Semidefinite programming: Difference between revisions

Revision as of 22:03, 24 May 2023

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=== Example 1 ===

Consider three random variables <math>A</math>, <math>B</math>, and <math>C</math>. By definition, theira given set of [[Correlation|correlation coefficients]] <math>\rho_{AB}, \ \rho_{AC}, \rho_{BC} </math> are validpossible if and only if

:<math>\begin{pmatrix}

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\rho_{AB} & 1 & \rho_{BC} \\

\rho_{AC} & \rho_{BC} & 1

\end{pmatrix} \succeq 0,.</math>

in which case thisThis matrix is called the [[correlation matrix]]. Suppose that we know from some prior knowledge (empirical results of an experiment, for example) that <math>-0.2 \leq \rho_{AB} \leq -0.1</math> and <math>0.4 \leq \rho_{BC} \leq 0.5</math>. The problem of determining the smallest and largest values that <math>\rho_{AC} \ </math> can take is given by:

:<math>\begin{array}{rl}