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

. We present a new trust region algorithm for solving nonlinear equality constrained optimization problems. At each iterate a change of variables is performed to improve the ability of the algorithm to follow the constraint level sets. The algorithm employs L 2 penalty functions for obtaining global convergence. Under certain assumptions we prove that this algorithm globally converges to a point satisfying the second order necessary optimality conditions; the local convergence rate is quadratic. Results of preliminary numerical experiments are presented. 1. Introduction. We consider the equality constrained optimization problem minimize f(x) subject to c(x) = 0 (1:1) where x 2 ! n and f : ! n ! !, and c : ! n ! ! m are smooth nonlinear functions. Problem (1.1) is often solved by successive quadratic programming (SQP) methods. At a current point x k 2 ! n , SQP methods determine a search direction d k by solving a quadratic programming problem minimize rf(x k ) T d + 1 2 ...

Abstract. This paper describes a software implementation of Byrd and Omojokun’s trust region algorithm for solving nonlinear equality constrained optimization problems. The code is designed for the efficient solution of large problems and provides the user with a variety of linear algebra techniques for solving the subproblems occurring in the algorithm. Second derivative information can be used, but when it is not available, limited memory quasi-Newton approximations are made. The performance of the code is studied using a set of difficult test problems from the CUTE collection.

This work presents a global convergence theory for a broad class of trust-region algorithms for the smooth nonlinear progro.mmln S problem with equality constraints. The main result generalizes Powell's 1975 result for unconstrained trust-region algorithms.

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...

Multidisciplinary design optimization (MDO) gives rise to nonlinear optimization problems characterized by a large number of constraints that naturally occur in blocks. We propose a class of multilevel optimization methods motivated by the structure and number of constraints and by the expense of the derivative computations for MDO. The algorithms are an extension to the nonlinear programming problem of the successful class of local Brown-Brent algorithms for nonlinear equations. Our extensions allow the user to partition constraints into arbitrary blocks to fit the application, and they separately process each block and the objective function, restricted to certain subspaces. The methods use trust regions as a globalization startegy, and they have been shown to be globally convergent under reasonable assumptions. The multilevel algorithms can be applied to all classes of MDO formulations. Multilevel algorithms for solving nonlinear systems of equations are a special case of the multil...

This paper discusses certain connections between nonlinear programming algorithms and the formulation of optimization problems for systems governed by state constraints. I work through the calculation of the sensitivities associated with the di erent formulations and present some useful relationships between them. These relationships have practical consequences; if one uses a reduced basis nonlinear programming algorithm, then the implementations for the di erent formulations need only di er in a single step.

In this paper we analyze inexact trust-region interior-point (TRIP) sequential quadratic programming (SQP) 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 nonlinear constraints often come from the discretization of partial differential equations. In such cases the calculation of derivative information and the solution of linearized equations is expensive. Often, the solution of linear systems and derivatives are computed inexactly yielding nonzero residuals. This paper analyzes the effect of the inexactness onto the convergence of TRIP SQP and gives practical rules to control the size of the residuals of these inexact calculations. It is shown that if the size of the residuals is of the order of both the size of the constraints and the trust-region radius, t...

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. The family of feasible methods for minimization with nonlinear constraints includes Rosen's Nonlinear Projected Gradient Method, the Generalized Reduced Gradient Method (GRG) and many variants of the Sequential Gradient Restoration Algorithm (SGRA). Generally speaking, a particular iteration of any of these methods proceeds in two phases. In the Restoration Phase, feasibility is restored by means of the resolution of an auxiliary nonlinear problem, generally a nonlinear system of equations. In the Minimization Phase, optimality is improved by means of the consideration of the objective function, or its Lagrangian, on the tangent subspace to the constraints. In this paper, minimal assumptions are stated on the Restoration Phase and the Minimization Phase that ensure that the resulting algorithm is globally convergent. The key point is the possibility of comparing two successive nonfeasible iterates by means of a suitable merit function that combines feasibility and optimality. The mer...