Quadratic programming with one negative eigenvalue is NP-hard Panos M. Pardalos Stephen A. Vavasis Article

Received: 06 February 1991 DOI :
10.1007/BF00120662

Cite this article as: Pardalos, P.M. & Vavasis, S.A. J Glob Optim (1991) 1: 15. doi:10.1007/BF00120662 Abstract We show that the problem of minimizing a concave quadratic function with one concave direction is NP-hard. This result can be interpreted as an attempt to understand exactly what makes nonconvex quadratic programming problems hard. Sahni in 1974 [8] showed that quadratic programming with a negative definite quadratic term (n negative eigenvalues) is NP-hard, whereas Kozlov, Tarasov and Hačijan [2] showed in 1979 that the ellipsoid algorithm solves the convex quadratic problem (no negative eigenvalues) in polynomial time. This report shows that even one negative eigenvalue makes the problem NP-hard.

Key words Global optimization quadratic programming NP-hard This author's work supported by the Applied Mathematical Sciences Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG02-86ER25013. A000 and in part by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS 8920550.

References 1.

Garey, M. R. and Johnson, D. S. (1979), Computers and Intractability, A Guide to the Theory of NP-Completeness , W. H. Freeman and Company, San Francisco.

2.

Kozlov, M. K., Tarasov, S. P., and Hačijan, L. G. (1979), Polynomial Solvability of Convex Quadratic Programming, Soviet Math. Doklady
20 , 1108–111.

3.

Murty, K. G. and Kabadi, S. N. (1987), Some NP-Complete Problems in Quadratic and Non-linear Programming, Mathematical Programming
39 , 117–129.

4.

Pardalos, P. M. (1990), Polynomial Time Algorithms for Some Classes of Nonconvex Quadratic Problems, To appear in Optimization .

5.

Pardalos, P. M. and Rosen, J. B. (1986), Global Concave Minimization: A Bibliographic Survey, SIAM Review
28 (3), 367–379.

6.

Pardalos, P. M. and Rosen, J. B. (1987), Constrained Global Optimization: Algorithms and Applications , Lecture Notes in Computer Science 268, Springer-Verlag, Berlin.

7.

Pardalos, P. M. and Schnitger, G. (1988), Checking Local Optimality in Constrained Quadratic Programming is NP -hard, Operations Research Letters
7 (1), 33–35.

8.

Sahni, S. (1974), Computationally Related Prolems, SIAM J. Comput.
3 , 262–279.

9.

Vavasis, S. A. (1990), Quadratic Programming Is in NP, Inf. Proc. Lett.
36 , 73–77.

© Kluwer Academic Publishers 1991

Authors and Affiliations Panos M. Pardalos Stephen A. Vavasis 1. Department of Computer Science Pennsylvania State University University Park U.S.A. 2. Department of Computer Science Cornell University Ithaca U.S.A.