Efficient method for finding globally optimal solution of problem with weighted L _p norm and L ₂ norm objective function

Abstract

This study proposes an iterative method to approximate an N-dimensional optimisation problem with a weighted L p and L 2 norm objective function by a sequence of N independent one-dimensional optimisation problems. Inspired by the existing weighted L 1 and L 2 norm separable surrogate functional (SSF) iterative shrinkage algorithm, there are N independent one-dimensional optimisation problems with weighted L p and L 2 norm objective functions. However, these optimisation problems are non-convex. Hence, they may have more than one locally optimal solutions and it is very difficult to find their globally optimal solutions. This paper 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. Here, 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. Computer numerical simulation results show that the proposed method outperforms the existing weighted L 1 and L 2 norm SSF iterative shrinkage algorithm.