Adaptive modification of objective function for lagrange multiplier method in constrained optimization problems

Abstract

Abstract This paper proposes an adaptive modification method to transform the objective function with a stationary point to an objective function with a minima point, such that search methods can be used to find the stationary point. The stationary point can be a saddle point in addition to a minima or a maxima. Therefore, this method can be used to transform a constrained optimization by applying Lagrange multipliers to an unconstrained optimization problem. A quadratic term, ½(X — XN ) T D (X—XN ), is added to the original function such that the modified function is a minima at the Newton point XN of the original function, where D is a diagonal matrix to make the modified Hessian matrix HO + D positive definite, and HO is the original Hessian matrix at the initial point XO .