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
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. 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.

[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-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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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.

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

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

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. 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.

[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-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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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.

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

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

[1]

Jinchuan Zhou, Changyu Wang, Naihua Xiu, Soonyi Wu. First-order optimality conditions for convex semi-infinite min-max programming with noncompact sets. Journal of Industrial & Management Optimization, 2009, 5 (4) : 851-866. doi: 10.3934/jimo.2009.5.851

[2]

Cheng Ma, Xun Li, Ka-Fai Cedric Yiu, Yongjian Yang, Liansheng Zhang. On an exact penalty function method for semi-infinite programming problems. Journal of Industrial & Management Optimization, 2012, 8 (3) : 705-726. doi: 10.3934/jimo.2012.8.705

[3]

Xiaoling Guo, Zhibin Deng, Shu-Cherng Fang, Wenxun Xing. Quadratic optimization over one first-order cone. Journal of Industrial & Management Optimization, 2014, 10 (3) : 945-963. doi: 10.3934/jimo.2014.10.945

[4]

Monika Laskawy. Optimality conditions of the first eigenvalue of a fourth order Steklov problem. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1843-1859. doi: 10.3934/cpaa.2017089

[5]

Zhi Guo Feng, Kok Lay Teo, Volker Rehbock. A smoothing approach for semi-infinite programming with projected Newton-type algorithm. Journal of Industrial & Management Optimization, 2009, 5 (1) : 141-151. doi: 10.3934/jimo.2009.5.141

[6]

Yanqun Liu, Ming-Fang Ding. A ladder method for linear semi-infinite programming. Journal of Industrial & Management Optimization, 2014, 10 (2) : 397-412. doi: 10.3934/jimo.2014.10.397

[7]

Mohammed Al Horani, Angelo Favini. First-order inverse evolution equations. Evolution Equations & Control Theory, 2014, 3 (3) : 355-361. doi: 10.3934/eect.2014.3.355

[8]

Jinchuan Zhou, Naihua Xiu, Jein-Shan Chen. Solution properties and error bounds for semi-infinite complementarity problems. Journal of Industrial & Management Optimization, 2013, 9 (1) : 99-115. doi: 10.3934/jimo.2013.9.99

[9]

Burcu Özçam, Hao Cheng. A discretization based smoothing method for solving semi-infinite variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 219-233. doi: 10.3934/jimo.2005.1.219

[10]

Ansgar Jüngel, Ingrid Violet. First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 861-877. doi: 10.3934/dcdsb.2007.8.861

[11]

Pierre Fabrie, Alain Miranville. Exponential attractors for nonautonomous first-order evolution equations. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 225-240. doi: 10.3934/dcds.1998.4.225

[12]

Cyril Joel Batkam. Homoclinic orbits of first-order superquadratic Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3353-3369. doi: 10.3934/dcds.2014.34.3353

[13]

Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281

[14]

B. Bonnard, J.-B. Caillau, E. Trélat. Second order optimality conditions with applications. Conference Publications, 2007, 2007 (Special) : 145-154. doi: 10.3934/proc.2007.2007.145

[15]

Liwei Zhang, Jihong Zhang, Yule Zhang. Second-order optimality conditions for cone constrained multi-objective optimization. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1-14. doi: 10.3934/jimo.2017089

[16]

Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483

[17]

Gábor Kiss, Bernd Krauskopf. Stability implications of delay distribution for first-order and second-order systems. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 327-345. doi: 10.3934/dcdsb.2010.13.327

[18]

Rafael del Rio, Mikhail Kudryavtsev, Luis O. Silva. Inverse problems for Jacobi operators III: Mass-spring perturbations of semi-infinite systems. Inverse Problems & Imaging, 2012, 6 (4) : 599-621. doi: 10.3934/ipi.2012.6.599

[19]

Igor Chueshov. Remark on an elastic plate interacting with a gas in a semi-infinite tube: Periodic solutions. Evolution Equations & Control Theory, 2016, 5 (4) : 561-566. doi: 10.3934/eect.2016019

[20]

Roman Chapko, B. Tomas Johansson. An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions. Inverse Problems & Imaging, 2008, 2 (3) : 317-333. doi: 10.3934/ipi.2008.2.317

2016 Impact Factor: 0.994

Article outline

[Back to Top]