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Optimization problems with orthogonal matrix constraints

  • * Corresponding author: Manil T. Mohan

    * Corresponding author: Manil T. Mohan

M. T. Mohan's current address Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247 667, INDIA

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  • The optimization problems involving orthogonal matrices have been formulated in this work. A lower bound for the number of stationary points of such optimization problems is found and its connection to the number of possible partitions of natural numbers is also established. We obtained local and global optima of such problems for different orders and showed their connection with the Hadamard, conference and weighing matrices. The application of general theory to some concrete examples including maximization of Shannon, Rény, Tsallis and Sharma-Mittal entropies for orthogonal matrices, minimum distance orthostochastic matrices to uniform van der Waerden matrices, Cressie-Read and K-divergence functions for orthogonal matrices, etc are also discussed. Global optima for all orders has been found for the optimization problems involving unitary matrix constraints.

    Mathematics Subject Classification: Primary: 15B51; Secondary: 65K05, 15B10, 15B34.

    Citation:

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  • Figure 1.  n versus d graph

    Figure 2.  Rényi entropy of $2\times 2$ orthogonal matrices

    Figure 3.  Sharma-Mittal entropy of $2\times 2$ orthogonal matrices

    Figure 4.  Tsallis entropy of $2\times 2$ orthogonal matrices

    Table 1.  1 $\leq n\leq $ 20.

    $n$ Orthogonal matrix $d({\rm J}_n, {\rm B}_n)$ $n$ Orthogonal matrix $d({\rm J}_n, {\rm B}_n)$
    $1$ ${\rm H}_1$ $0$ $11$ $\frac{1}{3}{\rm C}_{10}\oplus{\rm H}_1^*$ $1.054$
    $2$ $\frac{1}{\sqrt{2}}{\rm H}_2$ $0$ $12$ $\frac{1}{2\sqrt{3}}{\rm H}_{12}$ $0$
    $3$ ${\rm K}_3$ $0.471$ $13$ $\frac{1}{3}{\rm W}_{13, 9}$ $0.667$
    $4$ $\frac{1}{2}{\rm H}_4$ $0$ $14$ $\frac{1}{\sqrt{13}}{\rm C}_{14}$ $0.277$
    $5$ ${\rm K}_5$ $0.400$ $15$ ${\rm K}_3\otimes{\rm K}_5^{\dagger}$ $0.646$
    $6$ $\frac{1}{\sqrt{5}}{\rm C}_6$ $0.447$ $16$ $\frac{1}{4}{\rm H}_{16}$ $0$
    $7$ $\frac{1}{2}{\rm W}_{7, 4}$ $0.866$ $17$ $\frac{1}{3}{\rm W}_{17, 9}$ $0.943$
    $8$ $\frac{1}{2\sqrt{2}}{\rm H}(8)$ $0$ $18$ $\frac{1}{\sqrt{17}}{\rm C}_{18}$ $0.243$
    $9$ ${\rm K}_3\otimes{\rm K}_3$ $0.703$ $19$ $\frac{1}{\sqrt{17}}{\rm C}_{18}\oplus{\rm H}_1^*$ $1.029$
    $10$ $\frac{1}{3}{\rm C}_{10}$ $0.333$ $20$ $\frac{1}{2\sqrt{5}}{\rm H}_{20}$ $0$
    $^*$By Proposition 4.2, these orthogonal matrices are saddle points and not local minima. The weighing matrices ${\rm W}_{11, 4}$ and ${\rm W}_{19, 9}$ exists for orders $11$ and $19$, but they are also saddle points. The orthostochastic matrices corresponding to the orthogonal matrices $\frac{1}{2}{\rm W}_{11, 4}$ and $\frac{1}{3}{\rm W}_{19, 9}$ are at distances $1.323>1.054$ and $1.054>1.029$, respectively from the the uniform van der Waerden matrices ${\rm J}_{11}$ and ${\rm J}_{19}$. $^{\dagger}$For order $15$, weighing matrix ${\rm W}_{15, 9}$ exist, which are also local minima, by Proposition 4.11. But the orthostochastic matrix corresponding to the orthogonal matrix $\frac{1}{3}{\rm W}_{15, 9}$ is at a distance $0.817>0.646$, from the uniform van der Waerden matrix ${\rm J}_{15}$.
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