# The complexity of approximating a nonlinear program

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## 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

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## Copyright information

© The Mathematical Programming Society, Inc. 1995