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A computational study of global optimization solvers on two trust region subproblems.

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  • 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
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Summary
This summary is machine-generated.

This study analyzes branch-and-bound solvers for the two trust region subproblem (TTRS), a type of quadratically constrained quadratic programming. Findings suggest combining branch-and-bound with semidefinite programming (SDP) may overcome orthogonal difficulties for efficient problem-solving.

Keywords:
Branch-and-bound solversCelis–Dennis–Tapia subproblemCluster effectReliability analysisSDP-relaxations

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Area of Science:

  • Optimization
  • Mathematical Programming
  • Computational Science

Background:

  • Quadratically Constrained Quadratic Programming (QCQP) is a significant research area, with contributions from Chris Floudas.
  • The Two Trust Region Subproblem (TTRS) is a fundamental, yet challenging, instance of QCQP involving minimizing a quadratic function over the intersection of two ellipsoids.
  • The Lagrangian dual of TTRS is a Semidefinite Program (SDP), a method often employed for efficient problem-solving.

Purpose of the Study:

  • To numerically investigate the performance of state-of-the-art branch-and-bound solvers on TTRS.
  • To analyze the global search capabilities and measure the 'cluster effect' for different solvers.
  • To compare the efficacy of branch-and-bound methods against SDP approaches for TTRS.

Main Methods:

  • Numerical experiments were conducted on 212 challenging TTRS instances provided by Kurt Anstreicher.
  • Performance analysis focused on branch-and-bound solvers including Antigone 1.1, Baron 16.12.7, Lindo Global 10.0, Couenne 0.5, and SCIP 3.2.
  • Quantitative measures were used to assess solver performance and the 'cluster effect'.

Main Results:

  • Branch-and-bound solvers demonstrated varying abilities in completing the global search for TTRS solutions.
  • The study provides a quantitative approach to measure the 'cluster effect' impacting solver performance.
  • Semidefinite programming (SDP) relaxations and branch-and-bound methods exhibit complementary strengths and weaknesses.

Conclusions:

  • SDP relaxations and branch-and-bound approaches possess orthogonal difficulties when applied to TTRS.
  • A hybrid method combining SDP and branch-and-bound techniques may offer enhanced efficiency for solving TTRS.
  • Further research into integrated approaches is warranted to leverage the benefits of both methodologies.