Solving $L_{p}$-Norm Problems and Applications

Abstract

The $l_{p}$ norm discrete estimation problem min$_{x\in\Re^{n}} \Vert b-A^{T} x\Vert^{p}_{p}$ has been solved in many data analysis applications, e.g. geophysical modeling. Recently, a new globally convergent Newton method (called GNCS) has been proposed for solving $l{p}$ problems with 1 $\leq p \leq$ 2 [5]. This method is much faster than the widely used IRLS method when 1 $\leq p \leq$ 1.5 and comparable to it when $p greater than $ 1.5. In this paper, modification is made to the line search prodedure so that the GNCS method is applicable for $l_{p}$ problems with 1 $\leq p less than \infty$. The global convergence results for $l_{1}$ problems are obtained under weaker assumptions than required in [2]. In addition, the usefulness of $l_{p}$ norm solution with 1 $\leq p \leq$ 2 is demonstrated by applying the GNCS algorithm to a synthetic geophysical tomographic inversion problem. Additional numerical results are included to support the efficiency of GNCS. Key Words: linear regression, discrete estimation, tomographic inversion, IRLS, GNCS, linear programming, Newton method. Subject Classification: AMS/MOS: 65H10, 65K05, 65K10.