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<p>As a core class of continuous optimization models that encompasses numerous concrete problems from applications, even dating back centuries, second-order cone programming (SOCP) has been extensively studied over the past few decades. While it can be regarded as an adjacent generalization of conventional linear and nonlinear programming, it involves second-order cones in the constraints, and both the theoretical analysis and the algorithmic design pertain to these non-polyhedral convex cones, which distinguish the study of SOCP as a prominent research area in mathematical optimization. There are several landmark reviews and summaries on SOCP. Nevertheless, contributions have still been made in progress, involving a few important and interesting results during the past years that merit a new chapter of documentation. In this paper, we systematically review recent advances in linear and nonlinear SOCP, with an emphasis on novel theoretical findings, new algorithms, and promising applications. This review serves as a comprehensive pool of results and ideas that can facilitate research in both optimization and related fields. In addition, a collection of open problems in this area is summarized for future research.</p>

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A Survey on Some Recent Advances in Linear and Nonlinear Second-Order Cone Programming

  • 期刊论文

  • 发表日期：

    2025年11月6日

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  • Liang Chen,
  • Le-Xuan Yang,
  • Jun-Yuan Zhu

摘要

As a core class of continuous optimization models that encompasses numerous concrete problems from applications, even dating back centuries, second-order cone programming (SOCP) has been extensively studied over the past few decades. While it can be regarded as an adjacent generalization of conventional linear and nonlinear programming, it involves second-order cones in the constraints, and both the theoretical analysis and the algorithmic design pertain to these non-polyhedral convex cones, which distinguish the study of SOCP as a prominent research area in mathematical optimization. There are several landmark reviews and summaries on SOCP. Nevertheless, contributions have still been made in progress, involving a few important and interesting results during the past years that merit a new chapter of documentation. In this paper, we systematically review recent advances in linear and nonlinear SOCP, with an emphasis on novel theoretical findings, new algorithms, and promising applications. This review serves as a comprehensive pool of results and ideas that can facilitate research in both optimization and related fields. In addition, a collection of open problems in this area is summarized for future research.