The complexity of approximating a nonlinear program

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.

[1]  T. Motzkin,et al.  Maxima for Graphs and a New Proof of a Theorem of Turán , 1965, Canadian Journal of Mathematics.

[2]  Sartaj Sahni,et al.  Computationally Related Problems , 1974, SIAM J. Comput..

[3]  L. Khachiyan,et al.  The polynomial solvability of convex quadratic programming , 1980 .

[4]  Giorgio Ausiello,et al.  Structure Preserving Reductions among Convex Optimization Problems , 1980, J. Comput. Syst. Sci..

[5]  John Darzentas,et al.  Problem Complexity and Method Efficiency in Optimization , 1983 .

[6]  P. Hammer,et al.  Pseudo-Boolean functions and stability of graphs , 1984 .

[7]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.

[8]  Avi Wigderson,et al.  Multi-prover interactive proofs: how to remove intractability assumptions , 2019, STOC '88.

[9]  László Babai,et al.  Arthur-Merlin Games: A Randomized Proof System, and a Hierarchy of Complexity Classes , 1988, J. Comput. Syst. Sci..

[10]  John F. Canny,et al.  Some algebraic and geometric computations in PSPACE , 1988, STOC '88.

[11]  L. Fortnow,et al.  On the power of multi-power interactive protocols , 1988, [1988] Proceedings. Structure in Complexity Theory Third Annual Conference.

[12]  Silvio Micali,et al.  The Knowledge Complexity of Interactive Proof Systems , 1989, SIAM J. Comput..

[13]  Stephen A. Vavasis,et al.  Quadratic Programming is in NP , 1990, Inf. Process. Lett..

[14]  László Lovász,et al.  Approximating clique is almost NP-complete , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[15]  Uriel Feige.  On the success probability of the two provers in one-round proof systems , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[16]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[17]  S. Vavasis.  On approximation algorithms for concave quadratic programming , 1992 .

[18]  László Lovász,et al.  Two-prover one-round proof systems: their power and their problems (extended abstract) , 1992, STOC '92.

[19]  Sanjeev Arora,et al.  Probabilistic checking of proofs; a new characterization of NP , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[20]  Stephen A. Vavasis,et al.  Approximation algorithms for indefinite quadratic programming , 1992, Math. Program..

[21]  S. Vavasis.  Polynomial Time Weak Approximation Algorithms for Quadratic Programming , 1993 .

[22]  P. Pardalos.  Complexity in numerical optimization , 1993 .

[23]  Carsten Lund,et al.  Efficient probabilistically checkable proofs and applications to approximations , 1993, STOC.

[24]  Mihir Bellare,et al.  Interactive proofs and approximation: reductions from two provers in one round , 1993, [1993] The 2nd Israel Symposium on Theory and Computing Systems.

[25]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1993, STOC.

[26]  Gábor Tardos.  Multi-prover encoding schemes and three-prover proof systems , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[27]  Mihir Bellare,et al.  Improved non-approximability results , 1994, STOC '94.

[28]  Carsten Lund,et al.  Efficient probabilistic checkable proofs and applications to approximation , 1994, STOC '94.

[29]  Uriel Feige,et al.  Two prover protocols: low error at affordable rates , 1994, STOC '94.

[30]  Gábor Tardos.  Multi-prover Encoding Schemes and Three-prover Proof Systems , 1996, J. Comput. Syst. Sci..

[31]  Adi Shamir,et al.  Fully Parallelized Multi-Prover Protocols for NEXP-Time , 1997, J. Comput. Syst. Sci..