Local Theory of Optimization

In this chapter, we derive the local necessary, as well as sufficient conditions for the unconstrained optimization of a functional, the constrained optimization of a functional subject to equality constraints, as well as inequality constraints, on real Banach spaces. We define the concept of positive definite, positive semi-definite, negative definite, and negative semi-definite operators on a real Banach space. We derive the necessary and sufficient conditions for Fréchet differentiable functionals to be a convex functional. In constrained optimization problems, the constraints are mapping between Banach spaces, and the inequality constraints are in terms of positive cones. Lagrange multiplier theory is presented for constrained optimization problems. The generalized Kuhn–Tucker Theorem for optimization problems with infinite-dimensional inequality constraints is presented. The results are standard and inspired by Luenberger (Optimization by Vector Space Methods. Wiley, New York, 1969).