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.

中文翻译：

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多项式优化中基本问题的复杂性

在本论文中，我们解决了多项式优化中一些基本问题的计算复杂度。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) 决定是否达到最佳值。我们的结果表征了这三个问题对于先前文献留下的定义多项式的所有程度的复杂性。关于（i）和（ii），我们表明除非 P=NP，否则不可能有多项式时间算法在欧几里德距离 $c^n$（对于任何常数 $c$）的局部最小值内找到一个点$n$-variate 二次规划。相比之下，我们表明可以通过半定规划（SDP）有效地找到三次多项式的局部最小值。我们证明三次多项式的二阶点允许有效的半定表示，即使它们的临界点是 NP 难以找到的。我们还给出了三次多项式点的局部极小性的可有效检查的充分必要条件。关于（iii），我们证明测试具有有限最优值的二次约束二次规划是否具有最优解是NP-hard的。我们还表明，测试目标函数的矫顽力、可行集的紧凑性以及与可行集描述相关的阿基米德性质都是 NP 难的。我们还给出了适用于 SDP 层次结构的强制多项式的新特征。在我们的最后一章中，我们提出了在双矩阵博弈中寻找近似纳什均衡的 SDP 松弛。我们证明对于对称博弈，$1/3$-Nash 均衡可以从任何 rank-2 解中有效地恢复到这种松弛。我们还为与纳什均衡相关的 NP 难问题提出了 SDP 松弛，例如在任何纳什均衡下找到最高可实现的福利。