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Erschienen in:
2013 | OriginalPaper | Buchkapitel
13. Additional exercises for Part II
verfasst von : Francis Clarke
Erschienen in: Functional Analysis, Calculus of Variations and Optimal Control
Verlag: Springer London
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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 }.