We solve a class of convex infinite-dimensional optimization problems using a numerical approximation method that does not rely on discretization. Instead, we restrict the decision variable to a sequence of finite-dimensional linear subspaces of the original infinite-dimensional space and solve the corresponding finite-dimensional problems in a efficient way using structured convex optimization techniques. We prove that, under some reasonable assumptions, the sequence of these optimal values converges to the optimal value of the original infinite-dimensional problem and give an explicit description of the corresponding rate of convergence.
Solving Infinite-dimensional Optimization Problems by Polynomial Approximation, Olivier Devolder, François Glineur, Yurii Nesterov, in Recent Advances in Optimization and its Applications in Engineering, Springer, 2010, pp. 31-40. http://dx.doi.org/10.1007/978-3-642-12598-0_3