Convexification for a Class of Global Optimization Problems with C1,1 Functions

doi:10.1007/s40305-023-00521-0

Journal of the Operations Research Society of China ›› 2025, Vol. 13 ›› Issue (4): 900-918.doi: 10.1007/s40305-023-00521-0

Previous Articles Next Articles

Convexification for a Class of Global Optimization Problems with C^1,1 Functions

Qian Yan^1,2, Xin-Min Yang^3,4, Zhi-You Wu⁴

1 College of Sciences, Chongqing University of Technology, Chongqing 400054, China;
2 College of Mathematics, Sichuan University, Chengdu, Sichuan 610065, China;
3 National Center for Applied Mathematics of Chongqing, Chongqing 401331, China;
4 College of Mathematical Science, Chongqing Normal University, Chongqing 401331, China

Received:2022-11-16 Revised:2023-09-12 Online:2025-12-30 Published:2025-12-19
Contact: Qian Yan E-mail:qianyanmath@163.com
Supported by:
This work was supported by the Major Program of the National Natural Science Foundation of China (Nos. 11991020, 11991024), the National Natural Science Foundation of China (No. 11871128), Scientific Research Foundation of Chongqing University of Technology (No. 2023ZDZ021), Youth project of science and technology research program of Chongqing Education Commission of China (No. KJQN202301160).

Abstract

Abstract: A general convexification method via domain transformation scheme is presented for solving a class of global optimization problems with certain monotone properties. It is shown that this class of problems with C^1,1 functions can be converted into equivalent convex optimization problems by using the proposed convexification method. Finally, an example is shown to illustrate how a monotone non-convex optimization problem can be transformed into an equivalent convex minimization problem.

Key words: Global optimization, Monotone optimization, C^1,1 functions, Convexification

CLC Number:

90C26
90C30

Qian Yan, Xin-Min Yang, Zhi-You Wu. Convexification for a Class of Global Optimization Problems with C^1,1 Functions[J]. Journal of the Operations Research Society of China, 2025, 13(4): 900-918.

TrendMD

References

[1] Tzafestas, S.G.: Optimization of system reliability: a survey of problems and techniques. Int. J. Syst. Sci. 11, 455–486 (1980)
[2] Ibaraki, T., Katoh, N.: Resource Allocation Problems: Algorithmic Approaches. MIT Press, Cambridge (1988)
[3] Li, D., Sun, X.L., Biswal, M.P., Gao, F.: Convexification, concavification, and monotonization in global optimization. Ann. Oper. Res. 105, 213–226 (2001)
[4] Sun, X.L., Mckinnon, K.I.M., Li, D.: A convexification method for a class of global optimization problems with applications to reliability optimization. J. Glob. Optim. 21, 185–199 (2001)
[5] Xia, Y.: A survey of hidden convex optimization. J. Oper. Res. Soc. China 8, 1–28 (2020)
[6] Benson, H.P.: Deterministic algorithms for constrained concave minimization: a unified critical survey. Naval Res. Log. 43, 765–795 (1996)
[7] Hoffman, K.L.: A method for globally minimizing concave functions over convex sets. Math. Program. 20, 22–32 (1981)
[8] Horst, R., Pardalos, P.M., Van Thoai, N.: Introduction to Global Optimization. Springer (2000)
[9] Tuy, H.: Monotonic optimization: problems and solution approaches. SIAM J. Optim. 11, 464–494 (2000)
[10] Tuy, H.: Convex Analysis and Global Optimization. Spring, Berlin (2016)
[11] Horst, R.: On the convexification of nonlinear programming problems: an applications-oriented survey. Eur. J. Oper. Res. 15, 382–392 (1984)
[12] Xu, G.X.: Global optimization of signomial geometric programming problems. Eur. J. Oper. Res. 233, 500–510 (2014)
[13] Lu, H.C., Yao, L.M.: Efficient convexification strategy for generalized geometric programming problems. INFORMS J. Comput. 31, 226–234 (2019)
[14] Wu, Z.Y., Zhang, L.S., Bai, F.S., Yang, X.M.: Convexification and convexification methods for some global optimization problems. J. Syst. Sci. Complex 17, 421–436 (2004)
[15] Wu, Z.Y., Bai, F.S., Zhang, L.S.: Convexification and concavification for a general class of global optimization problems. J. Glob. Optim. 31, 45–60 (2005)
[16] Wu, Z.Y., Lee, H.W.J., Yang, X.M.: A class of convexification and concavification methods for nonmonotone optimization problems. Optimization 54, 605–625 (2005)
[17] Li, D., Wu, Z.Y., Lee, H.W.J., Yang, X.M., Zhang, L.S.: Hidden convex minimization. J. Glob. Optim. 31, 211–233 (2005)
[18] Sun, X.L., Luo, H.Z., Li, D.: Convexification of nonsmooth monotone functions. J. Optim. Theory Appl. 132, 339–351 (2007)
[19] Li, D.: Convexification of a noninferior frontier. J. Optim. Theory Appl. 88, 177–196 (1996)
[20] Zhang, L.S., Wu, D.H.: Convexification, concavification and monotonization in nonlinear programming. Chin. Ann. Math. Ser. A 23, 537–544 (2002)
[21] Yan, Q., Yang, X.M., Wu, Z.Y.: On convexification for a class of global optimization problems. J. Oper. Res. Soc. China. https://doi.org/10.1007/s40305-021-00379-0
[22] Rockafellar, R.T.: Solving a nonlinear programming problem by way of a dual problem. Symp. Math. 19, 135–160 (1976)
[23] Beck, A.: First-Order Methods in Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2017)
[24] Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)
[25] Bienstock, D.: Computational study of a family of mixed-integer quadratic programming problems. Math. Program. 74, 121–140 (1996)
[26] Chang, T.J., Meade, N., Beasley, J.E., Sharaiha, Y.M.: Heuristics for cardinality constrained portfolio optimisation. Comput. Oper. Res. 27, 1271–1302 (2000)
[27] La Torre, D., Vercellis, C.: C1, 1 approximations of generalized support vector machines. J. Concr. Appl. Math. 3, 41–54 (2005)
[28] Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer (2009)
[29] Ginchev, I., La Torre, D., Rocca, M.: Ck, 1 functions, characterization, Taylor’s formula and optimization: a survey. Real Anal. Exch. 35, 311–342 (2010)
[30] Yang, X.Q.: Generalized second-order directional derivatives and optimization conditions. Ph.D. Disseration, University of New South Wales, Australia (1994)
[31] Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
[32] Yang, X.Q., Jeyakumar, V.: Generalized second-order directional derivatives and optimization with C1, 1 functions. Optimization 26, 165–185 (1991)

