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Quadratic Growth and Linear Convergence of a DCA Method for Quartic Minimization over the Sphere

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  • Shenglong Hu

    (Hangzhou Dianzi University)

  • Zhifang Yan

    (Hangzhou Dianzi University)

Abstract

The quartic minimization over the sphere can be reformulated as a nonlinear nonconvex semidefinite program over the spectraplex. In this paper, under mild assumptions, we show that the reformulated nonlinear semidefinite program possesses the quadratic growth property at a rank one critical point which is a local minimizer of the quartic minimization problem. The quadratic growth property further implies the strong metric subregularity of the subdifferential of the objective function of the unconstrained reformulation of the nonlinear semidefinite program, from which we can show that the objective function is a Łojasiewicz function with exponent $$\frac{1}{2}$$ 1 2 at the corresponding critical point. With these results, we can establish the linear convergence of an efficient DCA method proposed for solving the nonlinear semidefinite program.

Suggested Citation

  • Shenglong Hu & Zhifang Yan, 2024. "Quadratic Growth and Linear Convergence of a DCA Method for Quartic Minimization over the Sphere," Journal of Optimization Theory and Applications, Springer, vol. 201(1), pages 378-395, April.
    • Handle: RePEc:spr:joptap:v:201:y:2024:i:1:d:10.1007_s10957-024-02401-w
    DOI: 10.1007/s10957-024-02401-w
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    References listed on IDEAS

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    1. Shenglong Hu & Liqun Qi, 2012. "Algebraic connectivity of an even uniform hypergraph," Journal of Combinatorial Optimization, Springer, vol. 24(4), pages 564-579, November.
    2. Jong-Shi Pang & Meisam Razaviyayn & Alberth Alvarado, 2017. "Computing B-Stationary Points of Nonsmooth DC Programs," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 95-118, January.
    3. Defeng Sun, 2006. "The Strong Second-Order Sufficient Condition and Constraint Nondegeneracy in Nonlinear Semidefinite Programming and Their Implications," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 761-776, November.
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