Portfolio Optimization

Chapter 1

Optimization

1.1 Quadratic Minimization

Throughout this chapter, we will deal with the minimization of a

quadratic function subject to linear equality constraints. In Chapters

2−4, this same type of optimization problem plays a central role. It is

therefore essential to have available necessary and sufficient conditions

for a point to minimize a quadratic function subject to linear equality

constraints. It is the purpose of this section to develop such conditions.

Throughout this text, we will rely heavily on vector and matrix

notation. Prime (

′

) will be used to denote matrix tra nsposition. All

unprimed vectors will be column vectors. The i−th compo nent of

the vector x

0

will be denoted by (x

0

)

i

. In a numerical example, we

sometimes use the sym ...