Efficient computation of the nearest polynomial by linearized alternating direction method

Abstract

Given multiple real polynomials f1,f2,…,fm and a real zero z, we investigate a problem of finding a real polynomial f˜ such that f˜ has z as a zero and the distance between f˜ and f1,f2,…,fm is minimal. By taking the distance as a pair of norms (ℓp,ℓq), the problem can be converted into a linearly constrained convex program via ℓp,q-norm minimization. We develop an efficient algorithm to compute the nearest polynomial following the framework of linearized alternating direction method (LADM). With adaptive penalty, we analyze the global convergency property of the proposed algorithm under a mild assumption, and also reveal the convergence rate in an ergodic sense by using a simple optimality measure. Moreover, we give detailed discussions to optimal solutions to the ℓp,q-norm regularized subproblems. Especially for the case of ℓp,∞-norm, we present a projection-based primal-dual method (PPD) to solve the subproblems. Two numerical examples with random and deterministic inputs are provided to validate the effectiveness of the proposed algorithm.