A Sequential Method for Mathematical Programming

A sequential programming method for constrained optimization, which is referred to as SP method, is presented. It is based on a sequence of linearized constrained problems. Convergence and convergence rate of the method are analyzed based on solutions of the first order optimality conditions and on Lagrange multiplier theory. Unlike the SQP method, the SP method does not rely on an approximation of the Hessian of the Lagrange functional and consequently avoids instabilities due to possible indefiniteness of its Hessian. The method can be used for non-smooth problems and it can efficiently be implemented by the use of saddle point solvers. The implementation of the algorithms is less involved when compared to the SQP method. The cost of performing each sequential step is very similar to the one for the gradient method with doubling number of unknowns and the method is stable when using damped updates. We also develop a second order convergent version which again be based only on sequential linearization of the equality constraints. The proposed methods are numerically tested for control in the coefficient problems or equivalently of bilinear optimal control problems.