Abstract
One of the most challenging optimization problems is determining the minimizer of a nonlinear nonconvex problem in which there are some discrete variables. Any such problem may be transformed to that of finding a global optimum of a problem in continuous variables. However, such transformed problems have astronomically large numbers of local minimizers, making them harder to solve than typical global optimization problems. Despite this apparent disadvantage we show the approach is not hopeless.
Since the technique requires finding a global minimizer we review the approaches to solving such problems. Our interest is in problems for differentiable functions, and our focus is on algorithms that utilize the large body of work available on finding local minimizers of smooth functions. The method we advocate convexifies the problem and uses an homotopy approach to find the required solution. To illustrate how well the new algorithm performs we apply it to a hard frequency assignment problem, and to binary quadratic problems taken from the literature.
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Murray, W., Ng, KM. (2002). Algorithms for Global Optimization and Discrete Problems Based on Methods for Local Optimization. In: Pardalos, P.M., Romeijn, H.E. (eds) Handbook of Global Optimization. Nonconvex Optimization and Its Applications, vol 62. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-5362-2_3
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DOI: https://doi.org/10.1007/978-1-4757-5362-2_3
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