2017, 13(4): 1859-1881. doi: 10.3934/jimo.2017022

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

Received  October 2015 Revised  October 2016 Published  December 2016

Fund Project: 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)

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.

Citation: Meixia Li, Changyu Wang, Biao Qu. Non-convex semi-infinite min-max optimization with noncompact sets. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1859-1881. doi: 10.3934/jimo.2017022
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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.

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S. C. Fang, C. J. Linb, 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.

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F. Guerra-Vazquez, H. Th. Jongen, 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.

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A. Hantoute, 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.

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A. Hantoute, M. A. López, 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.

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R. Hettich, G. Still, Second order optimality conditions for generalized semi-infinite programming problems, Optimization, 34 (1995), 195-211. doi: 10.1080/02331939508844106.

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H. Th. Jongen, J. J. Rückmann, O. Stein, Generalized semi-infinite optimization: A first order optimality condition and examples, Math. Program., 83 (1998), 145-158. doi: 10.1007/BF02680555.

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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.

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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.

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C. Ling, Q. Ni, L. Q. Qi, 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.

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Q. Liu, C. Y. Wang, 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.

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M. López, G. Still, Semi-infinite programming, Eur. J. Oper. Res., 180 (2007), 491-518. doi: 10.1016/j.ejor.2006.08.045.

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S. K. Mishra, M. Jaiswal, 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.

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Q. Ni, C. Ling, L. Q. Qi, 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.

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L. Q. Qi, S. Y. Wu, 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.
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J. J. Rückmann, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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.

show all references

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, 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, 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, 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, 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, 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, K. O. Kortanek, Semi-infinite programming: Theory, methods, and applications, SIAM Rev., 35 (1993), 380-429. doi: 10.1137/1035089.

[9]

R. Hettich, 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, 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, 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, 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, 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, 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, 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, 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, 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, R. J. Wets, Variational Analysis, Springer, New York, 1998. doi: 10.1007/978-3-642-02431-3.
[23]

J. J. Rückmann, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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.

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