# 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:
 [1] P.-A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, NJ, 2008. doi: 10.1515/9781400830244. [2] K. T. Arasu, M. T. Mohan, A. Pathak and R. J. Ramya, Entropy optimal orthogonal matrices, Submitted for Journal Publication, 2018. [3] K. T. Arasu and M. T. Mohan, Entropy of orthogonal matrices and minimum distance orthostochastic matrices from the uniform van der Waerden matrix, Submitted for Journal Publication, 2018. [4] R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics, Springer, New York, 1997 doi: 10.1007/978-1-4612-0653-8. [5] O. Chterental and D. Ž. Ɖoković, On orthostochastic, unistochastic and qustochastic matrices, Linear Algebra and its Applications, 428 (2008), 1178-1201. doi: 10.1016/j.laa.2007.09.022. [6] N. Cressie and T. R. C. Read, Multinomial goodness of fit, Journal of the Royal Statistical Society, B, 46 (1984), 440-464. [7] P. Delsarte, J. M. Goethals and J. J. Seidel, Orthogonal matrices with zero diagonal Ⅱ, Canadian Journal of Mathematics, ⅩⅩⅩⅢ (1971), 816-832. doi: 10.4153/CJM-1971-091-x. [8] A. Edelman, T. A. Arias and S. T. Smith, The geometry of algorithms with orthogonality constraints, SIAM Journal of Matrix Analysis and Applications, 20 (1999), 303-353. doi: 10.1137/S0895479895290954. [9] H. G. Gadiyar, K. M. Sangeeta Maini, R. Padma and H. S. Sharatchandra, Entropy and Hadamard matrices, Journal of Physics A: Mathematical and General, 36 (2003), 109-112. doi: 10.1088/0305-4470/36/7/103. [10] J. Gallier, Basics of Classical Lie Groups: The Exponential Map, Lie Groups, and Lie Algebras, Chapter 14, in "Geometric Methods and Applications", the series Texts in Applied Mathematicsolume, 38 (2001), 367–414. doi: 10.1007/978-1-4613-0137-0_14. [11] A. V. Geramita, J. M. Geramita and J. Seberry, Orthogonal designs, Linear and Multilinear Algebra, 3 (1976), 381-206. doi: 10.1080/03081087608817121. [12] A. V. Geramita and J. Seberry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices, Marcel Dekker, New York-Basel, 1979. [13] B. K. P. Horn, Closed form solution of absolute orientation using unit quaternions, Journal of the Optical Society A, 4 (1987), 629-642. doi: 10.1364/JOSAA.4.000629. [14] Z. Q. Ma, Group Theory for Physicists, World Scientific, Singapore 2007. doi: 10.1142/6596. [15] A. W. Marshall, I. Olkin and B. C. Arnold, Inequalities: Theory of Majorization and Its Applications, Springer Series in Statistics, New York, 2011. doi: 10.1007/978-0-387-68276-1. [16] M. T. Mohan, On some p−almost Hadamard matrices, Accepted in Operators and Matrices, 2018. [17] H. Nakzato, Set of 3 × 3 orthostochastic matrices, Nihonkai Mathematics Journal, 7 (1996), 83-100. [18] A. Nemirovski, Sums of random symmetric matrices and quadratic optimization under orthogonality constraints, Mathematical Programming, 109 (2007), 283-317. doi: 10.1007/s10107-006-0033-0. [19] J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering, 2nd ed. Springer, New York, 2006 [20] R. E. A. C. Paley, On orthogonal matrices, Journal of Mathematics and Physics, 12 (1933), 311-320. doi: 10.1002/sapm1933121311. [21] D. R. Stinson Combinatorial Designs: Constructions and Analysis, Springer-Verlag New York, Inc., 2004. [22] B. D. Sharma and D. P. Mittal, New non-additive measures of entropy for discrete probability distributions, Journal of Mathematical Sciences, 10 (1975), 28-40. [23] F. Szöllősi, Construction, classification and parametrization of complex Hadamard matrices, PhD Thesis, https://arXiv.org/abs/1110.5590, 2011. [24] V. Weber, J. Vande Vondele, J. Hütter and A. M. Niklasson, Direct energy functional minimization under orthogonality constraints, The Journal of Chemical Physics, 128 (2008), 84-113. doi: 10.1063/1.2841077. [25] Z. Wen and W. Yin, A feasible method for optimization with orthogonality constraints, Math. Program., Ser. A, 142 (2013), 397-434. doi: 10.1007/s10107-012-0584-1. [26] K. Życzkowski, M. Kuś, W. Słomczyński and H.-J. Sommers, Random unistochastic matrices, Journal of Physics A: Mathematical and General, 36 (2003), 3425-3450. doi: 10.1088/0305-4470/36/12/333.

