Mathematical Optimization for Engineers Course

Lesson 1 • Problem Formulation Fundamentals

Translates engineering scenarios into mathematical models. Connects real-world requirements to formal optimisation structures.

Lesson 2 • Mathematical Prerequisites Review

Reviews calculus, linear algebra, and set theory essentials. Ensures students can manipulate the mathematical objects used in optimisation.

Lesson 3 • Core Concepts and Terminology

Defines objective functions, decision variables, and constraints. Establishes the vocabulary used throughout all subsequent optimisation methods.

Lesson 1 • Sensitivity and Post-Optimality Analysis

Analyses how optimal solutions change with parameter perturbations. Supports robust engineering decision-making under uncertainty.

Lesson 2 • Duality Theory

Derives dual LP problems and proves strong duality. Enables sensitivity analysis and economic interpretation of constraints.

Lesson 3 • Standard Form and Geometry

Converts LP problems to standard form and visualises feasible polytopes. Establishes geometric intuition for the simplex method.

Lesson 4 • The Simplex Method

Develops the full simplex algorithm from pivot rules to termination. Provides the primary computational tool for LP problems.

Lesson 1 • Optimality Conditions for Smooth Functions

Derives first- and second-order necessary and sufficient conditions. Provides the theoretical basis for all gradient-based algorithms.

Lesson 2 • Derivative-Free Optimisation Methods

Covers Nelder-Mead, pattern search, and surrogate-based methods. Addresses problems where gradients are unavailable or unreliable.

Lesson 3 • Gradient Descent and Conjugate Gradient

Implements steepest descent and conjugate gradient algorithms. Compares convergence rates for quadratic and general objectives.

Lesson 4 • Line Search Strategies

Covers exact and inexact line search methods for step-size selection. Ensures sufficient decrease and curvature conditions in iterative methods.

Lesson 1 • Penalty and Barrier Methods

Converts constrained problems to unconstrained sequences via penalties. Connects constrained theory to unconstrained solvers already mastered.

Lesson 2 • Practical Constraint Handling Techniques

Addresses bound constraints, variable scaling, and warm starting. Improves solver robustness and efficiency in real engineering applications.

Lesson 3 • KKT Optimality Conditions

Derives Karush-Kuhn-Tucker conditions for constrained problems. Provides the theoretical foundation for all constrained optimisation algorithms.

Lesson 4 • Sequential Quadratic Programming

Solves constrained NLP by iterating quadratic subproblems. Achieves fast convergence for smooth engineering design problems.

Lesson 1 • Convex Sets and Functions

Defines convexity rigorously and identifies convex-preserving operations. Enables recognition of convex structure in engineering models.

Lesson 2 • Lagrangian Duality and Conjugate Functions

Develops Lagrangian relaxation and conjugate function theory. Provides tools for deriving dual problems and bounding primal objectives.

Lesson 3 • Subgradient and Proximal Methods

Extends gradient methods to non-smooth convex objectives. Handles L1 regularisation and other non-differentiable engineering objectives.

Lesson 4 • Conic Programming Formulations

Introduces second-order cone and semidefinite programs as convex generalisations of LP. Expands the class of tractable engineering problems.

Lesson 1 • Cutting Plane Methods

Strengthens LP relaxations with valid inequalities to accelerate MIP solving. Complements branch-and-bound in the branch-and-cut framework.

Lesson 2 • Heuristics and Metaheuristics

Covers greedy, local search, and population-based heuristics for large MIPs. Provides practical tools when exact methods are computationally infeasible.

Lesson 3 • Dynamic Programming for Discrete Problems

Applies Bellman's principle to sequential discrete optimisation. Solves shortest path, knapsack, and scheduling problems exactly.

Lesson 4 • Branch-and-Bound Algorithm

Develops the branch-and-bound tree search for exact integer solutions. Provides the core algorithmic engine behind modern MIP solvers.

Lesson 1 • Chance Constraints and CVaR

Encodes probabilistic feasibility requirements and tail-risk objectives. Handles safety and reliability constraints in engineering design.

Lesson 2 • Two-Stage Stochastic Programming

Formulates here-and-now vs. wait-and-see decisions across scenarios. Solves recourse problems arising in engineering planning under uncertainty.

Lesson 3 • Uncertainty Modelling in Engineering

Characterises parameter uncertainty via probability distributions and uncertainty sets. Motivates the need for stochastic and robust formulations.

Lesson 4 • Decomposition and Sampling Methods

Applies Benders decomposition and Monte Carlo sampling to large stochastic programs. Enables scalable solution of real engineering uncertainty problems.

Lesson 1 • Model Validation and Solution Reporting

Covers verification, benchmarking, and professional communication of optimisation results. Ensures engineering solutions meet quality and interpretability standards.

Lesson 2 • Supply Chain and Logistics Optimisation

Models facility location, routing, and inventory as MIP problems. Connects combinatorial optimisation theory to industrial engineering practice.

Lesson 3 • Structural and Mechanical Design Optimisation

Applies NLP and topology optimisation to structural weight and compliance problems. Demonstrates end-to-end optimisation workflow in mechanical engineering.

Lesson 4 • Energy Systems and Network Optimisation

Optimises power dispatch, flow networks, and energy storage scheduling. Applies LP, MIP, and stochastic methods to energy engineering problems.