October  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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D. Gale, A geometric duality theorem with economic applications, Rev. Econ. Stud., 34 (1967), 19-24.  doi: 10.2307/2296568.  Google Scholar

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F. Guerra-VazquezH. 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.  Google Scholar

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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.   Google Scholar

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A. HantouteM. 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.  Google Scholar

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

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H. Th. JongenJ. 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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[14]

C. LingQ. NiL. 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.  Google Scholar

[15]

Q. LiuC. 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.  Google Scholar

[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.  Google Scholar

[17]

S. K. MishraM. 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.  Google Scholar

[18]

Q. NiC. LingL. 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.  Google Scholar

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L. Q. QiS. 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.  Google Scholar

[21] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.   Google Scholar
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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.  Google Scholar

[24]

A. Shapiro, On duality theory of convex semi-infinite programming, Optimization, 54 (2005), 535-543.  doi: 10.1080/02331930500342823.  Google Scholar

[25]

A. Shapiro, Semi-infinite programming, duality, discretization and optimality condition, Optimization, 58 (2009), 133-161.  doi: 10.1080/02331930902730070.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

Y. TanakaM. 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.  Google Scholar

[32]

X. J. TongC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

C. Y. WangX. 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.  Google Scholar

[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.  Google Scholar

[38] C. Zǎlinescu, Convex Analysis in General Vector Spaces, World Scientific, Singapore, 2002.  doi: 10.1142/9789812777096.  Google Scholar
[39]

L. P. ZhangS. 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.  Google Scholar

[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.  Google Scholar

[41]

J. C. ZhouC. Y. WangN. 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.  Google Scholar

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.  Google Scholar

[2]

S. C. FangC. 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.  Google Scholar

[3]

D. Gale, A geometric duality theorem with economic applications, Rev. Econ. Stud., 34 (1967), 19-24.  doi: 10.2307/2296568.  Google Scholar

[4] M. A. Goberna and M. A. López, Semi-Infinite Programming-Recent Advances, Kluwer, Boston, 2001.  doi: 10.1007/978-1-4757-3403-4.  Google Scholar
[5]

F. Guerra-VazquezH. 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.  Google Scholar

[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.   Google Scholar

[7]

A. HantouteM. 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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

H. Th. JongenJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

C. LingQ. NiL. 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.  Google Scholar

[15]

Q. LiuC. 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.  Google Scholar

[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.  Google Scholar

[17]

S. K. MishraM. 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.  Google Scholar

[18]

Q. NiC. LingL. 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.  Google Scholar

[19] E. Polak, Optimization: Algorithms and Consistent Approximations, Springer-Verlag, New York, 1997.  doi: 10.1007/978-1-4612-0663-7.  Google Scholar
[20]

L. Q. QiS. 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.  Google Scholar

[21] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.   Google Scholar
[22] R. T. Rockafellar and R. J. Wets, Variational Analysis, Springer, New York, 1998.  doi: 10.1007/978-3-642-02431-3.  Google Scholar
[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.  Google Scholar

[24]

A. Shapiro, On duality theory of convex semi-infinite programming, Optimization, 54 (2005), 535-543.  doi: 10.1080/02331930500342823.  Google Scholar

[25]

A. Shapiro, Semi-infinite programming, duality, discretization and optimality condition, Optimization, 58 (2009), 133-161.  doi: 10.1080/02331930902730070.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

Y. TanakaM. 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.  Google Scholar

[32]

X. J. TongC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

C. Y. WangX. 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.  Google Scholar

[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.  Google Scholar

[38] C. Zǎlinescu, Convex Analysis in General Vector Spaces, World Scientific, Singapore, 2002.  doi: 10.1142/9789812777096.  Google Scholar
[39]

L. P. ZhangS. 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.  Google Scholar

[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.  Google Scholar

[41]

J. C. ZhouC. Y. WangN. 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.  Google Scholar

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