Smoothing Secant Line Slope Using Aggregation Fischer Burmeister Function

Abstract

As some of the objective functions are piecewise, so they are non-differentiable at specific points which have a significant impact on deep network rate and computational time. The non-differentiability issue increases the computational time dramatically. This issue is solved by the reformulation of the absolute value equation (AVE) through a parametrized single smooth equation. However, utilizing a single smoothing function is less effective to produce a better curve at the breaking points. Therefore, this work formulates a new smoothing function of Aggregation Fischer Burmister (AFB) via amalgamating of two popular smoothing functions: Aggregation (AGG) and Fischer-Burmeister (FB). These functions are having the ability of minimum estimation from both sides of the canonical piecewise function (CPF). If an amalgamation of smoothing functions can affect the differentiability of the piecewise objective function, then amalgamating the AGG and FB smoothing functions will produce a smooth secant line slope on both sides with less computational time. To evaluate the proposed technique, we implement a Newton algorithm using MATLAB, with random initial values. A new smoothing function is formulated by firstly converting the piecewise objective function to CPF. Then, we applied it to the Newton algorithm. Finally, to validate the AVE difficulty of the new piecewise function, we perform one run for each initial value, and 30 runs for time evaluation. The experimental analysis verified that the proposed technique outperformed other techniques of AGG and FB individually in terms of the natural logarithm, exponential, and square root. Hence, this novel technique yields promising smooth approximation for AVE with less computational time.

Keywords

• Absolute value equation
• artificial neural network
• objective function
• piecewise linear function
• smooth approximation