Conditional steepest descent directions over Cartesian product sets : With application to the Frank-Wolfe method

Abstract: We derive a technique for scaling the search directions of feasible direction methods when applied to optimization problems over Cartesian product sets. It is proved that when the scaling is included in a convergent feasible direction method, also the new method will be convergent. The scaling technique is applied to the Frank-Wolfe method, the partanized Frank-Wolfe method and a heuristic Frank-Wolfe method. The performance of  these algorithms with and without scaling is evaluated on the stochastic transportation problem. It is found that the scaling technique has the ability to improve the performance of some methods. In particular we observed a huge improvement in the performance of the partanized Frank-Wolfe method, especially when the scaling is used together with an exact line search and when the number of sets in the Cartesian product is large.