This chapter discusses the fundamentals of nonlinear optimization. Section 3.1 focuses on optimality conditions for unconstrained nonlinear optimization. Section 3.2 presents the first-order and second-order optimality conditions for constrained nonlinear optimization problems. This section presents the formulation and basic definitions of unconstrained nonlinear optimization along with the necessary, sufficient, and necessary and sufficient optimality conditions. An unconstrained nonlinear optimization problem deals with the search for a minimum of a nonlinear function f(x) of n real variables x = (x1, x2 , . . . , xn and is denoted as Each of the n nonlinear variables x1, x2 , . . . , xn are allowed to take any value from - ∞ to + ∞. Unconstrained nonlinear optimization problems arise in several science and engineering applications ranging from simultaneous solution of nonlinear equations (e.g., chemical phase equilibrium) to parameter estimation and identification problems (e.g., nonlinear least squares).
Unconstrained nonlinear optimization of a distributed SWE model using MODIS and in situ measurements over the French Alps