In this paper, we introduce a primal-dual algorithmic framework for solving Symmetric Cone Programs (SCPs), a versatile optimization model that unifies and extends Linear, Second-Order Cone (SOCP), and Semidefinite Programming (SDP). Our work generalizes the primal-dual framework for SDPs introduced by Arora and Kale, leveraging a recent extension of the Multiplicative Weights Update method (MWU) to symmetric cones. Going beyond existing works, our framework can handle SOCPs and mixed SCPs, exhibits nearly linear time complexity, and can be effectively parallelized. To illustrate the efficacy of our framework, we employ it to develop approximation algorithms for two geometric optimization problems: the Smallest Enclosing Sphere problem and the Support Vector Machine problem. Our theoretical analyses demonstrate that the two algorithms compute approximate solutions in nearly linear running time and with parallel depth scaling polylogarithmically with the input size. We compare our algorithms against CGAL as well as interior point solvers applied to these problems. Experiments show that our algorithms are highly efficient when implemented on a CPU and achieve substantial speedups when parallelized on a GPU, allowing us to solve large-scale instances of these problems.

翻译：本文提出一种求解对称锥规划（SCPs）的对偶-原始算法框架，对称锥规划是一种统一的优化模型，涵盖并推广了线性规划、二阶锥规划（SOCP）和半定规划（SDP）。我们的工作推广了Arora与Kale提出的半定规划对偶-原始框架，利用了乘法权重更新方法（MWU）近期在对称锥上的拓展。超越现有成果，本框架能处理SOCP与混合SCP问题，具有近线性时间复杂度，并支持高效并行化。为验证框架的有效性，我们将其应用于两个几何优化问题的近似算法设计：最小包围球问题与支持向量机问题。理论分析表明，这两个算法能在近线性运行时间内计算近似解，且并行深度随输入规模呈多对数增长。我们将算法与CGAL及求解此类问题的内点法求解器进行对比。实验证明，所提算法在CPU上实现时具有高效性，在GPU上并行化时获得显著加速，从而能够求解大规模实例问题。