Approximating Large-Scale Binary Integer Programs by Discrete Optimal Control
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Optimal control theory has been introduced as a powerful tool for approximately solving binary integer programming problems. In previous studies, an approach using continuous optimal control theory was developed, where the original binary integer problem was transformed into a continuous linear quadratic problem. However, the continualization process added approximation error. The primary objective of this dissertation is to develop a discrete optimal control technique to approximately solve large-scale binary integer programming problems efficiently. The basic idea of the new algorithm is to transform the original binary integer problem into the form of a discrete linear quadratic tracking problem to take advantage of control theory properties and build good numerical stability properties. Because the time index in the reformulation of the binary integer problem represents the dimension of the problem, a discrete time approach more accurately represents the partial summing reformulation than the continuous approach. Also, instead of iteratively solving the linear quadratic tracking problem based on a reduced set of constraints as done previously, the linear quadratic tracking problem is solved only once for the full-state space. The advantage of solving the linear quadratic tracking problem once is that no error is introduced by the constraint reduction. However, if the number of constraints is very large, the full-state space may require too much computation, so a stochastic decomposition method, which is an extension of the constraint reduction technique, is also developed. Binary integer problem is NP-hard, and even finding a feasible solution is considered NP-complete. In this dissertation, an idea of balancing the feasibility problem with optimizing the objective function has been explored to improve the approximate solution. The approach taken is to apply the feasibility pump idea in the form of discrete optimal control toward finding a feasible solution. Several other variations of the full-state space discrete control problem, including combinations of cross-entropy method, bang-bang control method, minimum fuel formulation, and minimum-variance formulation, are studied and presented to reveal their potential for approximately solving binary integer problems. The computational results show the effectiveness of the proposed approaches for approximating solutions to large-scale binary integer programs.