Gradient-Related Constrained Minimization Algorithms in Function Spaces: Convergence Properties and Computational Implications

Good finite-dimensional approximations to projected gradient and conditional gradient iterates in feasible sets of Lp functions u(-): [0,1] →U are relatively easy to compute when U is a simple closed convex set in Rm (e.g., an orthant, box, simplex, ball, etc.). Much is also known about the convergence behavior of the underlying infinite-dimensional iterative processes in these circumstances. Several novel features of this behavior are examined here, and the associated computational implications are explored with analytical tools and numerical experiments. The conclusions reached are immediately applicable to constrained input continuous-time optimal control problems.