This quite long chapter is concerned wit static optimization problems (the particular case of Linear Programming problems is treated separately in the next chapter). Indeed, much of economic analysis relies on static optimization problems (production theory, consumption theory, utility theory, etc.). We treat separately the cases of unconstrained and constrained optimization problems. These last problems are in turn particularized between “classical” constrained optimization problems (i.e. with equality constraints) and “modern” constrained optimization problems, i.e. mathematical programming problems. The treatment is concerned with the basic optimality conditions (necessary and sufficient), both under differentiability assumptions and in absence of differentiability (Sect. 4.10 is concerned with the so-called “saddle points conditions” of the Lagrangian function). As an application, the last Sect. 4.12 considers the so-called “portfolio selection problem” , a financial problem studied by the economist (and Nobel Prize recipient in 1990 in Economic Sciences) H. M. Markowitz.