This thesis explores algorithmic applications and limitations of convex relaxation hierarchies for approximating some discrete and continuous optimization problems. - We show a dichotomy of approximability of constraint satisfaction problems (CSPs) by linear programming (LP) relaxations: for every CSP, the approximation obtained by a basic LP relaxation, is no weaker than the approximation obtained using relaxations given by super-constant levels of the Sherali-Adams hierarchy on instances of size $n$. - For the problem of approximating the absolute maximum of an n-variate degree-d homogeneous polynomial f with real coefficients over the unit sphere, we analyze the optimum value of the level-t sum-of-squares (SoS) SDP relaxation of the problem. Our results offer a trade-off between the approximation ratio and running time, which can take advantage of additional structure in the polynomial, such as non-negativity or sparsity of the coefficients. - We study the problem of approximating the $p \to q$-norm of a matrix $A$, and prove the first NP-hardness result for approximating norms in the hypercontractive case $1< p < q < \infty$. We also prove almost tight algorithmic results for the case when $p \geq q$ (with $2 \in [q,p]$) where constant factor approximations for the matrix norms are possible. A common theme for these results is their connection to geometry. For the discrete optimization problem of CSP, geometry appears as a crucial tool for our lower bound proof. For the problem of polynomial optimization, we show that SDPs capture and extend earlier algorithms based on diameter estimation for convex bodies. For the matrix (operator) norm problem, the definition itself is geometric in nature and embedding theorems play a crucial role in our proofs.

翻译：本论文探讨了凸松弛层次结构在近似求解离散与连续优化问题中的算法应用与局限性。- 我们证明了约束满足问题（CSP）通过线性规划（LP）松弛可近似性的二分性：对于任意CSP，基础LP松弛获得的近似解不弱于在规模为$n$的实例上使用Sherali-Adams层次结构超常数层级松弛所获得的近似解。- 针对在单位球面上逼近n元d次实系数齐次多项式f的绝对最大值问题，我们分析了该问题第t层级平方和（SoS）半定规划（SDP）松弛的最优值。我们的结果提供了近似比与运行时间之间的权衡关系，能够利用多项式的附加结构（如系数的非负性或稀疏性）。- 我们研究了矩阵$A$的$p \to q$范数逼近问题，首次证明了在超压缩情形$1< p < q < \infty$下范数逼近的NP困难性结果。同时针对$p \geq q$（且$2 \in [q,p]$）情形证明了几乎紧界的算法结果，其中矩阵范数可获得常数因子近似解。这些结果的共同主题是其与几何的关联性。对于CSP这类离散优化问题，几何方法成为我们下界证明的关键工具；在多项式优化问题中，我们证明SDP能够涵盖并扩展基于凸体直径估计的早期算法；对于矩阵（算子）范数问题，其定义本身具有几何本质，而嵌入定理在我们的证明中起着至关重要的作用。