Overview:
- Credits:
- 5.0
- Level:
- 2
- Semester:
- Spring
- Subject:
- Mathematics
- School:
- Mathematics & Statistics
- Coordinator:
- Assoc Professor Christopher Boyd
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Curricular information is subject to change
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.
Indicative Module Content: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
Student Effort Type | Hours |
---|---|
Lectures | 30 |
Tutorial | 12 |
Specified Learning Activities | 32 |
Autonomous Student Learning | 32 |
Total | 106 |
Not applicable to this module.
Description | Timing | Component Scale | % of Final Grade | ||
---|---|---|---|---|---|
Continuous Assessment: Tutorial Quiz | Throughout the Trimester | n/a | Alternative linear conversion grade scale 40% | No | 10 |
Examination: end of semester exam | 2 hour End of Trimester Exam | No | Alternative linear conversion grade scale 40% | No | 70 |
Class Test: Midterm Examination | Varies over the Trimester | n/a | Alternative linear conversion grade scale 40% | No | 20 |
Resit In | Terminal Exam |
---|---|
Autumn | Yes - 2 Hour |
• Feedback individually to students, post-assessment
Correct solutions will be posted online. Corrected quizzes returned and students will have the possibility of viewing their midterms scripts.
Name | Role |
---|---|
Anton Sohn | Tutor |