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Equivalence of Minimal ℓ 0- and ℓ p -Norm Solutions of Linear Equalities, Inequalities and Linear Programs for Sufficiently Small p

Author

Listed:
  • G. M. Fung

    (Siemens Medical Solutions, Inc.)

  • O. L. Mangasarian

    (University of Wisconsin
    University of California at San Diego)

Abstract

For a bounded system of linear equalities and inequalities, we show that the NP-hard ℓ 0-norm minimization problem is completely equivalent to the concave ℓ p -norm minimization problem, for a sufficiently small p. A local solution to the latter problem can be easily obtained by solving a provably finite number of linear programs. Computational results frequently leading to a global solution of the ℓ 0-minimization problem and often producing sparser solutions than the corresponding ℓ 1-solution are given. A similar approach applies to finding minimal ℓ 0-solutions of linear programs.

Suggested Citation

  • G. M. Fung & O. L. Mangasarian, 2011. "Equivalence of Minimal ℓ 0- and ℓ p -Norm Solutions of Linear Equalities, Inequalities and Linear Programs for Sufficiently Small p," Journal of Optimization Theory and Applications, Springer, vol. 151(1), pages 1-10, October.
    • Handle: RePEc:spr:joptap:v:151:y:2011:i:1:d:10.1007_s10957-011-9871-x
    DOI: 10.1007/s10957-011-9871-x
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    Cited by:

    1. Guowei You & Zheng-Hai Huang & Yong Wang, 2019. "The sparsest solution of the union of finite polytopes via its nonconvex relaxation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(3), pages 485-507, June.
    2. Lu, Yisha & Hu, Yaozhong & Qiao, Yan & Yuan, Minjuan & Xu, Wei, 2024. "Sparse least squares via fractional function group fractional function penalty for the identification of nonlinear dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    3. Fassino, Claudia & Torrente, Maria-Laura & Uberti, Pierpaolo, 2022. "A singular value decomposition based approach to handle ill-conditioning in optimization problems with applications to portfolio theory," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    4. Chan, Felix & Pauwels, Laurent, 2019. "Equivalence of optimal forecast combinations under affine constraints," Working Papers BAWP-2019-02, University of Sydney Business School, Discipline of Business Analytics.
    5. Yue Xie & Uday V. Shanbhag, 2021. "Tractable ADMM schemes for computing KKT points and local minimizers for $$\ell _0$$ ℓ 0 -minimization problems," Computational Optimization and Applications, Springer, vol. 78(1), pages 43-85, January.
    6. Ziyan Luo & Linxia Qin & Lingchen Kong & Naihua Xiu, 2014. "The Nonnegative Zero-Norm Minimization Under Generalized Z-Matrix Measurement," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 854-864, March.

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