Convexification for a Class of Global Optimization Problems with C^1,1 Functions

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 10

Recommended Articles

Metrics

Comments

[1]	Qiao Chen, Xin-Min Yang, Qian Yan. A New Class of Filled Functions with Two Parameters for Solving Unconstrained Global Optimization Problems [J]. Journal of the Operations Research Society of China, 2024, 12(4): 921-936.
[2]	Hong-Wei Jiao, You-Lin Shang. Two-Level Linear Relaxation Method for Generalized Linear Fractional Programming [J]. Journal of the Operations Research Society of China, 2023, 11(3): 569-594.
[3]	Chen Wang, Dong-Hua Wu. A K-Means Clustering-Based Multiple Importance Sampling Algorithm for Integral Global Optimization [J]. Journal of the Operations Research Society of China, 2023, 11(1): 157-176.
[4]	Qian Yan, Xin-Min Yang, Zhi-You Wu. On Convexification for a Class of Global Optimization Problems [J]. Journal of the Operations Research Society of China, 2022, 10(3): 427-446.
[5]	Zhong-Yu Wang, Yong-Jian Yang. Novel Global Optimization Algorithm with a Space-Filling Curve and Integral Function [J]. Journal of the Operations Research Society of China, 2021, 9(3): 619-640.
[6]	Zhe Zhou, Fu-Sheng Bai. A Stochastic Adaptive Radial Basis Function Algorithm for Costly Black-Box Optimization [J]. Journal of the Operations Research Society of China, 2018, 6(4): 587-610.
[7]	Hong-Bin Yu, Wei-Jia Zeng, Dong-Hua Wu. A Stochastic Level-Value Estimation Method for Global Optimization [J]. Journal of the Operations Research Society of China, 2018, 6(3): 429-444.
[8]	Zhi-You Wu · Fu-Sheng Bai · Jing Tian. Optimization Methods for Box-Constrained Nonlinear Programming Problems Based on Linear Transformation and Lagrange Interpolating Polynomials [J]. Journal of the Operations Research Society of China, 2017, 5(2): 193-.
[9]	Yong-Jian Yang · Meng-Li He · Yue-Lin Gao. Discrete Global Optimization Problems with a Modified Discrete Filled Function [J]. Journal of the Operations Research Society of China, 2015, 3(3): 297-.
[10]	Zhao Sun. A Refined Error Analysis for Fixed-Degree Polynomial Optimization over the Simplex [J]. Journal of the Operations Research Society of China, 2014, 2(3): 379-394.