Linear matrix inequality representation of sets. (English) Zbl 1116.15016
One of the best techniques that are used successfully in many optimization problems is to convert those problems to linear matrix inequalities (LMIs). Unfortunately there is no systematic (or general) way to produce LMIs. Every problem has to be treated by some particular trick. Therefore, the following important question is discussed by the authors: Which types of constraint sets can be converted to LMIs and which do not. They obtain a necessary condition which must hold for a set \(C\subseteq \mathbb R^n\) in order for \(C\) to have an LMI representation. It is proved that the obtained condition is also sufficient for \(n=2\). Several examples are considered to illustrate the obtained results.
Reviewer: Fozi Dannan (Damascus)
MSC:
| 15A45 | Miscellaneous inequalities involving matrices |
References:
| [1] | ; Transversal mappings and flows. W. A. Benjamin, New York-Amsterdam, 1967. · Zbl 0171.44404 |
| [2] | Ball, Acta Appl Math 45 pp 239– (1996) |
| [3] | Ball, Amer J Math 121 pp 841– (1999) |
| [4] | ; ; Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 36. Springer, Berlin, 1998. · doi:10.1007/978-3-662-03718-8 |
| [5] | Cook, Quart J Math Oxford Ser (2) 30 pp 423– (1979) |
| [6] | Dixon, Proc Cambridge Phil Soc 2 pp 350– (1900) |
| [7] | Matrix finite-gap operators. Current Problems in Mathematics, vol. 23, 33–78. Itogi Nauki i Tekhniki. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983. |
| [8] | ; , eds. Advances in linear matrix inequality methods in control. Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2000. · Zbl 0932.00034 · doi:10.1137/1.9780898719833 |
| [9] | Manipulating matrix inequalities automatically. Mathematical systems theory in biology, communications, computation, and finance (Notre Dame, IN, 2002), 237–256. The IMA Volumes in Mathematics and Its Applications, 134, Springer, New York, 2003. · doi:10.1007/978-0-387-21696-6_8 |
| [10] | Lewis, Proc Amer Math Soc 133 pp 2495– (2005) |
| [11] | ; Interior-point polynomial algorithms in convex programming. SIAM Studies in Applied Mathematics, 13. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1994. · Zbl 0824.90112 · doi:10.1137/1.9781611970791 |
| [12] | Nuij, Math Scand 23 pp 69– (1968) |
| [13] | Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Doctoral dissertation, California Institute of Technology, May 2000. Available at: http://www.mit.edu/parrilo/pubs/index.html. |
| [14] | ; Minimizing polynomial functions. Algorithmic and quantitative real algebraic geometry (Piscataway, NJ, 2001). 83–99. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 60. American Mathematical Society, Providence, R.I., 2003. |
| [15] | ; ; A unified algebraic approach to linear control design. The Taylor & Francis Systems and Control Book Series. Taylor & Francis, London, 1998. |
| [16] | Vinnikov, Math Ann 296 pp 453– (1993) |
| [17] | Viro, Russian Math Surveys 41 pp 55– (1986) |
| [18] | Wall, Philos Trans Roy Soc London Ser A 289 pp 229– (1978) |
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