Generalized Benders Decomposition to solve a nonlinear routing problem with queueing delay goal function
Abstract
We address the multicommodity flow problem with a nonlinear goal function modeling queueing delay. It is well-known that linear programming solvers perform better than those used for nonlinear programming. We can leverage their performance by employing the Generalized Benders Decomposition (GBD) to partition the problem into master and primal subproblems. We prove that in the case of multiple subproblems, which is true in our case, we can split both the optimality and feasibility cuts and add them independently. Moreover, we extended a known proof of convergence to enable a wider range of problems to be solved using GBD. We use the split cuts technique to precompute feasibility cuts and analytically solve the subproblems to omit the use of nonlinear optimization software. Furthermore, we explore the possibilities of starting point selection through linear and quadratic approximation. We carry out tests on a classical network example to show that GBD can sometimes outperform nonlinear solvers, and also that quadratic approximation for starting point selection can provide strictly better solution times, dominating commercial solvers.
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