Conic Optimization Overview

Principles

Definition

Conic optimization is a subfield of convex optimization that studies the theory and algorithms for solving optimization problems defined by minimizing a convex function over the intersection of an affine subspace and a convex cone. This framework provides a general structure that unifies various important classes of convex problems, including linear, second-order cone, and semidefinite programming, serving as a powerful methodology for modeling and solving problems across diverse application domains.

Ontological type

Solution Algorithms

Core Cones

Applications

Key developments

Key Figures

Barrier Primal-Dual Era era

J.F. Sturm [1] is associated with the University of Groningen [3] and Erasmus University Rotterdam [4] during the Barrier Primal-Dual Era. His key contributions include the SeDuMi 1.02 toolbox for optimization over symmetric cones [7] and the demonstration of superlinear convergence of a symmetric primal–dual path-following algorithm for semidefinite programming [8], which advanced practical barrier-based optimization and SDP computation in this era. Dimitri P. Bertsekas [2] held positions at the Massachusetts Institute of Technology [5] and the University of California, Santa Barbara [6] during this period. His sole paper in the era, On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators [9], contributed foundational insights into operator splitting and proximal methods, underpinning modern primal–dual and splitting techniques central to barrier-based optimization.

Unified Scalable Algorithms era

Stephen Boyd [1] is a leading figure at Harvard University [3] and Stanford University [4] during the Unified Scalable Algorithms era. His key contributions in this era include ECOS: An SOCP solver for embedded systems [7], which demonstrated scalable, robust second-order cone programming for embedded and real-time applications. Masakazu Kojima [2] is affiliated with Tokyo Institute of Technology [5] and Chuo University [6] during this period. His work on Sums of Squares and Semidefinite Program Relaxations for Polynomial Optimization Problems with Structured Sparsity [8] and An Extension of Sums of Squares Relaxations to Polynomial Optimization Problems Over Symmetric Cones [9] extended SOS relaxations to structured sparsity patterns and symmetric cone settings, enabling more scalable conic optimization.

Self-Dual Operator-Splitting era

Stephen Boyd [1], affiliated with Stanford University [3] and the Technical University of Denmark [4], is a central figure in the Self-Dual Operator-Splitting era (2016–2022). This paper Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding [7] by Stephen Boyd [1] introduced a framework that enabled certifiable, scalable solvers for broad conic problems, a core capability for practical optimization in this era. Poom Kumam [2], affiliated with China Medical University [5] and China Medical University [6], is a prominent figure in this era. His works include A modified conjugate gradient method for monotone nonlinear equations with convex constraints [8], An Efficient Conjugate Gradient Method for Convex Constrained Monotone Nonlinear Equations with Applications [9], and A New Hybrid Spectral Gradient Projection Method for Monotone System of Nonlinear Equations with Convex Constraints [10], which collectively advanced iterative methods for large-scale monotone systems and contributed to robust, scalable solvers in high-dimensional settings.