Restricted simplicial decomposition: Computation and extensions

Restricted simplicial decomposition (RSD) is a very useful technique for certain large-scale pseudoconvex programming problems such as the traffic assignment problem and other network flow problems. The “restricted” version of this method allows the user to treat the maximum size of the generated simplices as a parameter. When the parameter is at its minimum value, the method reduces to the Frank-Wolfe algorithm; when at its maximum, the original simplicial decomposition of von Hohenbalken and Holloway results. Computer storage requirements increase linearly with the parameter, but our computational experiments on a variety of test problems show that there is a payoff in improved progress of the method as measured by the relative error in the objective function. Included in the tests are a number of real-world traffic networks, some large electrical networks, a water distribution network, and linearly constrained problems of the Colville study. Conditions are given for which RSD converges after a finite number of major cycles, and variations of RSD which have potential for real-world applications are developed. These include a quadratic approximation of the master problem and an extension to include the case of unbounded subproblems.