Optimisation & Operations Research

Making the best decisions. Mathematical optimisation for design and operations: unconstrained methods (gradient descent, Newton), constrained optimisation with Lagrange multipliers and KKT, linear programming with duality, and operations research — transportation, queueing, inventory and scheduling.

• Unconstrained Optimisation — Find the best design without constraints. Learn objective functions and decision variables, local vs global optima, gradient and Hessian conditions, convexity, gradient descent and learning rates, Newton's method for optimisation, and line searches — with a Python gradient-descent example.
• Constrained Optimisation: Lagrange & KKT — Real designs have limits: stresses, budgets, capacities. Learn equality constraints and Lagrange multipliers, the geometric meaning (gradients parallel), inequality constraints and KKT conditions, active constraints, shadow prices (sensitivity), and penalty and SQP methods.
• Linear Programming — Optimise a linear objective subject to linear constraints. Learn LP formulation, the feasible region and graphical method, corner-point theorem, the simplex method (slack variables, pivoting), duality and shadow prices, sensitivity analysis, and integer programming.
• Operations Research — Quantitative decision-making for operations. Learn transportation and assignment problems, network optimisation (critical path), scheduling, queueing models (M/M/1: utilisation, queue length, waiting time), inventory models (EOQ), and decision theory (expected value, decision trees).