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A Primal-Dual Framework for Symmetric Cone Programming

Jiaqi Zheng, Antonios Varvitsiotis, Tiow-Seng Tan, Wayne Lin

TL;DR

This work develops a primal–dual framework for symmetric cone programs by transplanting the multiplicative weights update method to symmetric cones (SCMWU) and integrating it into a meta-algorithm that solves α-feasibility tests to achieve near-optimal SCP solutions. It unifies LPs, SOCPs, and SDPs, enabling mixed-cone constraints and offering nearly linear-time, parallelizable algorithms. The authors instantiate the framework for two geometric problems—Smallest Enclosing Sphere and Support Vector Machine—producing provably efficient algorithms with parallel depth polylogarithmic in input size and near-linear time in the data. Experiments demonstrate strong GPU speedups and competitive performance against commercial solvers, underscoring the practical impact for large-scale SCPs and related geometric optimization tasks.

Abstract

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.

A Primal-Dual Framework for Symmetric Cone Programming

TL;DR

This work develops a primal–dual framework for symmetric cone programs by transplanting the multiplicative weights update method to symmetric cones (SCMWU) and integrating it into a meta-algorithm that solves α-feasibility tests to achieve near-optimal SCP solutions. It unifies LPs, SOCPs, and SDPs, enabling mixed-cone constraints and offering nearly linear-time, parallelizable algorithms. The authors instantiate the framework for two geometric problems—Smallest Enclosing Sphere and Support Vector Machine—producing provably efficient algorithms with parallel depth polylogarithmic in input size and near-linear time in the data. Experiments demonstrate strong GPU speedups and competitive performance against commercial solvers, underscoring the practical impact for large-scale SCPs and related geometric optimization tasks.

Abstract

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.
Paper Structure (12 sections, 8 theorems, 80 equations, 3 figures, 2 tables)

This paper contains 12 sections, 8 theorems, 80 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric cones and Euclidean Jordan algebras
  4. Symmetric cone programming
  5. Symmetric cone multiplicative weights update (SCMWU)
  6. Parallel computation
  7. Primal-dual meta algorithm for SCPs
  8. Applications
  9. Smallest enclosing sphere
  10. Support vector machine
  11. Experimental results
  12. Concluding remark

Key Result

Theorem 1

Let $(\mathcal{J}, \circ)$ be an EJA of rank $r$, $\mathcal{K}$ be its cone of squares, and ${\boldsymbol{x}}\bullet{\boldsymbol{y}} = \mathbf{Tr}({\boldsymbol{x}} \circ {\boldsymbol{y}})$ be the EJA inner product. For any $\eta \in (0, 1]$ and any sequence of loss vectors ${{\boldsymbol{m}}}^{(t)}$ where $\lambda_{\min}({\cdot})$ is the minimum eigenvalue of an EJA vector.

Figures (3)

  • Figure 1: Illustration of the process of the meta-algorithm, where $\alpha_1 > {\sf OPT} > \alpha_2$.
  • Figure 2: (Left) Running time of different solvers with inputs of different sizes, and (Right) the ratio of the running time of the best competitive solvers to that of PDSCP for the SES problem.
  • Figure 4: (Left) Running time of the solvers under different dimensionalities, and (Right) the ratio of the running time of the best competitive solvers to that of PDSCP for the SES problem.

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Lemma 6
  • ...and 6 more