##
**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.

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.

Reviewer: Nicolaie Luca (Iaşi)

### 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 |