多项式Optimization基本问题的复杂性
Complexity Aspects of Fundamental Questions in Polynomial Optimization
摘要：
在这篇论文中，我们解决了多项式优化中一些基本问题的计算复杂性。这些问题包括（i）找到局部最小值，（ii）测试点的局部最小值，以及（iii）确定是否达到最优值。我们的研究结果的复杂性，这三个问题的所有程度的定义多项式由以前的文献。关于（i）和（ii），我们表明，除非P=NP，不可能有一个多项式时间算法，找到一个点内的欧氏距离$c^n$（任何常数$c$）的局部极小值的$n$ -变量二次规划。通过对比，我们证明了三次多项式的局部极小值可以有效地由半定规划（SDP）找到。我们证明了三次多项式的二阶点允许一个有效的半定表示，即使它们的临界点是NP-难找到的。我们还给出了三次多项式点的局部极小的一个可有效检验的充要条件。关于（iii），我们证明了检验具有有限最优值的二次约束二次规划是否有最优解是NP-困难的。我们还表明，测试的目标函数，可行集的紧凑性，和阿基米德属性与描述的可行集都是NP-困难的。我们还给出了一个新的强制多项式的特点，本身的层次结构的SDP。在我们的最后一章中，我们提出了一个SDP松弛寻找近似纳什均衡的双矩阵游戏。我们表明，对于一个对称的游戏，一个$1/3$ -纳什均衡可以有效地恢复从任何秩2解决方案，这个放松。我们还提出了SDP松弛的NP-难问题的纳什均衡，如找到最高的可实现的福利下的任何纳什均衡。
英文摘要：
In this thesis, we settle the computational complexity of some fundamental questions in polynomial optimization. These include the questions of (i) finding a local minimum, (ii) testing local minimality of a point, and (iii) deciding attainment of the optimal value. Our results characterize the complexity of these three questions for all degrees of the defining polynomials left open by prior literature. Regarding (i) and (ii), 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 $c$) of a local minimum of an $n$-variate quadratic program. By contrast, we show that a local minimum of a cubic polynomial can be found efficiently by semidefinite programming (SDP). We prove that second-order points of cubic polynomials admit an efficient semidefinite representation, even though their critical points are NP-hard to find. We also give an efficiently-checkable necessary and sufficient condition for local minimality of a point for a cubic polynomial. Regarding (iii), we prove that testing whether a quadratically constrained quadratic program with a finite optimal value has an optimal solution is NP-hard. We also show that testing coercivity of the objective function, compactness of the feasible set, and the Archimedean property associated with the description of the feasible set are all NP-hard. We also give a new characterization of coercive polynomials that lends itself to a hierarchy of SDPs. In our final chapter, we present an SDP relaxation for finding approximate Nash equilibria in bimatrix games. We show that for a symmetric game, a $1/3$-Nash equilibrium can be efficiently recovered from any rank-2 solution to this relaxation. We also propose SDP relaxations for NP-hard problems related to Nash equilibria, such as that of finding the highest achievable welfare under any Nash equilibrium.
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