Optimisation problems arise in many areas of engineering, science, economic modelling, resource modelling and operations research. These problems can arise via the need to find a best approximation or 'fit' to a set of data or to use finite resources equitably and efficiently. Indeed, many physical laws are governed by the principle of least action. The solution of such problems often requires the use of modern digital computers and algorithms to approximate a solution. This course gives an introduction to the main ideas behind these algorithms and the mathematical theory underpinning their use. We consider what leads to convergence and efficiency for various algorithms. MATLAB, a modern computer programming language will be used to implement some algorithms to illustrate these principles. Skills in formulation and the reformulation of optimization problems will be introduced both in problem classes and using specific examples. Certain problem classes have associated with them an efficient solver and we will practise calling standard solvers for these problems and interpreting the output. Examples of problems will be sort from statistics, machine learning and mathematical modelling.