ABSTRACT
In this chapter we deal with unconstrained optimization problems in the form
minimize f(x),
where f : IRn → IR is a continuously differentiable function. We first develop optimality conditions, which characterize analytical properties of local optima and are fundamental for developing solution methods. Since computing not only global but even local optimal solutions is a very challenging task in general, we set a more realistic goal of finding a point that satisfies the first-order necessary conditions, which are the conditions that the first-order derivative (gradient) of the objective function must satisfy at a point of local optimum. We discuss several classical numerical methods converging to such a point.