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##### References:
 [1] P.-A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, NJ, 2008. doi: 10.1515/9781400830244. [2] K. T. Arasu, M. T. Mohan, A. Pathak and R. J. Ramya, Entropy optimal orthogonal matrices, Submitted for Journal Publication, 2018. [3] K. T. Arasu and M. T. Mohan, Entropy of orthogonal matrices and minimum distance orthostochastic matrices from the uniform van der Waerden matrix, Submitted for Journal Publication, 2018. [4] R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics, Springer, New York, 1997 doi: 10.1007/978-1-4612-0653-8. [5] O. Chterental and D. Ž. Ɖoković, On orthostochastic, unistochastic and qustochastic matrices, Linear Algebra and its Applications, 428 (2008), 1178-1201. doi: 10.1016/j.laa.2007.09.022. [6] N. Cressie and T. R. C. Read, Multinomial goodness of fit, Journal of the Royal Statistical Society, B, 46 (1984), 440-464. [7] P. Delsarte, J. M. Goethals and J. J. Seidel, Orthogonal matrices with zero diagonal Ⅱ, Canadian Journal of Mathematics, ⅩⅩⅩⅢ (1971), 816-832. doi: 10.4153/CJM-1971-091-x. [8] A. Edelman, T. A. Arias and S. T. Smith, The geometry of algorithms with orthogonality constraints, SIAM Journal of Matrix Analysis and Applications, 20 (1999), 303-353. doi: 10.1137/S0895479895290954. [9] H. G. Gadiyar, K. M. Sangeeta Maini, R. Padma and H. S. Sharatchandra, Entropy and Hadamard matrices, Journal of Physics A: Mathematical and General, 36 (2003), 109-112. doi: 10.1088/0305-4470/36/7/103. [10] J. Gallier, Basics of Classical Lie Groups: The Exponential Map, Lie Groups, and Lie Algebras, Chapter 14, in "Geometric Methods and Applications", the series Texts in Applied Mathematicsolume, 38 (2001), 367–414. doi: 10.1007/978-1-4613-0137-0_14. [11] A. V. Geramita, J. M. Geramita and J. Seberry, Orthogonal designs, Linear and Multilinear Algebra, 3 (1976), 381-206. doi: 10.1080/03081087608817121. [12] A. V. Geramita and J. Seberry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices, Marcel Dekker, New York-Basel, 1979. [13] B. K. P. Horn, Closed form solution of absolute orientation using unit quaternions, Journal of the Optical Society A, 4 (1987), 629-642. doi: 10.1364/JOSAA.4.000629. [14] Z. Q. Ma, Group Theory for Physicists, World Scientific, Singapore 2007. doi: 10.1142/6596. [15] A. W. Marshall, I. Olkin and B. C. Arnold, Inequalities: Theory of Majorization and Its Applications, Springer Series in Statistics, New York, 2011. doi: 10.1007/978-0-387-68276-1. [16] M. T. Mohan, On some p−almost Hadamard matrices, Accepted in Operators and Matrices, 2018. [17] H. Nakzato, Set of 3 × 3 orthostochastic matrices, Nihonkai Mathematics Journal, 7 (1996), 83-100. [18] A. Nemirovski, Sums of random symmetric matrices and quadratic optimization under orthogonality constraints, Mathematical Programming, 109 (2007), 283-317. doi: 10.1007/s10107-006-0033-0. [19] J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering, 2nd ed. Springer, New York, 2006 [20] R. E. A. C. Paley, On orthogonal matrices, Journal of Mathematics and Physics, 12 (1933), 311-320. doi: 10.1002/sapm1933121311. [21] D. R. Stinson Combinatorial Designs: Constructions and Analysis, Springer-Verlag New York, Inc., 2004. [22] B. D. Sharma and D. P. Mittal, New non-additive measures of entropy for discrete probability distributions, Journal of Mathematical Sciences, 10 (1975), 28-40. [23] F. Szöllősi, Construction, classification and parametrization of complex Hadamard matrices, PhD Thesis, https://arXiv.org/abs/1110.5590, 2011. [24] V. Weber, J. Vande Vondele, J. Hütter and A. M. Niklasson, Direct energy functional minimization under orthogonality constraints, The Journal of Chemical Physics, 128 (2008), 84-113. doi: 10.1063/1.2841077. [25] Z. Wen and W. Yin, A feasible method for optimization with orthogonality constraints, Math. Program., Ser. A, 142 (2013), 397-434. doi: 10.1007/s10107-012-0584-1. [26] K. Życzkowski, M. Kuś, W. Słomczyński and H.-J. Sommers, Random unistochastic matrices, Journal of Physics A: Mathematical and General, 36 (2003), 3425-3450. doi: 10.1088/0305-4470/36/12/333.
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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