Well‐behaved penalty functions for constrained optimization

Abstract

This paper proposes a penalty function to be used for constrained optimization problems. The proposed penalty function is based on two special types of the hyperbolic curve. For the equality constraint, the penalty function is √x ² + t² , where x= pg(X), g(X)=0 is the constraint, t is a shape parameter, p is a scale factor, and X contains the design variables. For the inequality constraint, the penalty function is √x ² + t² – x, where x = pg(X), and p > 0 is the constraint for g(X) > 0, or p < 0 for g(X)≤ 0. These penalty functions have the advantages of being defined everywhere, accurate and differentiable. Two extended penalty functions are also proposed to dominate the infinite negative objective function in some cases. One is of the power type, another is of the exponential type. These two extended penalty functions are continuous up to their second derivatives. Therefore, they can be used in almost all of the optimization methods.

Key words:

• penalty function
• constrained optimization