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First-Order Methods for Nonnegative Trigonometric Matrix Polynomials

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  • Daniel Cederberg

    (Stanford University)

Abstract

Optimization problems over the cone of nonnegative trigonometric matrix polynomials are common in signal processing. Traditionally, those problems are formulated and solved as semidefinite programs (SDPs). However, the SDP formulation increases the dimension of the problem, resulting in large problems that are challenging to solve. In this paper we propose first-order methods that circumvent the SDP formulation and instead optimize directly within the space of trigonometric matrix polynomials. Our methods are based on a particular Bregman proximal operator. We apply our approach to two fundamental signal processing applications: to rectify a power spectrum that fails to be nonnegative and for graphical modeling of Gaussian time series. Numerical experiments demonstrate that our methods are orders of magnitude faster than an interior-point solver applied to the corresponding SDPs.

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  • Daniel Cederberg, 2025. "First-Order Methods for Nonnegative Trigonometric Matrix Polynomials," Journal of Optimization Theory and Applications, Springer, vol. 204(2), pages 1-28, February.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:2:d:10.1007_s10957-024-02581-5
    DOI: 10.1007/s10957-024-02581-5
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    References listed on IDEAS

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    1. Brendan O’Donoghue & Eric Chu & Neal Parikh & Stephen Boyd, 2016. "Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 1042-1068, June.
    2. Xin Jiang & Lieven Vandenberghe, 2023. "Bregman Three-Operator Splitting Methods," Journal of Optimization Theory and Applications, Springer, vol. 196(3), pages 936-972, March.
    3. Brillinger, David R., 1996. "Remarks Concerning Graphical Models for Time Series and Point Processes," Brazilian Review of Econometrics, Sociedade Brasileira de Econometria - SBE, vol. 16(1), November.
    4. Xin Jiang & Lieven Vandenberghe, 2022. "Bregman primal–dual first-order method and application to sparse semidefinite programming," Computational Optimization and Applications, Springer, vol. 81(1), pages 127-159, January.
    5. GENIN, Yves & HACHEZ, Yvan & NESTEROV, Yu. & VAN DOOREN, Paul, 2003. "Optimizat problems over positive pseudopolynomial matrices," LIDAM Reprints CORE 1622, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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