Igor Griva

Linear and nonlinear optimization

Preface Part I. Basics: 1. Optimization models 2. Fundamentals of optimization 3. Representation of linear constraints Part II. Linear Programming: 4. Geometry of linear programming 5. The simplex method 6. Duality and sensitivity 7. Enhancements of the simplex method 8. Network problems 9. Computational complexity of linear programming 10. Interior-point methods of linear programming Part III. Unconstrained Optimization: 11. Basics of unconstrained optimization 12. Methods for unconstrained optimization 13. Low-storage methods for unconstrained problems Part IV. Nonlinear Optimization: 14. Optimality conditions for constrained problems 15. Feasible-point methods 16. Penalty and barrier methods Part V. Appendices: Appendix A. Topics from linear algebra Appendix B. Other fundamentals Appendix C. Software Bibliography Index.

Topics from Linear Algebra

The topics are: the determinant, the trace, and similarity classification of matrices using PID theory. In our discussion of the determinant it won’t be enough to consider matrices over fields; we will need the case R = F [x] for the application to linear algebra. So we might as well go all the way and allow general commutative rings R. At this point you might want to review the short section on algebras over a commutative ring in the algebra notes. The R-algebras we have in mind are MnR, RG for a group G, and R[x]. These are defined in the same way as for fields. For example, MnR is a free R-module on n generators, and the usual rules for matrix addition and multiplication make it an R-algebra.