Geometry of homogeneous convex cones, duality mapping, and optimal self-concordant barriers

Mathematical Programming

Volumn 100, Issue 2, 2004, Pages 295-316

  Truong, Van Anh ^a   Tunçel, Levent ^b  

^a Department of Molecular Biology and Genetics   (United States)

^b UNIVERSITY OF WATERLOO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; INTEGRATION; LINEAR PROGRAMMING; MATHEMATICAL MODELS; OPTIMIZATION; PROBLEM SOLVING;

CONVEX CONES; DUALITY MAPPING; INTERIOR-POINT METHODS; SELF-CONCORDANT BARRIERS;

FUNCTIONS;

EID: 21244456109     PISSN: 00255610     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10107-003-0470-y     Document Type: Article

References (26)

1. Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. Analysis, Algorithms, and Engineering Applications, MPS/SIAM Series on Optimization, Philadelphia, PA, USA, 2001
2. Chua, C.B.: Relating homogeneous cones and positive definite cones via T-algebras, manuscript. School of OR&IE, Cornell University, Ithaca, NY, USA, May 2002
3. Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Oxford University Press, NY, USA, 1994
4. Gindikin, S.G.: Tube Domains and the Cauchy Problem. Translation of Mathematical Monographs, vol. 111, AMS Providence, RI, USA, 1992
5. Güler, O.: Barrier functions in interior point methods. Math. Oper. Res. 21, 860-885 (1996)
6. Güler, O. : Hyperbolic polynomials and interior point methods for convex programming. Math. Oper. Res. 22, 350-377 (1997)
7. Güler, O., Tunçel, L.: Characterization of the barrier parameter of homogeneous convex cones. Math. Program. A 81, 55-76 (1998)
8. Hauser, R.A. : On Search Directions for Self-Scaled Conic Programming. Ph.D. Thesis, School of OR&IE, Cornell University, Ithaca, NY, USA, 2000
9. Hauser, R.A., Güler, O.: Self-scaled barrier functions on symmetric cones and their classification. Found. Comput. Math. 2, 121-143 (2002)
10. Hauser, R.A., Lim, Y.: Self-scaled barriers for irreducible symmetric cones. SIAM J. Optim. 12, 715-723 (2002)
11. n. Am. J. Math. 79, 575-596 (1957)
12. Nesterov, Yu.E., Nemirovskii, A.S.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia, PA, USA, 1994
13. Nesterov, Yu.E., Todd, M.J.: Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res. 22, 1-46 (1997)
14. Nesterov, Yu.E., Todd, M.J.: Primal-dual interior-point methods for self-scaled cones. SIAM J. Optim. 8, 324-364 (1998)
15. Ramana, M.V.: An exact duality theory for semidefinite programming and its complexity implications. Math. Program. B 77, 129-162 (1997)
16. Ramana, M.V., Tunçel, L., Wolkowicz, H.: Strong duality for semidefinite programming. SIAM J. Optim. 7, 641-662 (1997)
17. Renegar, J.: A Mathematical View of Interior-Point Methods in Convex Optimization. MPS/SIAM Series on Optimization, Philadelphia, PA, USA, 2001
18. Rothaus, O.S.: Domains of positivity. Abh. Math. Sem. Univ. Hamburg 24, 189-235 (1960)
19. Rothaus, O.S.: The construction of homogeneous convex cones. Ann. Math. 83, 358-376 (1966)
20. Schmieta, S.H.: Complete classification of self-scaled barrier functions. Tech. Report, Dept. of IEOR, Columbia Univ., NY, USA, 2000
21. Tam, B.-S.: On the duality operator of a convex cone. Linear Algebra and Its Applications 63, 33-56 (1985)
22. Tunçel, L.: Primal-dual symmetry and scale invariance of interior-point algorithms for convex optimization. Math. Oper. Res. 23, 708-718 (1998)
23. Tunçel, L.: Generalization of primal-dual interior-point methods to convex optimization problems in conic form. Found. Comput. Math. 1, 229-254 (2001)
24. Tunçel, L., Xu, S.: On homogeneous convex cones, the Carathéodory number, and the duality mapping. Math. Oper. Res. 26, 234-247 (2001)
25. Vinberg, È.B.: The theory of homogeneous cones. Trans. Moscow Math. Soc. 12, 340-403 (1965)
26. Vinberg, È.B.: The structure of the group of automorphisms of a homogeneous convex cone. Trans. Moscow Math. Soc. 13, 63-93 (1965)