Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available and that the constraint gradients are sparse.
We discuss an SQP algorithm that uses a smooth augmented Lagrangian merit function and makes explicit provision for infeasibility in the original problem and the QP subproblems. SNOPT is a particular implementation that makes use of a semidefinite QP solver. It is based on a limited-memory quasi-Newton approximation to the Hessian of the Lagrangian and uses a reduced-Hessian algorithm (SQOPT) for solving the QP subproblems. It is designed for problems with many thousands of constraints and variables but a moderate number of degrees of freedom (say, up to 2000). An important application is to trajectory optimization in the aerospace industry. Numerical results are given for most problems in the CUTE and COPS test collections (about 900 examples).

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SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization

This paper is an attempt to motivate and justify quasi-Newton methods as useful modifications of Newton''s method for general and gradient nonlinear systems of equations. References are given to ample numerical justification; here we give an overview of many of the important theoretical results and each is accompanied by sufficient discussion to make the results and hence the methods plausible. Key Words and Phrases: unconstrained minimization, nonlinear simultaneous equations, update methods, quasi-Newton methods.

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Quasi-Newton Methods, Motivation and Theory

An algorithm is described which appears to give an efficient solution of nonlinear finite element equations. It is a quisi-Nowton method, and we compare it with some of the alternatives. Initial tests of its application to both material and geometric nonlinearities are discussed.

The solution of nonlinear finite element equations

Some practical procedures for the solution of nonlinear finite element equations

New variable metric algorithms are presented with three distinguishing features:

(1)

They make no line searches and allow quite arbitrary step directions while maintaining quadratic termination and positive updates for the matrixH, whose inverse is the hessian matrix of second derivatives for a quadratic approximation to the objective function.




(2)

The updates fromH toH+ are optimally conditioned in the sense that they minimize the ratio of the largest to smallest eigenvalue ofH−1H+.




(3)

Instead of working with the matrixH directly, these algorithms represent it asJJT, and only store and update the Jacobian matrix J. A theoretical basis is laid for this family of algorithms and an example is given along with encouraging numerical results obtained with several standard test functions.

Optimally conditioned optimization algorithms without line searches

It is shown that certain rank-one and rank-two corrections to symmetric positive definite matrices may be expressed in the form of a product. This product form gives control over the positive definiteness, determinant value and conditioning of the corrected matrix. An application to updating formulae of quasi-Newton methods for unconstrained minimization is discussed.

J. Greenstadt

Papers

Rank-one and Rank-two Corrections to Positive Definite Matrices Expressed in Product Form