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All-Quadratic Mixed-Integer Problems: A Study on Evolution Strategies and Mathematical Programming.

Guy Zepko1, Ofer M Shir2

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Summary
This summary is machine-generated.

Black-box Evolution Strategies (ESs) and white-box solvers show competitive performance for mixed-integer quadratic programming. CPLEX excels unless optima are significantly translated, where ESs outperform it.

Keywords:
CMA-ES with integer handlingILOG-CPLEXUnbounded integer programsevolution strategiesinteger mutation distributions

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Area of Science:

  • Optimization
  • Computational Mathematics
  • Operations Research

Background:

  • All-Quadratic Mixed-Integer (MI) Programs are NP-complete, with unbounded integer variables making them undecidable.
  • Traditional white-box Mathematical Programming (MP) solvers struggle with the complexity of these problems.

Purpose of the Study:

  • To evaluate the effectiveness of black-box Evolution Strategies (ESs) against white-box solvers for minimizing MI convex quadratic objective and constraint functions.
  • To analyze solver performance under varying conditions, including Hessian forms, condition numbers, and unboundedness.

Main Methods:

  • Empirical assessment comparing CPLEX (white-box) with MI ESs (black-box) on all-quadratic test cases.
  • Investigation focused on higher dimensionality (D=64) where CPLEX often times out.
  • MI ESs utilized penalty methods for constraint handling.

Main Results:

  • CPLEX and MI ESs demonstrated comparable performance (67% similarity) in objective function values.
  • CPLEX outperformed or matched MI ESs in 98% of cases when not encountering timeouts.
  • CPLEX performance degraded significantly (81% inferior) when optima were translated, favoring MI ESs.

Conclusions:

  • Black-box and white-box solvers can be competitive for All-Quadratic MI Programs.
  • Problem characteristics like conditioning and separability do not intuitively predict difficulty.
  • The performance of solvers is highly sensitive to the translation of optima in unbounded scenarios.