Notes on Non-Linear Optimization

As it is well know convexity and non linear optimization has been increasingly important in the last years in the study of many areas of economics. Systems of inequalities the maximun or minimun of a concave function over a convex set are among the most useful topics in microeconomics, game theory, or general equilibrium. Certain these topics can be regarded as part of convex analysis, however since these notes are dedicate to economists with interest in mathematics models or mathematicians with interest in economics we made emphasis in the some topics and other has been omitted not because they lack interest, but because they would have required technical developments somewhat outside the mainstream of this focus. The exposition is limited to Rn, the space of the n - tuplas of real numbers, even though many of the results can easily be formulated in the more general setting of the functional analysis. The first part is dedicated to the structure of Rn as a topological vectorial space. Next we study the continuity of convex and concave function function an the structure of the convex set. Finally we study the problem of maximize a convex function in a convex set, and we give some applications to economics. As far as technical prerequisites are concerned the reader should be able to get by, with a sound of knowledge of linear algebra and elementary real analysis, nevertheless the style does presuppose a certain mathematical maturity