Introducción al análisis convexo y los problemas variacionales

Abstract

[eng] One of the most importants problems has been tominimize functions. The history of optimization and calculus of variations is tightly interwoven with the history of mathematics. The field has drawn the attention of a remarkable range ofmathematical luminaries, beginning with Newton and Leibniz, then initiated as a subject in its own right by the Bernoulli brothers Jakob and Johann. The first major developments appeared in the work of Euler, Lagrange, and Laplace. In the nineteenth century, Hamilton, Jacobi, Dirichlet, and Hilbert are but a few of the outstanding contributors. Nowadays, several related recent developments have stimulated new interest in the convex optimization problems. The first is the recognition that interior-point methods, developed in the 1980s to solve linear programming problems, can be used to solve convex optimization problems as well. The second development is the discovery that convex optimization problems (beyond least-squares and linear programs) are more prevalent in practice than was previously thought. Since 1990 many applications have been discovered in areas such as automatic control systems, estimation and signal processing, communications and networks, electronic circuit design, data analysis and modeling, statistics, and finance. Convex optimization has also foundwide application in combinatorial optimization and global optimization, where it is used to find bounds on the optimal value, as well as approximate solutions. Hopfully many other applications of convex optimization and calculus of variations are stillwaiting to be discovered. There are great advantages to formulating a problem as a convex optimization problem. The most basic advantage is that the problemcan then be solved, very reliably and efficiently, using algorithms for convex optimization. These solution methods are reliable enough to be solved with a computer. There are also theoretical or conceptual advantages of formulating a problem as a convex optimization problem.