Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.

Numerical Optimization

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).

/pdf/snopt-an-sqp-algorithm-for-large-scale-constrained-tznz7lt9u8.pdf

SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization

In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue problems, electronic structures computations, and signal processing. In addition to the new algorithms, we show how the geometrical framework gives penetrating new insights allowing us to create, understand, and compare algorithms. The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms. It is our hope that developers of new algorithms and perturbation theories will benefit from the theory, methods, and examples in this paper.

http://www-math.mit.edu/~edelman/publications/geometry_of_algorithms.pdf

The Geometry of Algorithms with Orthogonality Constraints

Since its popularization in the late 1970s, Sequential Quadratic Programming (SQP) has arguably become the most successful method for solving nonlinearly constrained optimization problems. As with most optimization methods, SQP is not a single algorithm, but rather a conceptual method from which numerous specific algorithms have evolved. Backed by a solid theoretical and computational foundation, both commercial and public-domain SQP algorithms have been developed and used to solve a remarkably large set of important practical problems. Recently large-scale versions have been devised and tested with promising results.

Sequential Quadratic Programming

The design and implementation of a new algorithm for solving large nonlinear programming problems is described. It follows a barrier approach that employs sequential quadratic programming and trust regions to solve the subproblems occurring in the iteration. Both primal and primal-dual versions of the algorithm are developed, and their performance is illustrated in a set of numerical tests.

An Interior Point Algorithm for Large-Scale Nonlinear Programming

Sequential quadratic programming for large-scale nonlinear optimization

In this paper some new theoretic results on piecewise differentiable exact penalty functions are presented. Sufficient conditions are given for the existence of exact penalty functions for inequality constrained problems more general than concave and several classes of such functions are presented.

Exact penalty functions in nonlinear programming

We consider the application of a general class of quasi-Newton methods to the solution of the classical equality constrained nonlinear optimization problem. Specifically, we develop necessary and sufficient conditions for the Q-superlinear convergence of such methods and present a companion linear convergence theorem. The essential conditions relate to the manner in which the Hessian of the Lagrangian function is approximated.

On the Local Convergence of Quasi-Newton Methods for Constrained Optimization

A weak qualification is given which insures that a broad class of constrained optimization problems satisfies the analogue of the Kuhn–Tucker conditions at optimality. The qualification is shown to be necessary and sufficient for these conditions to be valid for any objective function which is differentiable at the optimum.

/pdf/a-necessary-and-sufficient-qualification-for-constrained-37uq070x7f.pdf

Jon W. Tolle

Papers

Sequential Quadratic Programming

Sequential quadratic programming for large-scale nonlinear optimization

Exact penalty functions in nonlinear programming

On the Local Convergence of Quasi-Newton Methods for Constrained Optimization

A necessary and sufficient qualification for constrained optimization