Iterative projection algorithms for solving constraint satisfaction problems: Effect of constraint convexity

Many inverse problems in imaging involve solving an optimization problem. In many cases, the problem is high-dimensional and non-convex, requiring the solution of a difficult, non-convex, global optimization problem. Such problems can be made tractable by enforcing hard constraints and treating the problem as a constraint satisfaction problem to locate a global solution, which can be refined using soft constraints if necessary. Iterative projection algorithms are an effective way of solving non-convex constraint satisfaction problems. The difficulty of solution, and the performance of these algorithms, depends on the degree of non-convexity of the constraints. Here we use simulations of a phase retrieval problem to study the performance of an iterative projection algorithm, the difference map algorithm, to study performance as a function of non-convexity.