Introduction to Optimization, variables and Objective functions, Stationary values and Extrema, Relative and Absolute extrema, Equivalence of minimum and maximum, Convex, Concave and uni-model functions, Constraints, Mathematical programming problems. Optimization of one-dimensional functions. Optimization of two dimensional functions and derivatives of sufficient conditions for existence of optima for them Exercises. Optimization of function of several variables and necessary and sufficient conditions for it Exercises. Optimization by equality constraints: direct substitution method and Lagrange multiplier method. Behavior of the Lagrangian functions. Exercises. Necessary and sufficient conditions for an equality constrained optimum with bounded independent variables. Inequality constraints and Lagrange multipliers. Multidimensional optimization by Gradient method. Exercises. Convex and Concave programming Linearization. Exercises. Calculus of variation, Euler-Lagrange equations. Functional of several variable. Functionals depending on higher derivatives. Functionals depending on several independent variables. Variational problems in parametric form. Some applications, Constraints variational problem. A minimum path problem. Dynamic programming fundamentals. Generalized mathematical formulation of dynamic programming, Problems and exercises. Dynamic programming and variational calculus. Control theory.
