Course Name: Constrained Optimization and Linear Algebra

Course No. : MPE 111 Course title : Constrained Optimization and Linear Algebra Number of credits : 4 Number of Lectures–Tutorials–Practical : 50-6-0 Faculty Name : Prof. Badal Mukherji

Course Outlines

Constrained optimization is central to large parts of basic microeconomic theory. The consumer is visualized as maximizing satisfaction from consuming various goods and services but has to satisfy the constraint on consumption imposed by available income. The producer is the either trying to minimize cost subject to the need of producing at least a given level of output or trying to maximize profit subject to the limits imposed by technology and the market. The topics listed in group 1 below address these problems.

In group 2 we develop the mathematical tools required to handle the class of problems in which more than one variables interact with one another. The simplest of those can often be resolved to a stage wherein we need to simultaneously solve a set of linear or lienearized equations. Or, a class of problems of planning where a linear or nonlinear objective function is to be maximized over a feasible set described by a set of linear inequalities.

The course will make a serious attempt to drive home the point that both the topics are spanned by a common underlying theme of properties of convex sets and functions.

Evaluation Procedure

. Minor Exam: 40% . Major Exam: 60%

Details of course contents and allotted time

No. Details Allotted time (hours) Lectures Tutorials 1 Convex sets and concave functions 2 3 Theorem of separating hyperplanes 2 Constrained optimization 1 The Lagrange multiplier when constraints and 10 objective function are differentiable and objective function quasi concave. Nonlinear programmes, the Kuhn Tucker Theorem 6 The saddle value property of nonlinear programmes. 4 2 Vector spaces, linear dependence, rank, nullity and 5 3 basis Linear transforms and matrices 3 No. Details Allotted time (hours) Lectures Tutorials Solvability of systems of linear equations 3 Eigenvalues and eigenvectors 3 Consistency and optimizing planning models: the 11 Frobenieus Theorem and Leontief models. Total 50 6