On the complexity of finding a local minimizer of a quadratic function over a polytope

• URL: http://arxiv.org/abs/2008.05558v5
• Date: Thu, 14 Sep 2023 01:22:30 GMT
• Title: On the complexity of finding a local minimizer of a quadratic function over a polytope
• Authors: Amir Ali Ahmadi, Jeffrey Zhang
• Abstract summary: We show that unless P=NP, there cannot be a Euclidean-time algorithm that finds a point within $cn$ of a local minimizer of an $n$ quadratic function over a polytope. Our proof technique also implies that the problem of deciding whether a quadratic function has a local minimizer over anunbounded polyhedron, and that of deciding if a quartic has a local minimizer are NP-hard.
• Score: 1.0048921884287543
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We show that unless P=NP, there cannot be a polynomial-time algorithm that finds a point within Euclidean distance $c^n$ (for any constant $c \ge 0$) of a local minimizer of an $n$-variate quadratic function over a polytope. This result (even with $c=0$) answers a question of Pardalos and Vavasis that appeared in 1992 on a list of seven open problems in complexity theory for numerical optimization. Our proof technique also implies that the problem of deciding whether a quadratic function has a local minimizer over an (unbounded) polyhedron, and that of deciding if a quartic polynomial has a local minimizer are NP-hard.

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