An efficient surrogate optimization method for solving linear mixed-integer problems with cross-coupling constraints

Many optimization problems that frequently arise and have been extensively used in practice are modeled as linear mixed-integer programming problems. Among the many of such problems are transportation and assignment problems. In such problems, constraints that couple decision variables can be viewed as hyperplanes in their respective spaces. These hyperplanes frequently intersect at different angles thus making the feasible set of the problems complex and leading to difficulties defining the convex hull when using the cutting planes method. The efficiency of branch-and-cut is therefore low, since the branching tree grows quickly due to the combinatorial nature of these problems. This paper overcomes these difficulties by using additional cuts that can better define the convex hull. Thus when the Lagrangian relaxation and surrogate optimization method is used, the computational burden of obtaining the multiplier updating directions is significantly reduced. In the surrogate optimization framework, the constraints of the original problem are relaxed by introducing the Lagrange multipliers. After a good dual solution is found, it is then improved to obtain a good feasible solution, while the dual value provides a lower bound on the feasible cost. Numerical examples indicate that the surrogate optimization can obtain feasible solutions when the standard branch-and-cut method cannot. Additional cuts help improve the feasible solutions and tighten the lower bound.