Mounir Haddou

A Smoothing Method for Sparse Optimization over Polyhedral Sets

We investigate a method to solve NP-hard problem of minimizing l0-norm of a vector over a polyhedral set P. A simple approximation is to find a solution of the problem of minimizing l1-norm. We are concerned about finding improved results. Using a family of smooth concave functions θ r (.) depending on a parameter r we show that \(\min\limits_{x \in P} \sum_{i=1}^n \theta_r(x_i)\) is equivalent to \(\min\limits_{x \in P} ||x||_0\) for r sufficiently small and \(\min\limits_{x \in P} ||x||_1\) for r sufficiently large. This gives us an algorithm based on a homotopy-like method. We show convergence results, error estimates and numerical simulations.

Smoothing Methods for Nonlinear Complementarity Problems

In this paper, we present some new smoothing techniques to solve general nonlinear complementarity problems. Under a weaker condition than monotonicity as on the original problems, we prove convergence of our methods. We also present an error estimate under a general monotonicity condition. Some numerical tests confirm the efficiency of the proposed methods.

Solving Absolute Value Equation using Complementarity and Smoothing Functions

In this paper, we reformulate the NP-hard problem of the absolute value equation (AVE) as a horizontal linear complementarity one and then solve it using a smoothing technique. This approach leads to a new class of methods that are valid for general absolute value equation. An asymptotic analysis proves the convergence of our schemes and provides some interesting error estimates. This kind of error bound or estimate had never been studied for other known methods. The corresponding algorithms were tested on randomly generated problems and applications. These experiments show that, in the general case, one observes a reduction of the number of failures.