Sobolev seminorm of quadratic functions with applications to derivative-free optimization

Abstract

This paper studies the \(H^1\) Sobolev seminorm of quadratic functions. The research is motivated by the least-norm interpolation that is widely used in derivative-free optimization. We express the \(H^1\) seminorm of a quadratic function explicitly in terms of the Hessian and the gradient when the underlying domain is a ball. The seminorm gives new insights into least-norm interpolation. It clarifies the analytical and geometrical meaning of the objective function in least-norm interpolation. We employ the seminorm to study the extended symmetric Broyden update proposed by Powell. Numerical results show that the new thoery helps improve the performance of the update. Apart from the theoretical results, we propose a new method of comparing derivative-free solvers, which is more convincing than merely counting the numbers of function evaluations.

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Notes

  1. 1.

    Notice that \(f\) is not always equal to the objective function \(F\), which is the case in Powell [3741]. See Sect. 2 for details.

  2. 2.

    We define the convergence on \(\mathcal{Q }\) to be the convergence of coefficients.

  3. 3.

    Since the trust-region radii of the subsequent iterations vary dynamically according to the degree of success, there should be more elaborate ways to define the domain \(\fancyscript{B}\) adaptively. The definition given here might be the simplest one, and is enough for our experiment.

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Acknowledgments

This work is based on sections 4.3–4.5 of my PhD thesis, which was written under the supervision of Professor Ya-xiang Yuan, and I feel much more than grateful to Professor Yuan for his valuable guidance and suggestions. I thank Professor M. J. D. Powell for the encouragement and helpful discussions. Professor Powell helped improve the proof of Theorem 3.1. This work was partly done during a visit to Professor Klaus Schittkowski at Universität Bayreuth in 2010. I thank Alexander von Humboldt Foundation for supporting this visit, and I am much grateful to Professor Schittkowski for his warm hospitality. I appreciate the help of Professor Andrew R. Conn and an anonymous referee, whose comments have substantially improved the paper.

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Correspondence to Zaikun Zhang.

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Partially supported by Chinese NSF grants 10831006, 11021101, CAS grant kjcx-yw-s7, and Alexander von Humboldt Foundation.

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Zhang, Z. Sobolev seminorm of quadratic functions with applications to derivative-free optimization. Math. Program. 146, 77–96 (2014). https://doi.org/10.1007/s10107-013-0679-3

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Keywords

  • Sobolev seminorm
  • Least-norm interpolation
  • Derivative-free optimization
  • Extended symmetric Broyden update

Mathematics Subject Classification (2010)

  • 90C56
  • 90C30
  • 65K05