Projections onto convex sets (POCS) based optimization by lifting

Optimization steps


A new optimization technique based on the projections onto convex space (POCS) framework for solving convex and some non-convex optimization problems are presented. The dimension of the minimization problem is lifted by one and sets corresponding to the cost function are defined. If the cost function is a convex function in $R^N$ the corresponding set which is the epigraph of the cost function is also a convex set in $R^{N+1}$. The iterative optimization approach starts with an arbitrary initial estimate in $R^{N+1}$ and an orthogonal projection is performed onto one of the sets in a sequential manner at each step of the optimization problem. The method provides globally optimal solutions in total-variation, filtered variation, $\ell_1$, and entropic cost functions. It is also experimentally observed that cost functions based on $\ell_p$; $p \leq 1$ may be handled by using the supporting hyperplane concept. The new POCS based method can be used in image deblurring, restoration and compressive sensing problems.

In Global Conference on Signal and Information Processing (GlobalSIP), 2013 IEEE, Austin, TX, 2013, pp. 623-623.