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Setting Limits on Supersymmetry Using Simplified Models
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A manifold-based approach to sparse global constraint satisfaction problems.

Ali Baharev1, Arnold Neumaier1, Hermann Schichl1

  • 1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Journal of Global Optimization : an International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management and Engineering
|December 7, 2019
PubMed
Summary
This summary is machine-generated.

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This study introduces a novel algorithm for solving large, sparse nonlinear systems of equations. The method efficiently finds multiple solutions by decomposing systems, offering a linear computational growth advantage over traditional exponential approaches.

Area of Science:

  • Numerical analysis
  • Computational mathematics
  • Applied mathematics

Background:

  • Solving large, sparse nonlinear systems is computationally intensive.
  • Existing methods like multistart face exponential computational cost growth.
  • Numerical instability is a known challenge in these decomposition approaches.

Purpose of the Study:

  • To propose an efficient algorithm for approximating all well-separated solutions of square, sparse nonlinear systems.
  • To address and resolve the numerical instability issue in existing decomposition methods.
  • To reduce the computational complexity from exponential to linear with problem size.

Main Methods:

  • The algorithm assumes a bordered block lower triangular form of the Jacobian, achievable via automatic decomposition.
Keywords:
Decomposition methodsDiakopticsLarge-scale systems of equationsNumerical instabilitySparse matricesTearing

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  • It reduces solving large systems to solving a sequence of low-dimensional systems.
  • A novel backsolve step is introduced to mitigate numerical instability.

    • Main Results:

    • The proposed method demonstrates linear computational effort growth with problem size, contrasting with the exponential growth of traditional multistart.
    • The effectiveness of the decomposition and algorithm depends on problem size and hyperparameter settings.
    • Increasing the sample size hyperparameter enhances the method's robustness in finding solutions.

    Conclusions:

    • The developed algorithm offers a computationally efficient and more robust approach to solving large, sparse nonlinear systems.
    • The decomposition strategy and novel backsolve step effectively manage numerical challenges.
    • While not guaranteeing all solutions, the method's robustness can be tuned via hyperparameters.