Convex Optimization Problems

In the previous chapters we have introduced the notions of convex sets (Chapter

2) and convex functions (Chapter 3). Based on the foundation laid in the previ-

ous chapters, we now move forward to study the concepts of convex optimization.

An optimization problem to be solved may appear to be a nonconvex problem

due either to the nonconvexity of the objective function, to the the nonconvexity

of the constraint set, or to both. However, it may be potentially a convex opti-

mization problem. In this chapter, we will focus on formulation of such problems

into a standard convex optimization problem so that the optimal solution can be

obtained via either optimality conditions (to be presented partly in this chapter

and partly in Chapter 9) or available convex problem solvers. The quasiconvex

problem is also presented together with a sufficient optimality condition, and

then the widely used bisection method for solving it is introduced.