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Complex and quaternionic optimization
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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.
References:
| [1] |
L. M. B. C. Campos, Complex Analysis with Applications to Flows and Fields, CRC Press, 2011.
|
| [2] |
T. Cormen, C. Leiserson, R. Rivest and C. Stein, Introduction To Algorithms, The MIT Press, Cambridge, 2009.
|
| [3] |
C. A. Floudas and P. M. Pardalos, Encyclopedia of Optimization, Springer, New York, 2009.
doi: 10.1016/j.tcs.2009.07.038. |
| [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. |
| [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. |
| [6] |
G. James, Modern Engineering Mathematics, Trans-Atlantic Pubns Inc., 2015. |
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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. |
| [8] |
E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons Inc., 2011. |
| [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. |
| [10] |
V. Scheidemann, Introduction to Complex Analysis in Several Variables, Birkhäuser, 2005. |
| [11] |
W. T. Shaw, Complex Analysis with Mathematica, Cambridge, 2006.
doi: 10.1017/CBO9781316036549. |
| [12] |
L. Sorber and M. Van Barel,
Structured data fusion, IEEE Journal of Selected Topics in Signal Processing, 9 (2015), 586-600.
|
| [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. |
| [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. |
| [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.
|
show all references
References:
| [1] |
L. M. B. C. Campos, Complex Analysis with Applications to Flows and Fields, CRC Press, 2011.
|
| [2] |
T. Cormen, C. Leiserson, R. Rivest and C. Stein, Introduction To Algorithms, The MIT Press, Cambridge, 2009.
|
| [3] |
C. A. Floudas and P. M. Pardalos, Encyclopedia of Optimization, Springer, New York, 2009.
doi: 10.1016/j.tcs.2009.07.038. |
| [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. |
| [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. |
| [6] |
G. James, Modern Engineering Mathematics, Trans-Atlantic Pubns Inc., 2015. |
| [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. |
| [8] |
E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons Inc., 2011. |
| [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. |
| [10] |
V. Scheidemann, Introduction to Complex Analysis in Several Variables, Birkhäuser, 2005. |
| [11] |
W. T. Shaw, Complex Analysis with Mathematica, Cambridge, 2006.
doi: 10.1017/CBO9781316036549. |
| [12] |
L. Sorber and M. Van Barel,
Structured data fusion, IEEE Journal of Selected Topics in Signal Processing, 9 (2015), 586-600.
|
| [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. |
| [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. |
| [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.
|
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