. Y. Wang, The stability of the maximum entropy method for nonsmooth semi-infinite programming,, Sci. China Ser. A., 42, 1129 (1999) · Zbl 0959.90054 · doi:10.1007/BF02875980 |
| [36] | C. Y. Wang, Optimal value functions of generalized semi-infinite min-max programming on a noncompact set,, Sci. China Ser. A., 48, 261 (2005) · Zbl 1081.90064 · doi:10.1360/03YS0197 |
| [37] | J. J. Ye, First order optimality conditions for generalized semi-infinite programming problems,, J. Optim. Theory Appl., 137, 419 (2008) · Zbl 1152.90011 · doi:10.1007/s10957-008-9352-z |
| [38] | C. Zălinescu, <em>Convex Analysis in General Vector Spaces</em>,, World Scientific (2002) · Zbl 1023.46003 · doi:10.1142/9789812777096 |
| [39] | L. P. Zhang, An entropy based central plane algorithm for convex min-max semi-infinite programming problems,, Sci. China Math., 56, 201 (2013) · Zbl 1267.90160 · doi:10.1007/s11425-012-4502-z |
| [40] | X. Y. Zheng, Lagrange multipliers in nonsmooth semi-infinite optimization problems,, Math. Oper. Res., 32, 168 (2007) · Zbl 1278.90411 · doi:10.1287/moor.1060.0234 |
| [41] | J. C. Zhou, First-order optimality conditions for convex semi-infinite min-max programming with noncompact sets,, J. Ind. Manag. Optim., 5, 851 (2009) · Zbl 1196.90121 · doi:10.3934/jimo.2009.5.851 |
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