The Complexity of Approximating a Nonlinear Program

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.

Copositivity and constrained fractional quadratic problems

“…The difficulty arises as a rounding problem: the transition from X * to Z * in Theorem 1 requires to find an explicit factorization of a given completely positive matrix X * , which is not an easy task. This is in perfect concordance with the inapproximability result in [9]. For a general discussion of complexity over similar feasible sets we refer to [21].…”

The complexity of optimizing over a simplex, hypercube or sphere: a short survey

“…The next definition is based on this idea, and has been used by several authors, including Ausiello et al (1980), Bellare and Rogaway (1995), Bomze andDe Klerk (2002), de Klerk et al (2007b), Nesterov et al (2000), and Vavasis (1992).…”

“…A related negative result is due to Bellare and Rogaway (1995), who proved that if P = NP and ∈ (0, 1/3), there is no polynomial time (1− )-approximation algorithm in the weak sense for the problem of minimizing a polynomial of total degree d ≥ 2 over all sets of the form K = {x ∈ [0, 1] n | Ax ≤ b}.…”

Copositive optimization – Recent developments and applications

“…For the sake of conciseness, we concentrate here on the StQP case which forms an instance of NP-hard problems which admit a polynomial-time approximation scheme (PTAS). This is in sharp contrast to the case of box-constrained QPs [6] and may be intuitively explained by the fact that, unlike the hypercube, the volume of the standard simplex decreases exponentially fast in n as dimension n increases. Also, this poses no contradiction to the well-known inapproximability results [72] for the clique number ω(G), since the StQP formulation (30) in Section 6.3 below provides the inverse value 1 ω(G) .…”