幾乎囊括了所有主流的凸優化算法。包括梯度法、次梯度法、多面體逼近法、鄰近法和內點法等。這些方法通常依賴於代價函數和約束條件的凸性（而不一定依賴於其可微性），並與對偶性有着直接或間接的聯系。作者針對具體問題的特定結構，給出了大量的例題，來充分展示算法的應用。各章的內容如下：第1章，凸優化模型概述；第2章，優化算法概述；第3章，次梯度算法；第4章，多面體逼近算法；第5章，鄰近算法；第6章，其他算法問題。《凸優化算法》的一個特色是在強調問題之間的對偶性的同時，也十分重視建立在共軛概念上的算法之間的對偶性，這常常能為選擇合適的算法實現方式提供新的靈感和計算上的便利。

1.Convex Optimization Models： An Overview1.1.Lagrange Duality1.1.1.Separable Problems — Decomposition1.1.2.Partitioning1.2.Fenchel Duality and Conic Programming1.2.1.Linear Conic Problems1.2.2.Second Order Cone Programming1.2.3.Semidefinite Programming1.3.Additive Cost Problems1.4.Large Number of Constraints1.5.Exact Penalty Functions1.6.Notes， Sources， and Exercises2.Optimization Algorithms： An Overview2.1.Iterative Descent Algorithms2.1.1.Differentiable Cost Function Descent — Unconstrained Problems2.1.2.Constrained Problems — Feasible Direction Methods2.1.3.Nondifferentiable Problems — Subgradient Methods2.1.4.Alternative Descent Methods2.1.5.Incremental Algorithms2.1.6.Distributed Asynchronous Iterative Algorithms2.2.Approximation Methods2.2.1.Polyhedral Approximation2.2.2.Penalty， Augmented Lagrangian， and Interior Point Methods2.2.3.Proximal Algorithm， Bundle Methods， and Tikhonov Regularization2.2.4.Alternating Direction Method of Multipliers2.2.5.Smoothing of Nondifferentiable Problems2.3.Notes， Sources，and Exercises3.Subgradient Methods3.1.Subgradients of Convex Real—Valued Functions3.1.1.Characterization of the Subdifferential3.2.Convergence Analysis of Subgradient Methods3.3.∈—Subgradient Methods3.3.1.Connection with Incremental Subgradient Methods3.4.Notes， Sources，and Exercises4.Polyhedral Approximation Methods4.1.Outer Linearization — Cutting Plane Methods4.2.Inner Linearization — Simplicial Decomposition4.3.Duality of Outer and Inner Linearization4.4.Generalized Polyhedral Approximation4.5.Generalized Simplicial Decomposition4.5.1.Differentiable Cost Case4.5.2.Nondifferentiable Cost and Side Constraints4.6.Polyhedral Approximation for Conic Programming4.7.Notes， Sources， and Exercises5.Proximal Algorithms5.1.Basic Theory of Proximal Algorithms5.1.1.Convergence5.1.2.Rate of Convergence5.1.3.Gradient Interpretation5.1.4.Fixed Point Interpretation， Overrelaxation， and Generalization5.2.Dual Proximal Algorithms5.2.1.Augmented Lagrangian Methods5.3.Proximal Algorithms with Linearization5.3.1.Proximal Cutting Plane Methods5.3.2.Bundle Methods5.3.3.Proximal Inner Linearization Methods5.4.Alternating Direction Methods of Multipliers5.4.1.Applications in Ma.chine Learning5.4.2.ADMM Applied to Separable Problems5.5.Notes， Sources，and Exercises6.Additional Algorithmic Topics6.1.Gradient Projection Methods6.2.Gradient Projection with Extrapolation6.2.1.An Algorithm with Optimallteration Complexity6.2.2.Nondifferentiable Cost — Smoothing6.3.Proximal Gradient Methods6.4.Incremental Subgradient Proximal Methods6.4.1.Convergence for Methods with Cyclic Order6.4.2.Convergence for Methods with Randomized Order6.4.3.Applicationin Specially Structured Problems6.4.4.Incremental Constraint Projection Methods6.5.Coordinate Descent Methods6.5.1.Variants of Coordinate Descent6.5.2.Distributed Asynchronous Coordinate Descent6.6.Generalized Proximal Methods6.7.∈—Descent and Extended Monotropic Programming6.7.1.∈—Subgradients6.7.2.∈—Descent Method6.7.3.Extended Monotropic Programming Duality6.7.4.Special Cases ofStrong Duality6.8.InteriorPoint Methods6.8.1.Primal—DualMethods for Linear Programming6.8.2.Interior Point Methods for Conic Programming6.8.3.CentralCutting Plane Methods6.9.Notes， Sources，and ExercisesAppendixA：Mathematical BackgroundA.1.Linear AlgebraA.2.Topological PropertiesA.3.DerivativesA.4.Convergence TheoremsAppendix B： Convex Optimization Theory： A SummaryB.1.Basic Concepts of ConvexAnalysisB.2.Basic Concepts of Polyhedral ConvexityB.3.Basic Concepts of Convex OptimizationB.4.Geometric Duality FrameworkB.5.Duality and OptimizationReferences Index