  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.
