  We address a large-scale and nonconvex optimization problem, involving an
aggregative term. This term can be interpreted as the sum of the contributions
of N agents to some common good, with N large. We investigate a relaxation of
this problem, obtained by randomization. The relaxation gap is proved to
converge to zeros as N goes to infinity, independently of the dimension of the
aggregate. We propose a stochastic method to construct an approximate minimizer
of the original problem, given an approximate solution of the randomized
problem. McDiarmid's concentration inequality is used to quantify the
probability of success of the method. We consider the Frank-Wolfe (FW)
algorithm for the resolution of the randomized problem. Each iteration of the
algorithm requires to solve a subproblem which can be decomposed into N
independent optimization problems. A sublinear convergence rate is obtained for
the FW algorithm. In order to handle the memory overflow problem possibly
caused by the FW algorithm, we propose a stochastic Frank-Wolfe (SFW)
algorithm, which ensures the convergence in both expectation and probability
senses. Numerical experiments on a mixed-integer quadratic program illustrate
the efficiency of the method.
