This paper discusses the impact of the hierarchical master-worker paradigm on performance of an application program, which solves an optimization problem by a parallel branch and bound algorithm on a distributed computing system. The application program, which this paper addresses, solves the BMI Eigenvalue Problem, which is an optimization problem to minimize the greatest eigenvalue of a bilinear matrix function. This paper proposes a parallel branch and bound algorithm to solve the BMI Eigenvalue Problem with the hierarchical master-worker paradigm. The experimental results showed that the conventional algorithm with the master-worker paradigm significantly degraded performance on a Grid test bed, where computing resources were distributed on WAN via a firewall; however, the hierarchical master-worker paradigm sustained good performance.