Efficient global optimization for combinatorial problems

ABSTRACT

Real-world optimization problems may require time consuming and expensive measurements or simulations. Recently, the application of surrogate model-based approaches was extended from continuous to combinatorial spaces. This extension is based on the utilization of suitable distance measures such as Hamming or Swap Distance. In this work, such an extension is implemented for Kriging (Gaussian Process) models. Kriging provides a measure of uncertainty when determining predictions. This can be harnessed to calculate the Expected Improvement (EI) of a candidate solution. In continuous optimization, EI is used in the Efficient Global Optimization (EGO) approach to balance exploitation and exploration for expensive optimization problems. Employing the extended Kriging model, we show for the first time that EGO can successfully be applied to combinatorial optimization problems. We describe necessary adaptations and arising issues as well as experimental results on several test problems. All surrogate models are optimized with a Genetic Algorithm (GA). To yield a comprehensive comparison, EGO and Kriging are compared to an earlier suggested Radial Basis Function Network, a linear modeling approach, as well as model-free optimization with random search and GA. EGO clearly outperforms the competing approaches on most of the tested problem instances.

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Index Terms

• Mathematics of computing

  • Discrete mathematics

    • Combinatorics

      • Combinatorial algorithms

  • Mathematical analysis

    • Mathematical optimization

• Theory of computation

  • Design and analysis of algorithms

    • Mathematical optimization