## Large sparse numerical optimization.(English)Zbl 0536.65047

Lecture Notes in Computer Science, 165. Berlin etc.: Springer-Verlag. iv, 105 p. (1984).
In the first chapter of the book the author considers some methods for solving $$Ax=b$$ where $$A$$ is a large sparse matrix of order $$n$$. Two basic approaches – direct and iterative – are analysed. In the next chapter he considers the overdetermined linear least squares problem $\text{minimize } \| Ax-b\|_2$ where it will be assumed that $$A$$ is of full column rank. In connection with this problem the method of G. Peters and J. H. Wilkinson [Comput. J. 13, 309–316 (1970; Zbl 0195.44804)] is presented. At the end of this chapter he studies the following problem $\text{minimize } \| Ax-b\|_ 2,\quad Cx=d,$ where $$A$$ is $$m$$ by $$n$$ and $$C$$ is $$t$$ by $$n$$ with $$t\le n\le m$$. It is assumed that $$\mathrm{rank}(C)=t$$. Regarding this problem the following methods are presented: projection method, elimination method, method of Lagrange multipliers and infinite weights method.
Chapter 3 deals with some problems of linear programming. Consider the problem:
(1) minimize $$C^ Tx$$, $$Ax=b$$, $$\ell \leq x\leq \mu$$, where $$A$$ is $$m$$ by $$n$$ and $$m<n$$. Referring to problem (1) the author discusses some up-to-date variants of the simplex algorithm.
In chapter 4, entitled “Nonlinear equations and nonlinear least squares” he studies the problem
(2) minimize \| F(x)\|_ 2\), where $$F = R^m\to R^n$$, $$m\geq n$$, $$F$$ is at least continuously differentiable. In the case $$m=n$$ and $$F(x^*)=0$$, then (2) can be expressed at the zero-finding problem,
(3) solve $$F(x)=0$$.
For the problem (2) and (3) and for the computation of $$F'(x)$$ some algorithms are given.
The fifth chapter of the paper deals with the large unconstrained optimization problem:
Minimize $$f(x)$$, where $$f: R^n\to R^1$$ and $$f$$ by $$n$$ is twice continuously differentiable. It is supposed that: the Hessian $$\nabla^2f(x)$$ is large and sparse. In studying this problem the author exposes some new and interesting methods.

### MSC:

 65K05 Numerical mathematical programming methods 65F20 Numerical solutions to overdetermined systems, pseudoinverses 65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis 90C30 Nonlinear programming 90C05 Linear programming

Zbl 0195.44804

LSQR
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