Relaxation and exact algorithms for solving mixed integer-quadratic optimization problems

Abstract

We develop various algorithms for solving mixed integer-quadratic problems. These problems exhibit exponential complexity resulting from the presence of integer variables. Traditional approaches that apply in pure integer programming are not very helpful, since the existence of continuous variables in our problems complicates their use. Vie develop relaxation and heuristic algorithms designed so as to provide tight lower and upper bounds to the optimal solution of the mixed combinatorial problem. In some cases the obtained range, in which the optimum lies, is small enough to be considered satisfactory by itself. This has been accomplished in problems with up to 150 variables. Exact algorithms have also been developed and guarantee the optimal solution upon termination. The idea of Branch and Bound enhanced with the use of lower and upper bounds obtained with the aforementioned methods is implemented for that purpose. Problems with up to 70 variables have been solved. Our ideas and algorithms are applied to the Problem of Index and Portfolio Replication with a limited number of assets. This problem arises in Finance, but, in its more general form, can find application in various areas ranging from Statistics to Optimal Control and Manufacturing.