this paper we examine the underlying ideas of the SQP method and the theory that establishes it as a framework from which effective algorithms can

In this paper a family of trust-region interior-point SQP algorithms for the solution of a class of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise e.g. from the discretization of optimal control problems. The algorithms treat states and controls as independent variables. They are designed to take advantage of the structure of the problem. In particular they do not rely on matrix factorizations of the linearized constraints, but use solutions of the linearized state equation and the adjoint equation. They are well suited for large scale problems arising from optimal control problems governed by partial differential equations. The algorithms keep strict feasibility with respect to the bound constraints by using an affine scaling method proposed for a different class of problems by Coleman and Li and they exploit trust--region techniques for equality-constrained optimizatio...

In this paper we analyze incxact trust region interior point (TRIP) sequential quadr tic programming (SOP) algorithms for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on some of the variables. Such problems arise in many engineering applications, in particular in optimal control problems with bounds on the control. The nonhnear constraints often come from the discretization of partial differential equations. In such cases the calculation of derivative information and the solution of hncarizcd equations is expensive. Often, the solution of hncar systems and derivatives arc computed incxactly yielding nonzero residuals. This paper

In this paper we consider a class of nonlinear programming problems that arise from the discretization of optimal control problems with bounds on both the state and the control variables. For this class of problems, we analyze constraint qualifications and optimality conditions in detail. We derive an affine-scaling and two primal-dual interior-point Newton algorithms by applying, in an interior-point way, Newton's method to equivalent forms of the first-order optimality conditions. Under appropriate assumptions, the interior-point Newton algorithms are shown to be locally well-defined with a q-quadratic rate of local convergence. By using the structure of the problem, the linear algebra of these algorithms can be reduced to the null space of the Jacobian of the equality constraints. The similarities between the three algorithms are pointed out, and their corresponding versions for the general nonlinear programming problem are discussed.

. This paper investigates quasi-Newton updates for equality-constrained optimization in abstract vector spaces. Using a least-change argument we derive a class of rank-3 updates to approximations of the one-sided projection of the Hessian of the Lagrangian which keeps the symmetric part positive definite. By imposing the usual assumptions we are able to prove 1-step superlinear convergence for one of these updates. Encouraging numerical results and comparisons with other previously analyzed updates are presented. Key words. quasi-Newton update -- equality-constrained optimization -- superlinear convergence -- variable metric method 1. Introduction and Background Quasi-Newton methods for nonlinear optimization problems have been studied extensively since the late 60s. While there are a number of updates and convergence analyses for the unconstrained case (see, e.g.,[10] and [4]), the constrained case has only been discussed more recently, e.g., in [6],[22], [7], [19] and [14], and in Pe...