American Institute of Mathematical Sciences

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September  2020, 10(3): 249-255. doi: 10.3934/naco.2019051

Complex and quaternionic optimization

Yuly Shipilevsky

507-111 Ridelle Avenue, Toronto, Ontario, M6B 1J7, Canada

Received  February 2019 Revised  October 2019 Published  February 2020

We introduce and suggest to research a special class of optimization problems, wherein an objective function is a real-valued complex variables function under constraints, comprising complex-valued complex variables functions: "Complex Optimization". We demonstrate multiple examples to show a rich variety of problems, describing Complex Optimization as an optimization subclass as well as a Mixed Integer-Real-Complex Optimization.

Next, we introduce more general concept: "Quaternionic Optimization" for optimization over quaternion subsets.

Keywords: Complex variables, complexity, constraints, optimization, quaternion, target function.
Mathematics Subject Classification: Primary: 90C48; Secondary: 30D30, 30A99.
Citation: Yuly Shipilevsky. Complex and quaternionic optimization. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 249-255. doi: 10.3934/naco.2019051
References:
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[2] T. Cormen, C. Leiserson, R. Rivest and C. Stein, Introduction To Algorithms, The MIT Press, Cambridge, 2009.   Google Scholar
[3]

C. A. Floudas and P. M. Pardalos, Encyclopedia of Optimization, Springer, New York, 2009. doi: 10.1016/j.tcs.2009.07.038.  Google Scholar

[4]

I. Frenkel and M. Libine, Quaternionic analysis, representation theory and physics, Advances in Mathematics, 218 (2008), 1806-1877.  doi: 10.1016/j.aim.2008.03.021.  Google Scholar

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R. Hemmecke, M. Köppe, J. Lee and R. Weismantel, Nonlinear integer programming, in 50 Years of Integer Programming 1958–2008: The Early Years and State-of-the-Art Surveys (eds. M. Junger, T. Liebling, D. Naddef, W. Pulleyblank, W. Reinelt, G. Rinaldi, and L. Wolsey), Springer-Verlag, Berlin, (2010), 561–618. Google Scholar

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G. James, Modern Engineering Mathematics, Trans-Atlantic Pubns Inc., 2015. Google Scholar

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I. Kleiner, From numbers to rings: The early history of ring theory, Elem. Math., Birkhäuser, Basel, 53 (1998), 18–35. doi: 10.1007/s000170050029.  Google Scholar

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E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons Inc., 2011.  Google Scholar

[9]

J. Qian, C. Yang, A. Schirotzek, F. S. Maia and S. Marchesini, Efficient algorithms for ptychographic phase retrieval. Inverse problems and applications, Contemporary Mathematics, 615 (2014), 261-280.  doi: 10.1090/conm/615.  Google Scholar

[10]

V. Scheidemann, Introduction to Complex Analysis in Several Variables, Birkhäuser, 2005.  Google Scholar

[11]

W. T. Shaw, Complex Analysis with Mathematica, Cambridge, 2006. doi: 10.1017/CBO9781316036549.  Google Scholar

[12]

L. Sorber and M. Van Barel, Structured data fusion, IEEE Journal of Selected Topics in Signal Processing, 9 (2015), 586-600.   Google Scholar

[13]

L. Sorber, M. Van Barel and L. De. Lathauwer, Optimization-based algorithms for tensor decompositions: canonical polyadic decomposition, decomposition in rank-(l${}_{r}$, l${}_{r}$, 1) terms, and a new generalization, SIAM Journal on Optimization, 23 (2013), 695-720.  doi: 10.1137/120868323.  Google Scholar

[14]

L. Sorber, M. Van Barel and L. De. Lathauwer, Unconstrained optimization of real functions in complex variables, SIAM Journal on Optimization, 22 (2012), 879-898.  doi: 10.1137/110832124.  Google Scholar

[15]

Y. S. Xu, Q. Ye and G. X. Meng, Hybrid phase retrieval algorithm based on modified very fast simulated annealing, International Journal of Microwave and Wireless Technologies, 10 (2018), 1072-1080.   Google Scholar

show all references

References:
[1] L. M. B. C. Campos, Complex Analysis with Applications to Flows and Fields, CRC Press, 2011.   Google Scholar
[2] T. Cormen, C. Leiserson, R. Rivest and C. Stein, Introduction To Algorithms, The MIT Press, Cambridge, 2009.   Google Scholar
[3]

C. A. Floudas and P. M. Pardalos, Encyclopedia of Optimization, Springer, New York, 2009. doi: 10.1016/j.tcs.2009.07.038.  Google Scholar

[4]

I. Frenkel and M. Libine, Quaternionic analysis, representation theory and physics, Advances in Mathematics, 218 (2008), 1806-1877.  doi: 10.1016/j.aim.2008.03.021.  Google Scholar

[5]

R. Hemmecke, M. Köppe, J. Lee and R. Weismantel, Nonlinear integer programming, in 50 Years of Integer Programming 1958–2008: The Early Years and State-of-the-Art Surveys (eds. M. Junger, T. Liebling, D. Naddef, W. Pulleyblank, W. Reinelt, G. Rinaldi, and L. Wolsey), Springer-Verlag, Berlin, (2010), 561–618. Google Scholar

[6]

G. James, Modern Engineering Mathematics, Trans-Atlantic Pubns Inc., 2015. Google Scholar

[7]

I. Kleiner, From numbers to rings: The early history of ring theory, Elem. Math., Birkhäuser, Basel, 53 (1998), 18–35. doi: 10.1007/s000170050029.  Google Scholar

[8]

E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons Inc., 2011.  Google Scholar

[9]

J. Qian, C. Yang, A. Schirotzek, F. S. Maia and S. Marchesini, Efficient algorithms for ptychographic phase retrieval. Inverse problems and applications, Contemporary Mathematics, 615 (2014), 261-280.  doi: 10.1090/conm/615.  Google Scholar

[10]

V. Scheidemann, Introduction to Complex Analysis in Several Variables, Birkhäuser, 2005.  Google Scholar

[11]

W. T. Shaw, Complex Analysis with Mathematica, Cambridge, 2006. doi: 10.1017/CBO9781316036549.  Google Scholar

[12]

L. Sorber and M. Van Barel, Structured data fusion, IEEE Journal of Selected Topics in Signal Processing, 9 (2015), 586-600.   Google Scholar

[13]

L. Sorber, M. Van Barel and L. De. Lathauwer, Optimization-based algorithms for tensor decompositions: canonical polyadic decomposition, decomposition in rank-(l${}_{r}$, l${}_{r}$, 1) terms, and a new generalization, SIAM Journal on Optimization, 23 (2013), 695-720.  doi: 10.1137/120868323.  Google Scholar

[14]

L. Sorber, M. Van Barel and L. De. Lathauwer, Unconstrained optimization of real functions in complex variables, SIAM Journal on Optimization, 22 (2012), 879-898.  doi: 10.1137/110832124.  Google Scholar

[15]

Y. S. Xu, Q. Ye and G. X. Meng, Hybrid phase retrieval algorithm based on modified very fast simulated annealing, International Journal of Microwave and Wireless Technologies, 10 (2018), 1072-1080.   Google Scholar

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