Abstract
We present a potential reduction algorithm to approximate a Karush—Kuhn—Tucker (KKT) point of general quadratic programming (QP). We show that the algorithm is a fully polynomial-time approximation scheme, and its running-time dependency on accuracy ε ∈ (0, 1) is O((l/ε) log(l/ε) log(log(l/ε))), compared to the previously best-known result O((l/ε)2). Furthermore, the limit of the KKT point satisfies the second-order necessary optimality condition of being a local minimizer. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Bellare, P. Rogaway, The complexity of approximating a nonlinear program, Math. Programming 69 (1995) 429–442.
R.W. Cottle, G.B. Dantzig, Complementary pivot theory in mathematical programming, Linear Algebra Appl. 1 (1968) 103–125.
J.E. Dennis, Jr., R.E. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, NJ, 1983.
M.R. Garey, D.S. Johnson, Computers and Intractability, A Guide to the Theory of NP-completeness, Freeman, New York, 1968.
D.M. Gay, Computing optimal locally constrained steps, SIAM J. Sci. Statis. Comput. 2 (1981) 186–197.
P.E. Gill, W.M. Murray, M.H. Wright, Practical Optimization, Academic Press, London, 1981.
C.-G. Han, P.M. Pardalos, Y. Ye, On the solution of indefinite quadratic problems using an interiorpoint algorithm, Informatica 3(4) (1992) 474–496.
R. Horst, P.M. Pardalos, N.V. Thoai, Introduction to Global Optimization, Kluwer Academic Publishers, Boston, 1995.
D.S. Johnson, C. Papadimitriou, M. Yannakakis, How easy is local search, J. Comp. Syst. Sci. 37 (1988) 79–100.
A.P. Kamath, N.K. Karmarkar, K.G. Ramakrishnan, M.G.C. Resende, A continuous approach to inductive inference, Math. Programming 57 (1992) 215–238.
N.K. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984) 373–395.
K. Levenberg, A method for the solution of certain non-linear problems in least squares, Quart. Appl. Math. 2 (1963) 164–168.
D.W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. SIAM 11 (1963) 431–441.
J.J. Moré, The Levenberg—Marquardt algorithm: implementation and theory, in: G.A. Watson (Ed.), Numerical Analysis, Springer, New York, 1977.
K.G. Murty, Linear Complementarity, Linear and Nonlinear Programming, Heldermann, Berlin, 1988.
K.G. Murty, S.N. Kabadi, Some NP-complete problems in quadratic and nonlinear programming, Math. Programming 39 (1987) 117–129.
Y.E. Nesterov, A.S. Nemirovskii, Interior Point Polynomial Methods in Convex Programming: Theory and Algorithms, SIAM Publications, SIAM, Philadelphia, 1993.
P.M. Pardalos, J.B. Rosen, Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Sciences, vol. 268, Springer, Berlin, 1987.
P.M. Pardalos, S. Jha, Complexity of uniqueness and local search in quadratic 0–1 programming, Oper. Res. Lett. 11 (1992) 119–123.
P.M. Pardalos, G. Schnitger, Checking local optimality in constrained quadratic programming is NP-hard, Oper. Res. Lett. 7 (1988) 33–35.
F. Rendl, H. Wolkowicz, A semidefinite framework for trust region subproblems with applications to large scale minimization, CORR Report 94-32, Department of Combinatorics and Optimization, University of Waterloo, Ontario, Canada, 1994.
S. Sahni, Computationally related problems, SIAM J. Comput. 3 (1974) 262–279.
D.C. Sorenson, Newton's method with a model trust region modification, SIAM J. Numer. Anal. 19 (1982) 409–426.
S.A. Vavasis, Nonlinear Optimization: Complexity Issues, Oxford Science, New York, 1991.
S.A. Vavasis, Polynomial time weak approximation algorithms for quadratic programming, in: C.A. Floudas, P.M. Pardalos (Eds.), Complexity in Numerical Optimization, World Scientific, New Jersey, 1993.
S.A. Vavasis, R. Zippel, Proving polynomial time for sphere-constrained quadratic programming, Technical Report 90-1182, Department of Computer Science, Cornell University, Ithaca, NY, 1990.
Y. Ye, A fully polynomial-time approximation algorithm for computing a stationary point of the general LCP, Math. of Oper. Res. 18 (1993) 334–345.
Y. Ye, Combining binary search and Newton's method to compute real roots for a class of real functions, J. Complexity 10 (1994) 271–280.
Y. Ye, On affine scaling algorithms for nonconvex quadratic programming, Math. Programming 56 (1992) 285–300.
Author information
Authors and Affiliations
Additional information
Research support in part by NSF grants DDM-9207347 and DMI-9522507, and the Iowa Business School Summer Grant.
Rights and permissions
About this article
Cite this article
Ye, Y. On the complexity of approximating a KKT point of quadratic programming. Mathematical Programming 80, 195–211 (1998). https://doi.org/10.1007/BF01581726
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01581726