Convex optimization

Optimization problems are often specified in terms of an objective function to be minimized and a set that defines admissible decisions. The newsvendor in §1.C may not be able to order more than $$\alpha $$ newspapers and would need to select an order quantity from the set $$[0, \alpha ]$$ . A maximum likelihood problem (§1.E) permits only certain types of estimates. This chapter initiates our treatment of such optimization problems with constraints. As we’ll see, a problem of this general type can be formulated in terms of a modified function that incorporates the effect of both the set of admissible decisions and the original objective function. This perspective will allow us to achieve far-reaching extensions of the principles laid out in Chapter 1 and address nonsmooth problems broadly. Since it’s most accessible, we’ll concentrate on the convex case and leave a detailed treatment of more general settings for subsequent chapters.