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Analyzing the performance of simultaneous generalized hill climbing algorithms

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Abstract

Simultaneous generalized hill climbing (SGHC) algorithms provide a framework for using heuristics to simultaneously address sets of intractable discrete optimization problems where information is shared between the problems during the algorithm execution. Many well-known heuristics can be embedded within the SGHC algorithm framework. This paper shows that the solutions generated by an SGHC algorithm are a stochastic process that satisfies the Markov property. This allows problem probability mass functions to be formulated for particular sets of problems based on the long-term behavior of the algorithm. Such results can be used to determine the proportion of iterations that an SGHC algorithm will spend optimizing over each discrete optimization problem. Sufficient conditions that guarantee that the algorithm spends an equal number of iterations in each discrete optimization problem are provided. SGHC algorithms can also be formulated such that the overall performance of the algorithm is independent of the initial discrete optimization problem chosen. Sufficient conditions are obtained guaranteeing that an SGHC algorithm will visit the globally optimal solution for each discrete optimization problem. Lastly, rates of convergence for SGHC algorithms are reported that show that given a rate of convergence for the embedded GHC algorithm, the SGHC algorithm can be designed to preserve this rate.

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Authors and Affiliations

  1. Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Diane E. Vaughan

  2. Simulation and Optimization Laboratory, Department of Computer Science, University of Illinois, 201 North Goodwin Avenue (MC-258), Urbana, IL, 61801-2302, USA

    Sheldon H. Jacobson

  3. Department of Applied Mathematics, Illinois Institute of Technology, 10 West 32nd Street, Chicago, IL, 60616, USA

    Hemanshu Kaul

Authors
  1. Diane E. Vaughan
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  2. Sheldon H. Jacobson
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  3. Hemanshu Kaul
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Correspondence to Diane E. Vaughan.

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Vaughan, D.E., Jacobson, S.H. & Kaul, H. Analyzing the performance of simultaneous generalized hill climbing algorithms. Comput Optim Appl 37, 103–119 (2007). https://doi.org/10.1007/s10589-007-9019-y

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  • DOI: https://doi.org/10.1007/s10589-007-9019-y

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