On the hardness and smoothed complexity of quasi-concave minimization

Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

Volumn , Issue , 2007, Pages 472-482

  Kelner, Jonathan A ^a   Nikolova, Evdokia ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

(0 , 1) VECTORS; (2+1)-DIMENSIONAL; (ALGORITHMIC) COMPLEXITY; (I ,J) CONDITIONS; APPROXIMATION SCHEMES; CONCAVE FUNCTIONS; CONCAVE MINIMIZATION; EXTREME POINTS; GENERAL (CO); HARDNESS OF APPROXIMATION; NON CONVEXITY; PERTURBATION SIZE; POLYTOPE; POLYTOPES; SMOOTHED ANALYSIS; TWO-DIMENSIONAL SUBSPACES; UPPER AND LOWER BOUNDS; UPPER BOUNDS;

COMPUTER SCIENCE; CONTROL THEORY; FIBER OPTIC CHEMICAL SENSORS; FOUNDATIONS; HARDNESS; MILITARY DATA PROCESSING; POLYNOMIALS; TOPOLOGY; TWO DIMENSIONAL;

POLYNOMIAL APPROXIMATION;

EID: 46749146536     PISSN: 02725428     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/FOCS.2007.4389517     Document Type: Conference Paper

References (21)

1. N. Amenta and G. Ziegler. Deformed Products and Maximal Shadows of Polytopes, volume 223 of Contemporary Mathematics. pages 57-90. American Mathematics Society, Providence, 1999.
2. D. Bertsekas, A. Nedić. and A. Ozdaglar. Convex Analysis and Optimization. Athena Scientific, Belmont, MA, 2003.
3. T. Bonnesen and W. Fenchel. Theory of Convex Bodies. B C S Associates, Moscow, Idaho, 1988.
4. J. Conway. A Course in Functional Analysis. Springer-Verlag, 1990.
5. I. Dumitriu. Eigenvalue Statistics for Beta-Ensembles. PhD thesis. MIT, 2003.
6. U. Feige. Personal communication. 2006.
7. A. D. Flaxman, A. Frieze, and M. Krivelevich. On the random 2-stage minimum spanning tree. Random Structures & Algorithms, 28(1):24-36, 2006.
8. D. Goldfarb. Worst case complexity of the shadow vertex simplex algorithm. Technical report, Columbia University, 1983.
9. G. Golub and C. V. Loan. Matrix Computations. Johns Hopkins University Press, 1996.
10. R. Horst and P. M. Pardalos. Handbook of Global Optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995.
11. R. Horst, P. M. Pardalos, and N. V. Thoai. Introduction to Global Optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
12. J. A. Kelner and D. A. Spielman. A randomized polynomial-time simplex algorithm for linear programming. In Proceedings of the Symposium on the Theory of Computing. 2006.
13. H. Konno, C. Gao, and I. Saitoh. Cutting plane/tabu search algorithms for low rank concave quadratic programming problems. Journal of Global Optimization, 13:225-240, 1998.
14. H. Konno, T. Thach, and H. Tuy. Optimization on Low Rank Nonconvex Structures. Kluwer Academic Publishers, Dordrecht. The Netherlands, 1996.
15. L. Lovász. Submodular functions and convexity. In Mathematical Programming - The State of the Art, A. Bachem, M, Grotschel and B. Korte, eds., pages 235-257. Springer-Verlag, 1983.
16. M. Mehta. Matrix Theory. 1990.
17. E. Nikolova, J. A. Kelner, M. Brand, and M. Mitzenmacher. Stochastic shortest paths via quasi-convex maximization. In Lecture Notes in Computer Science 4168 (ESA 2006). pages 552-563, Springer-Verlag, 2006.
18. P. M. Pardalos and G. Schnitger. Checking local optimality in constrained quadratic programming is np-hard. Operations Research Letters, 7:33-35, 1988.
19. M. Porembski. Cutting planes for low-rank-like concave minimization problems. Operations Research, 52(6):942-953, 2004.
20. D. A. Spielman and S.-H. Teng. Smoothed analysis of termination of linear programming algorithms. Math. Prog.: B, 97(1/2):375, 2003.
21. D. A. Spielman and S.-H. Teng. Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. Journal of the ACM, 51:385-463, 2004.