Catálogo de publicaciones - libros

Operations Research, in a narrow sense, is the application of scientific models, especially mathematical and statistical ones, to decision making problems. The present course material is devoted to parts of mathematics that are used in Operations Research.

This chapter includes a rapid review of basic concepts of Linear Algebra. After defining fields and vector spaces, we are going to cover bases, dimension and linear transformations. The theory of simultaneous equations and triangular factorization are going to be discussed as well. The chapter ends with the fundamental theorem of linear algebra.

In this chapter, we will analyze distance functions, inner products, projection and orthogonality, the process of finding an orthonormal basis, QR and singular value decompositions and conclude with a final discussion about how to solve the general form of Ax = b .

In this chapter, we will analyze determinant and its properties, definition of eigen values and vectors, different ways how to diagonalize square matrices and finally the complex case with Hermitian, unitary and normal matrices.

Positive definite matrices are of both theoretical and computational importance in a wide variety of applications. They are used, for example, in optimization algorithms and in the construction of various linear regression models. As an initiation of our discussion in this chapter, we investigate first the properties for maxima, minima and saddle points when we have scalar functions with two variables. After introducing the quadratic forms, various tests for positive (semi) definiteness are presented.

For square matrices, we can measure the sensitivity of the solution of the linear algebraic system Ax = b with respect to changes in vector b and in matrix A by using the notion of the condition number of matrix A . If the condition number is large, then the matrix is said to be ill-conditioned. Practically, such a matrix is almost singular, and the computation of its inverse or solution of a linear system of equations is prone to large numerical errors. In this chapter, we will investigate computational methods for solving Ax = b , and obtaining eigen values/vectors of A .

This chapter is compiled to present a brief summary of the most important concepts related to convex sets. Following the basic definitions, we will concentrate on supporting and separating hyperplanes, extreme points and polytopes.

A Linear Programming problem, or LP, is a problem of optimizing a given linear objective function over some polyhedron. We will present the forms of LPs in this chapter. Consequently, we will focus on the simplex method of G. B. Dantzig, which is the algorithm most commonly used to solve LPs; in practice it runs in polynomial time, but the worst-case running time is exponential. Following the various variants of the simplex method, the duality theory will be introduced. We will concentrate on the study of duality as a means of gaining insight into the LP solution. Finally, the series of Farkas’ Lemmas, the most important theorems of alternatives, will be stated.

In this chapter, we will review the basic concepts in real analysis: order relations, ordered sets and fields, construction and properties of the real and the complex fields, and finally the theory of countable and uncountable sets together with the cardinal numbers. The known sets of numbers that we will use in this chapter are ℕ: Natural ℤ: Integer ℚ: Rational ℝ: Real ℂ: Complex

In this chapter, basic notions in general topology will be defined and the related theorems will be stated. This includes the following: metric spaces, open and closed sets, interior and closure, neighborhood and closeness, compactness and connectedness.