The purpose of this thesis is to review work carried out by Professor W. V. Candler of the Department of Agricultural Economics and Farm Management at Massey University, leading to the development of a Generalised Convex Quadratic Programming Algorithm. However the responsibility for the following manner of presenting the material and forming the arguments rests with the candidate. The first chapter gives a brief summary of the algebra of quadratic functions which will form a background for future developments. At the end of Chapter 1 is a bibliography for further and more detailed reading. Chapter 2 classifies the problem in the title within the frame-work of the more general mathematical programming problem. Chapter 3 describes and developes the mathematical conditions which any successful algorithm must satisfy, and Chapter 4 developes the algorithm, in the form of three separate algorithms, as a form of presentation. The last chapter provides a brief discussion.