  We consider a method of pairwise variations for smooth optimization problems,
which involve polyhedral constraints. It consists in making steps with respect
to the difference of two selected extreme points of the feasible set together
with special threshold control and tolerances whose values reduce sequentially.
The method is simpler and more flexible than the well-known conditional
gradient method, but keeps its useful sparsity properties and is very suitable
for large dimensional optimization problems. We establish its convergence under
rather mild assumptions. Efficiency of the method is confirmed by its
convergence rates and results of computational experiments.
