# The complexity of approximating a nonlinear program

Article

Received:

Revised:

## Abstract

We consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope. We show that there is no polynomial time approximation algorithm for this problem, even one with a very poor guarantee, unless P = NP. We show that even when the polynomial is quadratic (i.e. quadratic programming) there is no polynomial time approximation unless NP is contained in quasi-polynomial time.

Our results rely on recent advances in the theory of interactive proof systems. They exemplify an interesting interplay of discrete and continuous mathematics—using a combinatorial argument to get a hardness result for a continuous optimization problem.

## Keywords

Approximation Optimization Probabilistically checkable proofs Quadratic programming Nonlinear programming## Preview

Unable to display preview. Download preview PDF.

## References

- [1]S. Arora, C. Lund, R. Motwani, M. Sudan and M. Szegedy, “Proof verification and hardness of approximation problems,” in:
*Proceedings of the Thirty-Third Annual Symposium on the Foundations of Computer Science*, IEEE, 1992.Google Scholar - [2]S. Arora and S. Safra, “Probabilistic checking of proofs; A new characterization of NP,” in:
*Proceedings of the Thirty-Third Annual Symposium on the Foundations of Computer Science*, IEEE, 1992.Google Scholar - [3]G. Ausiello, A. D'Atri and M. Protasi, “Structure preserving reductions among convex optimization problems,”
*Journal of Computer and System Sciences*21 (1980) 136–153.Google Scholar - [4]L. Babai, L. Fortnow and C. Lund, “Non-deterministic exponential time has two-prover interactive protocols,”
*Computational Complexity*1 (1991) 3–40.Google Scholar - [5]L. Babai and S. Moran, “Arthur-Merlin games: A randomized proof system and a hierarchy of complexity classes,”
*Journal of Computer and System Sciences*36 (1988) 254–276.Google Scholar - [6]M. Bellare, “Interactive proofs and approximation: reductions from two provers in one round,” in:
*Proceedings of the Second Israel Symposium on Theory and Computing Systems*, 1993.Google Scholar - [7]M. Bellare, S. Goldwasser, C. Lund and A. Russell, “Efficient probabilistically checkable proofs and applications to approximation,” in:
*Proceedings of the Twenty-Fifth Annual Symposium on the Theory of Computing*, ACM, 1993.Google Scholar - [8]M. Bellare and P. Rogaway, “The complexity of approximating a nonlinear program,” in: P.M. Pardalos, ed.
*Complexity of Numerical Optimization*(World Scientific, Singapore, 1993).Google Scholar - [9]M. Bellare and M. Sudan, “Improved non-approximability results,” in:
*Proceedings of the Twenty-Sixth Annual Symposium on the Theory of Computing*, ACM, 1994.Google Scholar - [10]M. Ben-Or, S. Goldwasser, J. Kilian and A. Wigderson, “Multi-prover interactive proofs: how to remove intractability assumptions,” in:
*Proceedings of the Twentieth Annual Symposium on the Theory of Computing*, ACM, 1988.Google Scholar - [11]J. Canny, “Some algebraic and geometric computations in PSPACE,” in:
*Proceedings of the Twentieth Annual Symposium on the Theory of Computing*, ACM, 1988.Google Scholar - [12]A. Condon, “The complexity of the max word problem and the power of one-way interactive proof systems,” in:
*Proceedings of the Eighth Annual Symposium on Theoretical Aspects of Computer Science*, Lecture Notes in Computer Science, Vol. 480 (Springer, Berlin, 1991).Google Scholar - [13]C. Ebenegger, P. Hammer and D. de Werra, “Pseudo-boolean functions and stability of graphs,” in
*Algebraic and Combinatorial Methods in Operations Research*, Annals of Discrete Mathematics, Vol. 19 (North-Holland, Amsterdam, 1984) 83–97.Google Scholar - [14]U. Feige, “NEXPTIME has two-provers one-round proof systems with exponentially small error probability,” Manuscript, 1991.Google Scholar
- [15]U. Feige, “On the success probability of the two provers in one round proof systems,” in:
*Proceedings of the Sixth Annual Conference on Structure in Complexity Theory*, IEEE, 1991.Google Scholar - [16]U. Feige, S. Goldwasser, L. Lovász, S. Safra and M. Szegedy, “Approximating clique is almost NP-complete,” in:
*Proceedings of the Thirty-Second Annual Symposium on the Foundations of Computer Science*, IEEE, 1991.Google Scholar - [17]U. Feige and J. Kilian, “Two prover protocols—Low error at affordable rates,” in:
*Proceedings of the Twenty-Sixth Annual Symposium on the Theory of Computing*, ACM, 1994.Google Scholar - [18]U. Feige and L. Lovász, “Two-prover one round proof systems: their power and their problems,” in:
*Proceedings of the Twenty-Fourth Annual Symposium on the Theory of Computing*, ACM, 1992.Google Scholar - [19]L. Fortnow, J. Rompel and M. Sipser, “On the power of multiprover interactive protocols,” in:
*Proceedings of the Third Annual Conference on Structure in Complexity Theory*, IEEE, 1988.Google Scholar - [20]S. Goldwasser, S. Micali and C. Rackoff, “The knowledge complexity of interactive proofs,”
*SIAM Journal of Computing*18 (1) (1989) 186–208.Google Scholar - [21]M. Kozlov, S. Tarasov and L. Hačijan, “Polynomial solvability of convex quadratic programming,”
*Doklady Akademii Nauk SSSR*248 (1979) 1049–1051; Translation in:*Soviet Mathematics Doklady*20 (1979) 1108–1111.Google Scholar - [22]D. Lapidot and A. Shamir, “Fully parallelized multi-prover protocols for NEXP-time,” in:
*Proceedings of the Thirty-Second Annual Symposium on the Foundations of Computer Science*, IEEE, 1991.Google Scholar - [23]C. Lund and M. Yannakakis, “On the hardness of approximating minimization problems,” in:
*Proceedings of the Twenty-Fifth Annual Symposium on the Theory of Computing*, ACM, 1993.Google Scholar - [24]T. Motzkin and E. Straus, “Maxima for graphs and a new proof of a theorem by Tuán,”
*Notices of the American Mathematical Society*11 (1964) 533–540.Google Scholar - [25]A. Nemirovsky and D. Yudin,
*Slozhnost' Zadach i Effektivnost' Metodov Optimizatsii*(1979); Translated by E. Dawson as*Problem Complexity and Method Efficiency in Optimization*(Wiley, New York, 1983).Google Scholar - [26]S. Sahni, “Computationally related problems,”
*SIAM Journal of Computing*3 (1974) 262–279.Google Scholar - [27]G. Tardos, “Multi-prover encoding schemes and three prover proof systems,” in:
*Proceedings of the Ninth Annual Conference on Structure in Complexity Theory*, IEEE, 1994.Google Scholar - [28]S. Vavasis, “Quadratic programming is in NP,”
*Information Processing Letters*36 (1990) 73–77.Google Scholar - [29]S. Vavasis, “Approximation algorithms for indefinite quadratic programming,” TR 91-1228, Dept. of Computer Science, Cornell University, August 1991.Google Scholar
- [30]S. Vavasis, “On approximation algorithms for concave programming,” in: C.A. Floudas and P.M. Pardalos, eds.,
*Recent Advances in Global Optimization*(Princeton University Press, 1992) pp. 3–18.Google Scholar - [31]S. Vavasis, “Polynomial time weak approximation algorithms for quadratic programming,” in: P. Pardalos, ed.,
*Complexity in Numerical Optimization*(1992).Google Scholar

## Copyright information

© The Mathematical Programming Society, Inc. 1995