Optimisation and decision-making under uncertainty

Lecturers

A/Prof Nam Ho-Nguyen, The University of Sydney
Dr Li Chen, The University of Sydney

Synopsis

The most interesting and challenging applications of mathematics often begin when uncertainty enters the picture: real decisions must be made with incomplete information, noisy data, and competing objectives. This course develops a rigorous approach to decision-making in that setting, using mathematical optimisation to formulate decision problems precisely and solve them systematically. The emphasis is on optimisation problems driven by random or uncertain inputs, where the aim is to maximise expected performance, manage downside risk, or ensure reliable performance across plausible scenarios. These questions arise across areas such as finance, data science, operations, energy systems, and the design of large-scale engineered and social systems.

The course begins by building core optimisation foundations, with a focus on modelling techniques that transfer across a wide range of problem classes. It then introduces risk: how to define it, quantify it, and incorporate it into optimisation models in a principled way. Finally, it covers modern methods for optimisation under uncertainty such as stochastic programming and robust optimisation, equipping you with both model-building skills and algorithmic tools for tackling uncertainty in practice. By the end of the course, students will be able to model and interpret optimisation problems under uncertainty using a unified view of risk, stochastic, and robust methods, and to articulate the trade-offs between performance, risk, and robustness.

Course Overview

Week 1: Optimisation foundations for decision models

• Modelling with optimisation: variables, objectives, constraints; modelling assumptions and interpretation
• Convexity, LP, QP, conic programming
• Duality and optimality conditions
• Gradient descent
• Examples: regression, classification, portfolio optimisation, resource allocation

Week 2: Understanding uncertainty

• Modelling of uncertainty: risk and ambiguity,
• Decision criteria: utility theory, certainty equivalent, prospect theory, satisficing
• Axiomatic development of risk measures (properties + intuition, VaR, expected shortfall, optimized certainty equivalent, coherence, dual representation)
• Risk, modelling and convexity (chance constraints, CVaR approximations)

Week 3: stochastic optimisation

• Stochastic programs, scenarios, sample average approximations (SAA) and asymptotic optimality
• Recourse in decision-making (fixed vs random, complete vs incomplete), two-stage models
• Decomposition algorithms (progressive hedging)
• Stochastic gradient descent

Week 4: robust optimisation

• Uncertainty sets and robust counterparts
• Distributionally robust optimisation and ambiguity sets
• Wasserstein DRO (out-of-sample guarantee, reformulation, equivalence of robustification and regularization)
• Two-stage DRO

Prerequisites

Linear algebra, real analysis, basic probability, and basic programming skills. (The course will use Python.)

This course will be taught at a foundational level, i.e., no optimisation knowledge will be assumed. However, it can also be taken as a second course in optimisation, with the first week treated as revision. The material from Week 2 onwards is not typically covered in traditional optimisation coursework.

Assessment

• TBC

Resources/pre-reading

Core references

• [BV] Stephen Boyd and Lieven Vandenberghe, Convex Optimization (free PDF; publisher).
• [KKN] Fatma Kılınç-Karzan and Arkadi Nemirovski, Essential Mathematics for Convex Optimization—the more advanced reference (free PDF; publisher).
• [MML] Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon Ong, Mathematics for Machine Learning—a supplementary reference (free PDF; publisher).

Essential refresher

1. Linear algebra and quadratic forms
   • [BV], Appendix A.5: range, nullspace, eigenvalue decomposition, and definiteness.
   • [KKN], Appendix A, Sections A.1–A.3; Appendix D, Sections D.1–D.2: vectors, matrices, linear mappings, symmetric matrices, eigenvalues, and quadratic forms.
2. Multivariable and matrix calculus.
   • [BV], Appendix A.3–A.4, especially Sections A.4.1–A.4.4: functions, continuity, Jacobians, gradients, chain rules, and Hessians.
   • [KKN], Appendix C, Sections C.1–C.2; Section 10.2.1: derivatives, higher-order derivatives, and their connection with convexity.
3. Probability and elementary statistics.
   • [MML], Chapter 6, Sections 6.1–6.4: random variables and distributions; sum, product, and Bayes’
     rules; expectation, variance, covariance, independence, sample means, and sample covariance.
4. Sequences and convergence.
   • [BV], Sections A.2.1 and A.3.2: sequences, limits, and continuity.
   • [KKN], Sections B.1.2 and B.2.1–B.2.3: convergence of sequences and preservation of limits.

Optimisation

Both [BV] and [KKN] cover the fundamentals of convex optimisation. This material is not a pre-requisite and will be covered in the course, but both books are useful references.

Programming

The course uses the standard scientific Python stack (NumPy, SciPy, and pandas), commonly used in data-science workflows, together with state-of-the-art optimisation solvers Gurobi and/or MOSEK through their Python APIs.

• Anaconda Distribution: Python, conda, Jupyter, and commonly used scientific-computing packages.
• IBM: Python Basics for Data Science (edX): free through the audit track; Python fundamentals, Jupyter, pandas, and NumPy.
• Gurobi with Python: install gurobipy, activate a licence, then work through the first-model tutorial.
• MOSEK Fusion with Python: install mosek, add the licence file, verify from mosek.fusion
  import *, then work through thelinear optimisation tutorial.Licensing. Gurobi and MOSEK provide free academic licences for eligible students, faculty, and staff. If you cannot obtain an academic licence, contact the instructors.