This paper presents a detailed derivation and description of a new method for solving equality constrained optimization problem. The new method is based upon the quadratic penalty function, but uses orthogonal transformations, derived from the Jacobian matrix of the constraints, to deal with the numerical ill-conditioning that affects the methods of this type.

At each iteration of the new algorithm, the orthogonal search direction is determined by a quasi-Newton method which can avoid the necessity of solving a set of equations and the step-length is chosen by an Armijo line search. The matrix which approaches the inverse of the projected Hessian of composite function is updated by means of the BFGS formula from iteration to iteration. As the penalty parameter approaches zero, the projected inverse Hessian has special structure which can guarantee us to obtain the search direction accurately even if the Hessian of composite function is ill-conditioned in the former quadratic penalty methods.

quadratic penalty function method, orthogonal transformation, Jacobian matrix, equality constrained optimizations.