Improved Frank-Wolfe directions with applications to traffic problems

Abstract: The main contribution of this thesis is the development of some new efficient algorithms for solving structured linearly constrained optimization problems. The conventional Frank-Wolfe method is one of the most frequently used methods for solving such problems. We develop algorithms based on conjugate directions methods and aim to improve the performance of the pure Frank-Wolfe method by choosing better search directions.In the conjugate direction Frank-Wolfe method for linearly constrained convex optimization problems, one performs line search along a direction, which is conjugate to the previous one with respect to the hessian of the objective function at the current point. The new method is applied to the single-class traffic equilibrium problem. The convergence of the presented method is also proved. In a limited set of computational tests the algorithm turns out to be quite efficient, outperforming the pure and "PARTANized" Frank-Wolfe methods.One further refinement of the conjugate direction Frank-Wolfe methods. is derived by applying conjugation with respect to the last two directions instead of only the last one.We also extend the conjugate direction Frank-Wolfe method to nonconvex optimization problems with linear constraints. We apply this extension to the multi-class traffic equilibrium problem under social marginal cost pricing.