Optimization

In this chapter, we will start from basic concepts and examples, which contain some well-known optimization problems used in a wide fields. Then, for general optimization, we introduce optimal condition satisfied by local or global minimum. Well-known Karush-Kuhn-Tucker condition is introduced in detail. In the following, dual optimization is discussed, which includes weak and strong forms. Condition satisfied by strong duality is formulated through the framework of constraint qualification. Finally, we stress on the convex optimization and introduce the corresponding version of optimal condition. In terms of numerical algorithm of convex optimization, we mention active set method to solve quadratic convex problem. For nonconvex optimization, we include sequential quadratic programming for quadratic optimization system.