We consider a wide class of penalty and barrier methods for convex programming which includes a number of specific functions proposed in the literature. We provide a systematic way to generate penalty and barrier functions in this class, and we analyze the existence of primal and dual optimal paths generated by these penalty methods, as well as their convergence to the primal and dual optimal sets. For linear programming we prove that these optimal paths converge to single points.
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.
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