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An algorithm for nonlinear optimization problems with binary variables

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Abstract

One of the challenging optimization problems is determining the minimizer of a nonlinear programming problem that has binary variables. A vexing difficulty is the rate the work to solve such problems increases as the number of discrete variables increases. Any such problem with bounded discrete variables, especially binary variables, may be transformed to that of finding a global optimum of a problem in continuous variables. However, the transformed problems usually have astronomically large numbers of local minimizers, making them harder to solve than typical global optimization problems. Despite this apparent disadvantage, we show that the approach is not futile if we use smoothing techniques. The method we advocate first convexifies the problem and then solves a sequence of subproblems, whose solutions form a trajectory that leads to the solution. To illustrate how well the algorithm performs we show the computational results of applying it to problems taken from the literature and new test problems with known optimal solutions.

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Authors and Affiliations

  1. Systems Optimization Laboratory, Department of Management Science and Engineering, Stanford University, Stanford, CA, 94305-4022, USA

    Walter Murray

  2. Department of Industrial & Systems Engineering, National University of Singapore, Singapore, Singapore

    Kien-Ming Ng

Authors
  1. Walter Murray
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  2. Kien-Ming Ng
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Correspondence to Walter Murray.

Additional information

The research of W. Murray was supported by Office of Naval Research Grant N00014-08-1-0191 and Army Grant W911NF-07-2-0027-1.

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Murray, W., Ng, KM. An algorithm for nonlinear optimization problems with binary variables. Comput Optim Appl 47, 257–288 (2010). https://doi.org/10.1007/s10589-008-9218-1

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  • DOI: https://doi.org/10.1007/s10589-008-9218-1

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