Sequential quadratic programming.

*(English)*Zbl 0828.65060
Iserles, A. (ed.), Acta Numerica 1995. Cambridge: Cambridge University Press. 1-51 (1995).

The paper presents a review on sequential quadratic programming (SQP) methods, that belong to the most popular and efficient algorithms for solving constrained optimization problems. In particular theoretical aspects are discussed, i.e. local and global convergence behaviour.

First the authors describe the basic ideas and several possibilities to formulate the quadratic programming subproblem. In another chapter, the most interesting local convergence results are summarized for equality constrained problems (without loss of generality). Newton’s method, general assumptions leading to superlinear convergence speed and quasi- Newton methods are discussed, also the reduced Hessian case.

Next two merit functions are introduced, the augmented Lagrangian and the \(L_1\)-penalty function, that are needed to get global convergence theorems. Also some comments are found how to combine local and global results. Moreover two sections about trust region approaches and practical considerations are added.

The paper consists of 50 pages and comes with a large number of references. For some theorems also proofs are presented or at least sketched.

It is a pleasure to read the paper, since the authors succeeded to find a comprehensive and unified way to overview the large variety of different approaches that are available.

For the entire collection see [Zbl 0817.00007].

First the authors describe the basic ideas and several possibilities to formulate the quadratic programming subproblem. In another chapter, the most interesting local convergence results are summarized for equality constrained problems (without loss of generality). Newton’s method, general assumptions leading to superlinear convergence speed and quasi- Newton methods are discussed, also the reduced Hessian case.

Next two merit functions are introduced, the augmented Lagrangian and the \(L_1\)-penalty function, that are needed to get global convergence theorems. Also some comments are found how to combine local and global results. Moreover two sections about trust region approaches and practical considerations are added.

The paper consists of 50 pages and comes with a large number of references. For some theorems also proofs are presented or at least sketched.

It is a pleasure to read the paper, since the authors succeeded to find a comprehensive and unified way to overview the large variety of different approaches that are available.

For the entire collection see [Zbl 0817.00007].

Reviewer: K.Schittkowski (Bayreuth)