多凸规划罚函数的部分精确性研究

doi:10.15960/j.cnki.issn.1007-6093.2024.04.005

Abstract

Abstract:

Multi-convex programming(MCP) is an important model in solving many engineering optimization problems in areas like machine learning and signal and information processing. In this paper, some new concepts of partial optimum, partial KKT condition, partial KKT ponit, partial Slater constraint qualification, partial exactness and partial stableness for the penalty function of multi-convex programming are defined. Under the partial Slater constraint qualification, a partial optimum of MCP is proved to be equivalent to partial KKT condition of MCP. The partial exactness of MCP is proved to be equivalent to partial KKT condition of MCP. The partial exactness of MCP is proved to be equivalent to partial stableness of MCP. These results are important for studying the exact penalty function of multi convex programming.

Key words: multi-convex programming, partial optimum, partial KKT condition, partial exactness, partial stableness

CLC Number:

O122.2

Yichen LAI, Zhiqing MENG. Partial exactness of penalty function of multi-convex programming[J]. Operations Research Transactions, 2024, 28(4): 57-65.

References 26

1	Shen X, Diamond S, Udell M, et al. Disciplined multi-convex programming [C]//Chinese Control and Decision Conference, 2017: 895-900.
2	Grant M, Boyd S, Ye Y. Disciplined convex programming [M]//Global Optimization: From Theory to Implementation, Nonconvex Optimization and Its Applications, New York: Springer, 2006: 155-210.
3	Chiu W Y . Method of reduction of variables for bilinear matrix inequality problems in system and control designs[J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2017, 47 (7): 1241- 1256. doi: 10.1109/TSMC.2016.2571323
4	Ichihara H, Nobuyama E. Difference of multiconvex relaxation of parameterized LMIs: Control applications [C]//SICE 2003 Annual Conference, IEEE Xplore, 2003.
5	Hours J, Jones C. A parametric multiconvex splitting technique with application to real-time NMPC [C]//53rd IEEE Conference on Decision and Control, 2014: 5052-5057.
6	Suh S , Shin S , Lee J , et al. Localized user-driven topic discovery via boosted ensemble of nonnegative matrix factorization[J]. Knowledge and Information Systems, 2018, 56, 503- 531. doi: 10.1007/s10115-017-1147-9
7	Udell M , Horn C , Zadeh R , et al. Generalized low rank models[J]. Foundations and Trends in Machine Learning, 2016, 9 (1): 1- 118. doi: 10.1561/2200000055
8	Fu X , Huang K , Sidiropoulos N D . On identifiability of nonnegative matrix factorization[J]. IEEE Signal Processing Letters, 2018, 25 (3): 328- 332. doi: 10.1109/LSP.2018.2789405
9	Hong M , Luo Z Q . On the linear convergence of the alternating direction method of multipliers[J]. Mathematical Programming, 2017, 162, 165- 199. doi: 10.1007/s10107-016-1034-2
10	Serbetli S , Yener A . Transceiver optimization for multiuser MIMO systems[J]. IEEE Transactions on Signal Processing, 2004, 52 (1): 214- 226. doi: 10.1109/TSP.2003.819988
11	Al-Shatri H , Li X , Ganesa R S S . Maximizing the sum rate in cellular networks using multiconvex optimization[J]. IEEE Transactions on Wireless Communications, 2016, 15 (5): 3199- 3211. doi: 10.1109/TWC.2016.2517632
12	Wen Z , Yin W , Zhang Y . Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm[J]. Mathematical Programming Computation, 2012, 4 (4): 333- 361. doi: 10.1007/s12532-012-0044-1
13	Xu Y , Yin W . A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion[J]. SIAM Journal on Imaging Sciences, 2013, 6 (3): 1758- 1789. doi: 10.1137/120887795
14	Gorski J , Pfeuffer F , Klamroth K . Biconvex sets and optimization with biconvex functions: A survey and extensions[J]. Mathematical Methods of Operations Research, 2007, 66, 373- 407. doi: 10.1007/s00186-007-0161-1
15	Li G , Wen C , Zheng W X , et al. Iterative identification of block-oriented nonlinear systems based on biconvex optimization[J]. Systems & Control Letters, 2015, 79, 68- 75.
16	Shah S, Yadav A K, Castillo C D, et al. Biconvex relaxation for semidefinite programming in computer vision [C]//European Conference on Computer Vision: Computer Vision, 2016: 717-735.
17	Al-Khayyaltt F A , Falk J E . Jointly constrained biconvex programming[J]. Mathematics of Operations Research, 1983, 8 (2): 273- 286. doi: 10.1287/moor.8.2.273
18	Liang X , Bai J . Preconditioned ADMM for a class of bilinear programming problems[J]. Mathematical Problems in Engineering, 2018, 2018, 1- 9.
19	Hajinezhad D , Shi Q J . Alternating direction method of multipliers for a class of nonconvex bilinear optimization: Convergence analysis and applications[J]. Journal of Global Optimization, 2018, 70, 261- 88. doi: 10.1007/s10898-017-0594-x
20	Charkhgard H , Savelsbergh M , Talebian M . A linear programming based algorithm to solve a class of optimization problems with a multi-linear objective function and affine constraints[J]. Computers & Operations Research, 2018, 89, 17- 30.
21	Pardalos P M , Resende M G C . Handbook of Applied Optimization[M]. Oxford: Oxford University Press, 2002.
22	Zangwill W I . Nonlinear programming via penalty function[J]. Manangement Science, 1967, 13, 334- 358.
23	Rosenberg E . Globally convergent algorithms for convex programming[J]. Mathematics of Operations Rresearch, 1981, 6, 437- 443. doi: 10.1287/moor.6.3.437
24	Di Pillo G , Grippo L . An exact penalty function method with global conergence properties for nonlinear programming problems[J]. Mathemathical Programming, 1981, 36, 1- 18.
25	Clarke F H . Optimization and Nonsmooth Analysis[M]. New York: Wiley, 1983.
26	Burke J V . Calmness and exact penalization[J]. SIAM Journal on Control and Optimization, 1991, 29, 493- 497.

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Partial exactness of penalty function of multi-convex programming

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