In this thesis modelling and solution methods for portfolio optimisation are presented. The investigations reported in this thesis extend the Markowitz mean-variance model to the domain of quadratic mixed integer programming (QMIP) models which are 'NP-hard' discrete optimisation problems. In addition to the modelling extensions a number of challenging aspects of solution algorithms are considered. The relative performances of sparse simplex (SSX) as well as the interior point method (IPM) are studied in detail. In particular, the roles of 'warmstart' and dual simplex are highlighted as applied to the construction of the efficient frontier which requires processing a family of problems; that is, the portfolio planning model stated in a parametric form. The method of solving QMIP models using the branch and bound algorithm is first developed; this is followed up by heuristics which improve the performance of the (discrete) solution algorithm. Some properties of the efficient frontier with discrete constraints are considered and a method of computing the discrete efficient frontier (DEF) efficiently is proposed. The computational investigation considers the efficiency and effectiveness in respect of the scale up properties of the proposed algorithm. The extensions of the real world models and the proposed solution algorithms make contribution as new knowledge.