Instantaneous magnitudes and instantaneous frequencies of signals with their positivity constraints via non‐smooth non‐convex functional constrained optimisation

Abstract

This study proposes an iterative method to approximate an N-dimensional optimisation problem with a weighted Lp norm and L2 norm objective function by a sequence of N independent one-dimensional optimisation problems. This iterative method is inspired by the existing weighted L1 norm and L2 norm separable surrogate functional (SSF) iterative shrinkage algorithm. However, as these independent one-dimensional optimisation problems consist of weighted Lp norm and L2 norm objective functions, these optimisation problems are non-convex and they may have more than one locally optimal solutions. In general, it is very difficult to find their globally optimal solutions. To address this difficulty, this study proposes to partition the feasible set of each approximated problem into various regions such that the sign of the convexity of the objective function in each region remains unchanged. In this case, there is no more than one stationary point in each region. By finding the stationary point in each region, the globally optimal solution of each approximated optimisation problem can be found. Besides, this study also shows that the sequence of the globally optimal solutions of the approximated problems converge to the globally optimal solution of the original optimisation problem.