NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems

关于判定Lasserre层级有限收敛性的困难性

On the hardness of deciding the finite convergence of Lasserre hierarchies

本文证明除非P=NP，否则不存在多项式时间算法能判定Lasserre层级是否有限收敛，并基于Motzkin-Straus公式对图稳定数的刻画完成证明。

On the Difficulty of Deciding Asymptotic Stability of Cubic Homogeneous Vector Fields

It is well-known that asymptotic stability (AS) of homogeneous polynomial vector fields of degree one (i.e., linear systems) can be decided in polynomial time e.g. by searching for a quadratic Lyapunov function. Since homogeneous vector fields of even degree can never be AS, the next interesting degree to consider is equal to three. In this paper, we prove that deciding AS of homogeneous cubic vector fields is strongly NP-hard and pose the question of determining whether it is even decidable. As a byproduct of the reduction that establishes our NP-hardness result, we obtain a Lyapunov-inspired…

Deciding positivity of multisymmetric polynomials

The question how to certify non-negativity of a polynomial function lies at the heart of Real Algebra and also has important applications to Optimization. In this article we investigate the question of non-negativity in the context of multisymmetric polynomials. In this setting we generalize the characterization of non-negative symmetric polynomials by adapting the method of proof developed by the second author. One particular case where our results can be applied is the question of certifying that a (multi-)symmetric polynomial defines a convex function. As a direct corollary of our main resu…

四元四次多项式非负性证书的高效求和平方方法

An efficient sum of squares nonnegativity certificate for quaternary quartic

研究四元四次非负多项式通过二次形式分解为平方和，利用Hilbert定理并探讨单个二次形式是否足够分解的问题，发现非负多项式分解可能需更优方法。

关于在多面体上寻找二次函数局部极小点复杂度的研究

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

本文证明了除非P=NP，否则不存在多项式时间算法能找到多面体上二次函数局部极小点的近似点，回答了1992年提出的开放问题，并证明了相关判定问题为NP-hard。

在并行机器上最小化随机作业完成时间是困难的

Minimizing Completion Times of Stochastic Jobs on Parallel Machines is Hard

本文证明了在并行机器上最小化随机作业完成时间的难题，展示了该问题的内在不可处理性。