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An objective penalty function method for biconvex programming

Author

Listed:
  • Zhiqing Meng

    (Zhejiang University of Technology)

  • Min Jiang

    (Zhejiang University of Technology)

  • Rui Shen

    (Zhejiang University of Technology)

  • Leiyan Xu

    (Nanjing Vocational College of Information Technology)

  • Chuangyin Dang

    (City University of Hong Kong)

Abstract

Biconvex programming is nonconvex optimization describing many practical problems. The existing research shows that the difficulty in solving biconvex programming makes it a very valuable subject to find new theories and solution methods. This paper first obtains two important theoretical results about partial optimum of biconvex programming by the objective penalty function. One result holds that the partial Karush–Kuhn–Tucker (KKT) condition is equivalent to the partially exactness for the objective penalty function of biconvex programming. Another result holds that the partial stability condition is equivalent to the partially exactness for the objective penalty function of biconvex programming. These results provide a guarantee for the convergence of algorithms for solving a partial optimum of biconvex programming. Then, based on the objective penalty function, three algorithms are presented for finding an approximate $$\epsilon $$ ϵ -solution to partial optimum of biconvex programming, and their convergence is also proved. Finally, numerical experiments show that an $$\epsilon $$ ϵ -feasible solution is obtained by the proposed algorithm.

Suggested Citation

  • Zhiqing Meng & Min Jiang & Rui Shen & Leiyan Xu & Chuangyin Dang, 2021. "An objective penalty function method for biconvex programming," Journal of Global Optimization, Springer, vol. 81(3), pages 599-620, November.
    • Handle: RePEc:spr:jglopt:v:81:y:2021:i:3:d:10.1007_s10898-021-01064-5
    DOI: 10.1007/s10898-021-01064-5
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    References listed on IDEAS

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    1. Jochen Gorski & Frank Pfeuffer & Kathrin Klamroth, 2007. "Biconvex sets and optimization with biconvex functions: a survey and extensions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(3), pages 373-407, December.
    2. Davood Hajinezhad & Qingjiang Shi, 2018. "Alternating direction method of multipliers for a class of nonconvex bilinear optimization: convergence analysis and applications," Journal of Global Optimization, Springer, vol. 70(1), pages 261-288, January.
    3. Eric Rosenberg, 1981. "Globally Convergent Algorithms for Convex Programming," Mathematics of Operations Research, INFORMS, vol. 6(3), pages 437-444, August.
    4. Xiaobo Liang & Jianchao Bai, 2018. "Preconditioned ADMM for a Class of Bilinear Programming Problems," Mathematical Problems in Engineering, Hindawi, vol. 2018, pages 1-9, January.
    5. Zhiqing Meng & Chuangyin Dang & Min Jiang & Xinsheng Xu & Rui Shen, 2013. "Exactness and algorithm of an objective penalty function," Journal of Global Optimization, Springer, vol. 56(2), pages 691-711, June.
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