2. Unconstrained Optimization

In optimization theory, the optimality conditions for interior points are usually much simpler than the optimality conditions for boundary points. In this chapter, we deal with the former, easier case. Boundary points appear more prominently in constrained optimization, when one tries to optimize a function, subject to several functional constraints. For this reason, the optimality conditions for boundary points are generally discussed in constrained optimization, whereas the optimality conditions for interior points are discussed in unconstrained optimization, regardless of whether the optimization problem at hand has constraints. In this chapter, we first establish some basic results on the existence of global minimizer or maximizers of continuous functions on a metric space. These are the famous Weierstrass theorem and its variants, which are essentially the only general tools available for establishing the existence of optimizers. The rest of the chapter is devoted to obtaining the fundamental first-order and second-order necessary and sufficient optimality conditions for minimizing or maximizing differentiable functions. Since the tools here are based on differentiation, and differentiation is a local theory, the optimality conditions generally apply to local optimizers. The necessary and sufficient conditions play different, usually complementary, roles. A typical necessary condition for