The ill-conditioning issue in the penalty function method

The penalty function method has been widely used in many constrained optimization problems.However, a companion of such method is the ill-conditioning problem in the corresponding Hessian matrix.In this paper, a thorough investigation is provided to address this ill-conditioning problem.Our study reveals that this illconditioning is mathematically tractable.Furthermore, the resulting ill-conditioned system can be converted into a well-conditioned one.Consequently, this ill-conditioning problem can be overcome.Some theoretical results will be also provided.Our preliminary numerical tests have confirmed our findings.