Two-phase gradient based search method for global optimum of constrained mixed integer programming problems using a tunneling function & comparison

In this paper, a gradient-based optimization approach with two stages is proposed that is designed to efficiently solve mixed-integer programming problems with linear and nonlinear objective functions and constraints. To find a viable local minimum in the first step, a gradient method with hemstitching modifications is used, which guarantees that the trajectory is redirected into the feasible region whenever constraint(s) violation take place. The purpose of the second phase is to identify another point on the opposite side of the barrier where the function value is approximately the same, by constructing a tunneling function at the current local minimum. This newly discovered point serves as the starting vector for the next cycle, which repeats the process of searching for the global optimum. The iterative procedure is repeated until no further reduction in the objective function value is observed. Several benchmark problems involving mixed-integer variables are tested. The gradient method involving 3n enumeration that the author previously proposed is compared with the results from this new optimization technique, with a minor change made in the first phase of the previous published method. The results are compared with other published results.