Semidefinite programming: Difference between revisions

Revision as of 07:31, 1 December 2021

Revision as of 01:59, 20 January 2022

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in which case this 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:

:

:minimize/maximize <math>x_{13}</math>

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

:subject to

{\displaystyle\min/\max} & x_{13} \\

:<math>-0.2 \leq x_{12} \leq -0.1</math>

:<math>\text{subject to} & -0.42 \leq x_{2312} \leq -0.5</math>1\\

:<math>-& 0.24 \leq x_{1223} \leq -0.1</math>5\\

:<math>\begin{pmatrix}

& \begin{pmatrix}

1 & x_{12} & x_{13} \\

x_{12} & 1 & x_{23} \\

x_{13} & x_{23} & 1

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

\end{array}</math>

We set <math>\rho_{AB} = x_{12}, \ \rho_{AC} = x_{13}, \ \rho_{BC} = x_{23} </math> to obtain the answer. This can be formulated by an SDP. We handle the inequality constraints by augmenting the variable matrix and introducing [[slack variable]]s, for example