Optimal correction of infeasible equations system as Ax + B|x|= b using ℓ p-norm regularization

Fakhrodin Hashemi,Saeed Ketabchi

doi:10.5269/bspm.44437

Abstract

Optimal correction of an infeasible equations system as Ax + B|x|= b leads into a non-convex fractional problem. In this paper, a regularization method(ℓp-norm, 0 < p < 1), is presented to solve mentioned fractional problem. In this method, the obtained problem can be formulated as a non-convex and nonsmooth optimization problem which is not Lipschitz. The objective function of this problem can be decomposed as a difference of convex functions (DC). For this reason, we use a special smoothing technique based on DC programming. The numerical results obtained for generated problem show high performance and the effectiveness of the proposed method.

Highlights

Optimal correction of an infeasible equations system as Ax + B|x| = b, leads into a non-convex fractional problem
A few algorithms were applied without considering non-convexity and unsmoothness properties of the objective function [15,16,20]
In addition to focousing on non-convexity and unsmoothness properties of the objective function, the present study proposes the exact regularization method with the following general form min x∈Rn

Summary

Then for any x

To compute the mentioned generalized gradient, a solution for a non-convex and non-smooth optimization subproblem is required Before solving this subproblem, we first state some of the theorems and lemmas in the following. The objective function of problem (2.8) is smooth and according to Lemmas 2.1 and 2.2, its gradient is continuously Lipschitz. Algorithms and their convergence Consider the following problem min x∈Rn gμ(x) g1(x) g2(x),. To find a solution for problem (3.1), by considering xk ∈ domf, in each iteration, the following relaxation problem will be solved min x∈Rn gμ(x). Since the objective function of problem (3.2) is convex and continuously differentiable and its Hessian is available almost every where, to solve it, a second order classic algorithm (i.e., smoothing difference of convex) is proposed here: Algorithm 1 Smoothing difference of convex algorithm.

MiMfM error M time

Conclusion