Non-convex semi-infinite min-max optimization with noncompact sets

Meixia Li; Changyu Wang; Biao Qu

doi:10.3934/jimo.2017022

Article Contents

2017, Volume 13, Issue 4: 1859-1881. Doi: 10.3934/jimo.2017022

This issue Previous Article Incremental gradient-free method for nonsmooth distributed optimization Next Article A new analytical model for optimized cognitive radio networks based on stochastic geometry

Non-convex semi-infinite min-max optimization with noncompact sets

1.
School of Mathematics and Information Science, Weifang University, Weifang Shandong, 261061, China
2.
School of Management Science, Qufu Normal University, Rizhao Shandong, 276826, China

* Corresponding author: Meixia Li
* Corresponding author: Meixia Li

Received: October 2015

Revised: October 2016

Published: November 2017

This project is supported by the Natural Science Foundation of China (Grant No. 11571120,11271226,11271233,11401438,11401331), Shandong Provincial Natural Science Foundation (Grant No. ZR2013FL032) and the Project of Shandong Province Higher Educational Science and Technology Program (Grant No. J14LI52).

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 90C34, 90C46; Secondary: 65K10.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	B. Betrò, An accelerated central cutting plane algorithm for linear semi-infinite programming, Math. Program., 101 (2004), 479-495. doi: 10.1007/s10107-003-0492-5.
[2]	S. C. Fang, C. J. Linb and S. Y. Wu, Solving quadratic semi-infinite programming problems by using relaxed cutting-plane scheme, J. Comput. Appl. Math., 129 (2001), 89-104. doi: 10.1016/S0377-0427(00)00544-6.
[3]	D. Gale, A geometric duality theorem with economic applications, Rev. Econ. Stud., 34 (1967), 19-24. doi: 10.2307/2296568.
[4]	M. A. Goberna and M. A. López, Semi-Infinite Programming-Recent Advances, Kluwer, Boston, 2001. doi: 10.1007/978-1-4757-3403-4.
[5]	F. Guerra-Vazquez, H. Th. Jongen and V. Shikhman, General semi-infinite programming: Symmetric Mangasarian-Fromovitz constraint qualification and the closure of the feasible set, SIAM J. Optim., 20 (2010), 2487-2503. doi: 10.1137/090775294.
[6]	A. Hantoute and M. A. López, A complete characterization of the subdifferential set of the supremum of an arbitrary family of convex functions, J. Convex Anal., 15 (2008), 831-858.
[7]	A. Hantoute, M. A. López and C. Zǎlinescu, Subdifferential calculus rules in convex analysis: A unifying approach via pointwise supremum functions, SIAM J. Optim., 19 (2008), 863-882. doi: 10.1137/070700413.
[8]	R. Hettich and K. O. Kortanek, Semi-infinite programming: Theory, methods, and applications, SIAM Rev., 35 (1993), 380-429. doi: 10.1137/1035089.
[9]	R. Hettich and G. Still, Second order optimality conditions for generalized semi-infinite programming problems, Optimization, 34 (1995), 195-211. doi: 10.1080/02331939508844106.
[10]	H. Th. Jongen, J. J. Rückmann and O. Stein, Generalized semi-infinite optimization: A first order optimality condition and examples, Math. Program., 83 (1998), 145-158. doi: 10.1007/BF02680555.
[11]	N. Kanzi and S. Nobakhtian, Necessary optimality conditions for nonsmooth generalized semi-infinite programming problems, Eur. J. Oper. Res., 205 (2010), 253-261. doi: 10.1016/j.ejor.2009.12.025.
[12]	N. Kanzi, Necessary optimality conditions for nonsmooth semi-infinite programming problems, J. Global Optim., 49 (2011), 713-725. 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 (2013), 417-430. doi: 10.1007/s10898-011-9828-5.
[14]	C. Ling, Q. Ni, L. Q. Qi and S. Y. Wu, A new smoothing Newton-type algorithm for semi-infinite programming, J. Global Optim., 47 (2010), 133-159. doi: 10.1007/s10898-009-9462-7.
[15]	Q. Liu, C. Y. Wang and X. M. Yang, On the convergence of a smoothed penalty algorithm for semi-infinite programming, Math. Methods Oper. Res., 78 (2013), 203-220. doi: 10.1007/s00186-013-0440-y.
[16]	M. López and G. Still, Semi-infinite programming, Eur. J. Oper. Res., 180 (2007), 491-518. doi: 10.1016/j.ejor.2006.08.045.
