Basics of Convex Programming

The solution procedure of the previous chapter relies crucially on the ability to easily identify what constraints are active at the solution of the optimization problem under study. This works fine for problems with only two design variables, but when trying to solve real-life problems, where the number of design variables may vary from the order of 10 to the order of 100 000 or more, one needs more systematic solution methods. In this and the following chapter we will study methods from the field of mathematical programming that are applicable for large-scale problems. We begin by reviewing some fundamental results of mathematical programming, with focus on convex programming. Actually, most problems of structural optimization are in fact nonconvex, but this does not imply that convex programming is of little importance in structural optimization: we will see in Chap. 4 that convex approximations play a very important role in the solution algorithms for nonconvex problems. All theorems are presented without proofs; these may be found in any good book on nonlinear mathematical programming such as Bazaraa, Sherali and Shetty (Nonlinear Programming—Theory and Algorithms, Wiley, 1993) or Bertsekas (Nonlinear Programming, Athena Scientific, 1995).