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.

翻译：我们证明，除非P=NP，否则不存在多项式时间算法能够在多面体上找到n元二次函数局部极小点的欧几里得距离c^n（对于任意常数c≥0）内的点。这一结果（即使c=0）回答了Pardalos与Vavasis于1992年在数值优化复杂性理论七大开放问题列表中提出的问题。我们的证明技术还表明：判定二次函数在（无界）多面体上是否存在局部极小点的问题，以及判定四次多项式是否存在局部极小点的问题均为NP难问题。