Guaranteed ε-approximate solution for indefinite quadratic global minimization

Abstract

The global minimization of large-scale indefinite quadratic problems over a bounded polyhedral set using a parallel branch-and-bound approach is considered. The objective function consists of both a nonlinear part (nonlinear variables) and a strictly linear part, which are coupled by the linear constraints. These large-scale problems are characterized by having the number of linear variables much greater than the number of nonlinear variables. A convex quadratic underestimating function to the indefinite quadratic objective function is easily constructed and minimized over the feasible domain to get both upper and lower bounds on the global minimum function value. At each minor iteration of the algorithm, the feasible domain is divided into subregions and convex quadratic underestimating problems over each subregion are solved in parallel. Branch-and-bound techniques can then be used to eliminate parts of the feasible domain from consideration and improve the upper and lower bounds. It is shown that the algorithm guarantees that a solution is obtained to within any specified tolerance in a finite number of steps. Computational results are presented for problems with the number of nonlinear variables varying from 25 to 125 and up to 400 linear variables. These results were obtained on a four-processor CRAY 2 using both sequential and parallel implementations of the algorithm. The computation times using a four-processor CRAY 2 ranged from less than one second to a maximum of 703 seconds with a relative stopping tolerance of 0.001. The average (ten problems) parallel solution time for a problem with 125 nonlinear variables and 200 purely linear variables (the largest problem solved in terms of the number nonlinear variables) was approximately 35 seconds. The parallel implementation is very efficient as shown by speedups greater than the number of processors in certain cases.