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Semi-definite Representations for Sets of Cubics on the Two-dimensional Sphere

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  • Roland Hildebrand

    (Université Grenoble Alpes, CNRS, Grenoble INP, LJK)

Abstract

The compact set of homogeneous quadratic polynomials in n real variables with modulus bounded by 1 on the unit sphere is trivially semi-definite representable. The compact set of homogeneous ternary quartics with modulus bounded by 1 on the unit sphere is also semi-definite representable. This suggests that the compact set of homogeneous ternary cubics with modulus bounded by 1 on the unit sphere is semi-definite representable. We deduce an explicit semi-definite representation of this norm ball. More generally, we provide a semi-definite description of the cone of inhomogeneous ternary cubics which are nonnegative on the unit sphere. This allows to incorporate nonnegativity conditions on polynomials in this space into semi-definite programs by transforming them into semi-definite constraints on the coefficient vector.

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  • Roland Hildebrand, 2022. "Semi-definite Representations for Sets of Cubics on the Two-dimensional Sphere," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 666-675, November.
    • Handle: RePEc:spr:joptap:v:195:y:2022:i:2:d:10.1007_s10957-022-02104-0
    DOI: 10.1007/s10957-022-02104-0
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    References listed on IDEAS

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    1. NESTEROV, Yu., 2000. "Squared functional systems and optimization problems," LIDAM Reprints CORE 1472, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Roland Hildebrand, 2021. "Optimal step length for the Newton method: case of self-concordant functions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(2), pages 253-279, October.
    3. Faizan Ahmed & Georg Still, 2019. "Maximization of Homogeneous Polynomials over the Simplex and the Sphere: Structure, Stability, and Generic Behavior," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 972-996, June.
    4. NESTEROV, Yu, 2003. "Random walk in a simplex and quadratic optimization over convex polytopes," LIDAM Discussion Papers CORE 2003071, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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