Simplicial decomposition of variational inequalities with multiple nonlinear column generation

Abstract

Simplicial decomposition (SD) of variational inequalities experiences the long-tail convergence property. That is, the equilibrium solution rapidly progresses at first but then tails off, making only a tiny amount of progress per column generation iteration, which is a drawback of SD-VI. In the context of Dantzig-Wolfe of LP, it is reported that the more proposals are used to initialize the algorithm, the faster the solution can be found by reducing the number of decomposition steps. Therefore, I proposed to solve multiple nonlinear column generation (mNCG) subproblems in each SD-VI iteration (SD-VI-mNCG) instead of solving only one subproblem as in SD-VI. Generating multiple column generation subproblem solutions in each SD-VI iteration enabled the corresponding convex hull to be rapidly enlarged. Consequently, the number of SD-VI iterations could be greatly reduced. A transportation network equilibrium problem was used to study the performance of the SD-VI-mNCG.

Keywords

• column generation
• simplicial decomposition
• convergence rate
• variational inequalities
• nonlinear programming