A 10-week course covering the mathematical theory of optimisation that underpins modern economics. Each module develops a core concept — from unconstrained calculus to dynamic programming — and pairs it with computational exercises in Python so you can see the theory at work. Designed as a companion to Programming & Numerical Methods: that course teaches you to implement algorithms; this one teaches you the mathematics behind them. No prior programming experience is assumed; all materials are open access.
How to use these materials
Each week has a notebook you can read online, download, or run in the cloud, plus a slide deck for the lecture:
- Read online — click the notebook link; it renders as a webpage with all code and output.
- Run in Google Colab — click the Colab badge and the notebook opens ready to run in your browser. Nothing to install.
- Run locally — install Anaconda (includes Python, Jupyter, NumPy, and Matplotlib), then download the
.ipynbfile and open it in Jupyter.
Companion course: this course provides the mathematical foundations for Programming & Numerical Methods. Taking them together is recommended but not required.
Materials
| Week | Topic | Notebook | Slides | Colab |
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| 1 | Mathematical Foundations | View · Download |  |
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| 2 | Unconstrained Optimisation I | View · Download | |
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| 3 | Unconstrained Optimisation II | View · Download | |
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| 4 | Constrained Optimisation I | View · Download | |
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| 5 | Constrained Optimisation II | View · Download | |
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| 6 | Convex Optimisation | View · Download | |
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| 7 | Linear Programming | View · Download | |
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| 8 | Dynamic Optimisation I | View · Download | |
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| 9 | Dynamic Optimisation II | View · Download | |
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| 10 | Frontiers | View · Download | |
Notebooks include interactive exercises with hidden solutions — try first, then click to check.
Syllabus
- Mathematical foundations — sets, real analysis essentials, vector spaces, matrices, eigenvalues, and positive-definiteness; Python refresher with NumPy
- Unconstrained optimisation I — multivariate calculus review, gradient and Hessian, first- and second-order conditions, convexity and concavity, economic examples (profit maximisation, cost minimisation)
- Unconstrained optimisation II — iterative algorithms: steepest descent, Newton’s method, quasi-Newton (BFGS), convergence rates; implementing gradient descent in Python
- Constrained optimisation I — equality constraints, Lagrange multipliers, the bordered Hessian, envelope theorem; applications to utility maximisation and the expenditure function
- Constrained optimisation II — inequality constraints, Karush–Kuhn–Tucker conditions, complementary slackness, constraint qualification; portfolio choice and production planning
- Convex optimisation — convex sets and functions, convex programs, Lagrangian duality, strong duality and Slater’s condition; solving convex problems with CVXPY
- Linear programming — the standard form, the simplex method, LP duality and shadow prices, sensitivity analysis; applications to input–output models, transportation, and diet problems using SciPy and PuLP
- Dynamic optimisation I — calculus of variations, Euler–Lagrange equation, transversality conditions, optimal control theory, Pontryagin’s maximum principle; the Ramsey–Cass–Koopmans growth model
- Dynamic optimisation II — the principle of optimality, Bellman’s equation, contraction mapping theorem, policy and value function iteration; consumption–savings with borrowing constraints
- Frontiers — stochastic dynamic programming, introduction to Bayesian optimisation, optimisation in machine learning (SGD, Adam), connections to computational economics and current research
Assessment
- Weekly problem sets: 20%
- Computational essays (3 over the term): 40%
- Final project: 30%
- Participation: 10%
References
Main texts
- Boyd, S. & Vandenberghe, L. Convex Optimization (2004) — freely available; main reference for Weeks 2–6
- Kochenderfer, M. J. & Wheeler, T. A. Algorithms for Optimization (2019) — freely available; algorithms and implementation
- Sundaram, R. K. A First Course in Optimization Theory (1996) — rigorous treatment of constrained optimisation and duality
- Chiang, A. C. & Wainwright, K. Fundamental Methods of Mathematical Economics (4th ed.) — accessible introduction for economics students
Supplementary
- Judd, K. L. Numerical Methods in Economics (1998) — bridge to computational implementation (also used in PNM)
- Bertsekas, D. P. Convex Optimization Theory — freely available; advanced convex analysis
- Bertsekas, D. P. Dynamic Programming and Optimal Control, Vols. I & II — definitive reference for Weeks 8–9
- Stokey, N., Lucas, R. & Prescott, E. Recursive Methods in Economic Dynamics (1989) — dynamic programming for economists
- Simon, C. P. & Blume, L. Mathematics for Economists (1994) — comprehensive maths-for-econ reference
- QuantEcon — T. Sargent and J. Stachurski’s open lectures on computational economics
Video lectures (all free)
- Stanford EE364A: Convex Optimization — Stephen Boyd (2023)
- CMU: Convex Optimization — Ryan Tibshirani
- Arizona Math Camp: Optimization
- Dimitri Bertsekas: Dynamic Programming Lectures
- KIT: Optimization Methods for ML and Engineering — Julius Pfrommer
Python tools used in this course
- SciPy Optimize — general-purpose optimisation routines
- CVXPY — disciplined convex programming in Python
- PuLP — linear programming modelling
Questions
Found an error in a notebook, or stuck on setup? Email juan.zurita@ed.ac.uk.