040100 VO Mathematics 2 (2020S)

The lecture is concerned with differential calculus for functions of several variables, convex analysis, and static optimization techniques (unconstrained, equality-constrained, and inequality-constrained problems)
as well as their application in business and economics.

Contents:
1. Introduction: optimization problems in business and economics
2. Differential calculus for functions of several variables
(real valued functions of several variables, some basics terms of topoloy, partial derivatives, derivative, tangent plane, gradient, vector functions, Jacobian, chain rule, directional derivatives, total differential, geometric interpretation of the gradient, second derivatives, Hessian, second directional derivative)
3. Convexity
(convex sets, convex and concave functions in several variables)
4. Optimization of scalar valued functions
(stationary points, second order conditions, comparative statics, envelope theorem)
Inverse and implicit functions
5. Optimization with equality constraints: Lagrange's method
(first and second order conditions, interpretation of the Lagrange multipliers, conditions for global optima, quasiconcavity and quasiconvexity, economic applications)
6. Nonlinear programming
(convex programs, Kuhn-Tucker conditions, constraint qualifications, saddle point condition)
7. Linear programming
(model formulation, assumptions underlying a linear planning model, graphic solution of two-variable programs, basic solutions, characterization of the sets of feasible and optimal solutions, simplex method, formal structure of the simplex tabelaus, alternative optimal solutions, duality, complementary slackness, economic interpretation of the dual programme, interpretation of a computer solution)