On successful completion of this module, the student will(i) be familiar with basic mathematical techniques of constrained and unconstrained extrema in financial models;(ii) be aware of the relationship between constraint parameters and shadow prices;(iii) be able to set up and solve linear optimization problems by the simplex method and duality;(iv) be able to apply convexity methods to certain extremum problems.

Contents
1 Unconstrained Optimization
1.1 Introduction
1.2 Background and Notation. . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 The Second Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 The Hessian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.2 Second Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Lagrange Multipliers
2.1 Lagrange Multipliers with one constraint . . . . . . . . . . . . . . . . . . 17
2.2 Lagrangian method with n variables and one constraint . . . . . . . . . 20
2.3 The Lagrangian Method: n variables, m constraints . . . . . . . . . . . 23
2.4 Economic interpretation of Lagrange Multipliers . . . . . . . . . . . . . 26

3 Linear Programming: The Graphical Method
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Linear Programming (General Problem) . . . . . . . . . . . . . . . . . . . 28
3.3 The Graphical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 No Solution to Linear Programming Problem . . . . . . . . . . . . . . . . 34

4 Linear Programming: The Simplex Method
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Standard Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Gaussian reduction from linear algebra (Recall) . . . . . . . . . . . . . . 41
4.4 The Simplex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5 Two Choices for Optimal row . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 The Simplex Method with Mixed Constraints
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 Transforming the Objective Function . . . . . . . . . . . . . . . . . . . . 52
5.3 The Simplex Method with Mixed Constraints . . . . . . . . . . . . . . . 54
5.4 Solving a Minimisation Problem . . . . . . . . . . . . . . . . . . . . . . . 57
5.5 When the Simplex Method breaks down . . . . . . . . . . . . . . . . . . 61
5.5.1 When the Feasible set is Empty . . . . . . . . . . . . . . . . . . . 62
5.5.2 When the Feasible set is Unbounded . . . . . . . . . . . . . . . . 63

6 Linear Programming: The Dual Problem
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.2 The Dual Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.3 Interpretation of the Dual Problem . . . . . . . . . . . . . . . . . . . . . 70

7 Nonlinear Programming
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.2 Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.2.1 Extreme-Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . 77
7.2.2 The general case: Kuhn-Tucker theory . . . . . . . . . . . . . . . 81

8 Convex and Concave Functions
28.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
8.2 Concave and Convex Functions . . . . . . . . . . . . . . . . . . . . . . . 92
8.3 Positive and negative (semi)definite matrices . . . . . . . . . . . . . . . . 95
8.4 The n-variable case: convex and concave . . . . . . . . . . . . . . . . . . 99
8.5 Quasi-concave and quasi-convex functions . . . . . . . . . . . . . . . . 102
8.5.1 Upper Level Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.5.2 Quasi-concave and quasi-convex functions . . . . . . . . . . . . 104
8.5.3 A determinant criterion for quasi-concavity . . . . . . . . . . . . 106

Not applicable to this module.

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