Additional exercises for Part II

Abstract

This chapter consists of a collection of problems bearing upon optimization and nonsmooth analysis, the first of which is the following: Let M be an n×n symmetric matrix. Consider the optimization problem

$$ \text{Minimize }\: \langle\, x , M x \,\rangle\:\:\: \text{subject to}\:\: \: h(x)\,:=\, 1-|\,x\,|^{\,2}\: =\:0 ,\:\;x\in\, {\mathbb{R}}^{ n}. $$

• (a) Observe that a solution x _∗ exists, and write the conclusions of the multiplier rule for this problem. Show that they cannot hold abnormally. It follows that the multiplier in this case is of the form (1,λ).

• (b) Deduce that λ is an eigenvalue of M, and that λ =〈 x _∗,Mx _∗ 〉.

• (c) Prove that λ coincides with the first, or least eigenvalue λ ₁ of M (this statement makes sense because the eigenvalues are real). Deduce the Rayleigh formula, which asserts that λ ₁ is given by min {〈 x,Mx 〉:| x |=1 }.