# American Institute of Mathematical Sciences

December  2018, 8(4): 413-440. doi: 10.3934/naco.2018026

## Optimization problems with orthogonal matrix constraints

 1 Department of Mathematics and Statistics, Wright State University, 3640 Colonel Glenn Highway Dayton, OH 45435, U.S.A 2 Department of Mathematics and Statistics, Air Force Institute of Technology, 2950 Hobson Way, Wright Patterson Air Force Base, OH 45433, USA

* 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

Received  May 2017 Revised  August 2018 Published  September 2018

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.

Citation: K. T. Arasu, Manil T. Mohan. Optimization problems with orthogonal matrix constraints. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 413-440. doi: 10.3934/naco.2018026
##### References:

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##### References:
n versus d graph
Rényi entropy of $2\times 2$ orthogonal matrices
Sharma-Mittal entropy of $2\times 2$ orthogonal matrices
Tsallis entropy of $2\times 2$ orthogonal matrices
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 ${\rm J}_{15}$.
 $n$ 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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