Contemporary Mathematics in Economics
Prerequisites: Linear Algebra, Real Analysis, multivariate calculus, linear programming.
Course Description
This course is for master students which are specialized on “quantitative economics”. It
covers the basic mathematical tools used in economic theory. Knowledge of elementary
calculus is assumed; some of the prerequisite material is reviewed in the first section. The
main topics are multivariate calculus, concavity and convexity, nonlinear programming
and optimization theory. The emphasis throughout is on techniques rather than abstract
theory.
Many real-life problems involve nonlinearities: many mechanical and chemical systems
are nonlinear (drag force is nonlinear in the velocity), people's attitude towards risk is
nonlinear, and shipping/ticket costs are nonlinear in weight/lenght of the trip. This course
presents theoretical foundation and methods for solving the nonlinear programming
(NLP) problems.
Solving NLP problems is hard, and we discuss why. Linear programming (LP) problems
are usually considered as the "easy" problems and they have many important applications
in real-life, but on this course we show that the class of easy problems (at least in theory)
should be convex problems. We introduce the basics of convex analysis, which is itself
one beautiful and complete field of mathematics. We develop and prove optimality
conditions by starting from a geometric condition "there is no better solution nearby" and
end up with algebraic equations that determine whether the point in question can or
cannot be the optimum.
Course objectives
 To introduce students to some basic concepts and theories of optimization.
 To develop capacity to think strategically about a company, its present business
position, its long-term direction, its resources and competitive capabilities, and its
opportunities for gaining sustainable competitive advantage.
 To build skills in conducting mathematical analysis in a variety of competitive
situations.
 At the end of the course, students will be able to formulate economic problems in forms of
NLP and solve them with suitable software.

Course content
Introduction.
The nature of mathematical economics. Mathematical economics versus econometrics.
Importance of models in economics. Types of models in economics.
1 Things to know
1.1 Sets.
1.2 The space we work in.
1.3 Facts from real analysis.
1.4 Facts from linear algebra.
2 Feasibility
2.1 Fundamental theorem of linear algebra.
2.2 Linear inequalities.
2.3 Non-negative solutions (cone, finite cone, Farkas Lemma, Fakas alternative).
2.4 The general case (slack and surplus variables, standard form).
2.5 Application: arbitrage (short and long position in the stock, discret Black–Scholes
formula).
3 Convex sets
3.1 Separating hyperplane theorem (strict separating hyperplane theorem, weak
separating hyperplane theorem, geometrical interpretation of the Farkas lemma, prove of
the Farkas lemma-geometrical approach)
3.2 Polyhedrons and polytopes (Farcas-Minkowski-Weyl theorem, convex combination,
convex hull, resolution theorem, extreme point, extreme ray)
3.3 Dimension of a set (affinely independent collection and dimension)
3.4 Properties of convex sets (Krein-Millman theorem, Helly’s theorem, topologically
equivalent sets.
3.5 Application: linear production model.
4. Basic properties of linear programming.
4.1 Standard form, size of problems.
4.2 Examples of linear programming problems (the diet problem, the transport
problem)
4.3 Basic solutions, basic feasible solution, degenerate basic feasible solution,
fundamental theorem of linear programming, equivalence of extreme points and basic
solutions.
5. Non-linear programming.
Problem classification- unconstrained optimization, linear programming, quadratic
programming, linear constrained problem, separable programming, convex programming.
5.1 Problem statement. Examples: portfolio selection, water resources planning,
constrained regression. Local and global maximum/minimum.
5.1 Necessary and sufficient conditions – general definitions. Optimality criteria.
5.2 Optimality conditions – unconstrained case. Necessary conditions for local
optimality. Feasible directions. Examples of unconstrained problems: Production,
approximation, control. Hessian matrix. Positive definite matrix. Positive semidefinite
matrix. Cholesky factorization. Second-order necessary conditions-unconstrained case.
5.3 Karush-Kuhn-Tucker (KKT) conditions in constrained optimization. Active,
inactive constrains. The LI constrain qualifications. Lagrange multipliers. Lagrangian.
First order necessary conditions to a local extremum. Second order necessary conditions
to a local extremum. Second order sufficient conditions. Constrained Case – KKT
Conditions – minimization with inequality constrain, Necessary KKT Conditions (if
g(x)≥0). Necessary KKT Conditions (General Case). KKT Sufficiency Theorem (Special
Case). Limitations of the Kuhn–Tucker conditions.
6. Concave functions.
Continuity of concave bounded function on convex set (Theorem). Local and global
minimum and maximum of concave and convex functions. Hypograph. Combinations of
convex functions. Properties of differentiable convex functions. Twice continuously
differentiable functions - characterization of convexity. Jensen’s inequality.
7. Particular problems of NLP.
Convex NLP problems. Feasibility problems. Maximization problems. Equivalent
problems.Change variables. Transformation of objective and constraint functions.
Separable programming. Reformulation as a linear programming problem. Quadratic
programming. Nonconvex programming. Fractional programming. Example -The
Wyndor Glass Company (WGC).
8. Numerical methods for nonlinear programming problems.
Algorithm, iterative algorithm. Descent, descent function. Global convergence, local
convergence. Order of convergence. Linear convergence. Quadratic convergence. Closed
mappings. Composition of closed mappings. Global convergence theorem. Fixed point
problem. Contraction mappings. Banach fixed point theorem (proof). Brouwer fixed
point theorem.
8.1 Basic descent methods.
Linear search method. Exhaustive search. Dichotomous (Bi-section) search.
Fibonacci and golden section search. Line search by curve fitting. Newton’s method.
Theorem about Newton’s method convergence. Method of false position. Inaccurate line
search – percentage test, Armijo’s rule, Goldstein test.
8.2 The method of steepest descent. Global convergence. The quadratic case.
Kantorovich inequality. Condition number of matrix. The nonquadratic case. Scaling.
Newton-like algorithms for nonlinear systems.
8.3 Constrained nonlinear optimization. Penalty function methods. Interior-point
methods - barrier functions.
Assessment
The assessments’ process include both final exam and some points of current control
based on such tools as essays, case analysis, tests, group work, comments on an article's
theoretical perspective and so on.
Main reading
1.
Rakesh V. Vohra, Advanced Mathematical Economics
2.
David G. Luenberger, Stanford University Linear and Nonlinear Programming,
Third Edition, Stanford University