[17]	S. K. Mishra, M. Jaiswal and H. A. Le Thi, Nonsmooth semi-infinite programming problem using limiting subdifferentials, J. Global Optim., 53 (2012), 285-296. doi: 10.1007/s10898-011-9690-5.
[18]	Q. Ni, C. Ling, L. Q. Qi and K. L. Teo, A truncated projected Newton-type algorithm for large-scale semi-infinite programming, SIAM J. Optim., 16 (2006), 1137-1154. doi: 10.1137/040619867.
[19]	E. Polak, Optimization: Algorithms and Consistent Approximations, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0663-7.
[20]	L. Q. Qi, S. Y. Wu and G. L. Zhou, Semismooth Newton methods for solving semi-infinite programming problems, J. Global Optim., 27 (2003), 215-232. doi: 10.1023/A:1024814401713.
[21]	R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.
[22]	R. T. Rockafellar and R. J. Wets, Variational Analysis, Springer, New York, 1998. doi: 10.1007/978-3-642-02431-3.
[23]	J. J. Rückmann and A. Shapiro, First-order optimality conditions in generalized semi-infinite programming, J. Optim. Theory and Appl., 101 (1999), 677-691. doi: 10.1023/A:1021746305759.
[24]	A. Shapiro, On duality theory of convex semi-infinite programming, Optimization, 54 (2005), 535-543. doi: 10.1080/02331930500342823.
[25]	A. Shapiro, Semi-infinite programming, duality, discretization and optimality condition, Optimization, 58 (2009), 133-161. 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 (1998), 807-832. 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 (2001), 99-121. doi: 10.1070/im2001v065n01ABEH000323.
[28]	O. Stein and G. Still, On optimality conditions for generalized semi-infinite programming problems, J. Optim. Theory Appl., 104 (2000), 443-458. 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 (2001), 565-582. doi: 10.1287/moor.26.3.565.10583.
[30]	O. Stein and A. Winterfeld, Feasible method for generalized semi-infinite programming, J. Optim. Theory Appl., 146 (2010), 419-443. doi: 10.1007/s10957-010-9674-5.
[31]	Y. Tanaka, M. Fukushima and T. Ibaraki, A globally convergent SQP method for semi-infinite nonlinear optimization, J. Comput. Appl. Math., 23 (1988), 141-153. doi: 10.1016/0377-0427(88)90276-2.
[32]	X. J. Tong, C. Ling and L. Q. Qi, A semi-infinite programming algorithm for solving optimal power flow with transient stability constraints, J. Comput. Appl. Math., 217 (2008), 432-447. doi: 10.1016/j.cam.2007.02.026.
[33]	A. I. F. Vaz and E. C. Ferreira, Air pollution control with semi-infinite programming, Appl. Math. Modelling, 33 (2009), 1957-1969. doi: 10.1016/j.apm.2008.05.008.
[34]	F. G. Vázquez and J. J. Rückmann, Extensions of the Kuhn-Tucker constraint qualification to generalized semi-infinite programming, SIAM J. Optim., 15 (2005), 926-937. doi: 10.1137/S1052623403431500.
[35]	C. Y. Wang and J. Y. Han, The stability of the maximum entropy method for nonsmooth semi-infinite programming, Sci. China Ser. A., 42 (1999), 1129-1136. doi: 10.1007/BF02875980.
[36]	C. Y. Wang, X. Q. Yang and X. M. Yang, Optimal value functions of generalized semi-infinite min-max programming on a noncompact set, Sci. China Ser. A., 48 (2005), 261-276. doi: 10.1360/03YS0197.
[37]	J. J. Ye and S. Y. Wu, First order optimality conditions for generalized semi-infinite programming problems, J. Optim. Theory Appl., 137 (2008), 419-434. doi: 10.1007/s10957-008-9352-z.
[38]	C. Zǎlinescu, Convex Analysis in General Vector Spaces, World Scientific, Singapore, 2002. doi: 10.1142/9789812777096.
[39]	L. P. Zhang, S. C. Fang and S. Y. Wu, An entropy based central plane algorithm for convex min-max semi-infinite programming problems, Sci. China Math., 56 (2013), 201-211. doi: 10.1007/s11425-012-4502-z.
[40]	X. Y. Zheng and X. Q. Yang, Lagrange multipliers in nonsmooth semi-infinite optimization problems, Math. Oper. Res., 32 (2007), 168-181. doi: 10.1287/moor.1060.0234.
[41]	J. C. Zhou, C. Y. Wang, N. H. Xiu and S. Y. Wu, First-order optimality conditions for convex semi-infinite min-max programming with noncompact sets, J. Ind. Manag. Optim., 5 (2009), 851-866. doi: 10.3934/jimo.2009.5.851.