Convergence Proof, Stopping Criterion

This module focuses on convergence proofs and stopping criteria in optimization, providing a rigorous framework for evaluating the efficiency of optimization algorithms. Key topics include:

• Proof of convergence for subgradient methods.
• Establishing stopping criteria based on convergence results.
• Techniques for locating points within the intersection of convex sets using alternating projections.
• Examples highlighting the application of subgradient methods in constrained environments, such as least L₁-norm problems.

Course Lectures

• Basic Rules for Subgradient Calculus

  Stephen Boyd

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  This module covers the foundational aspects of subgradient calculus, providing a thorough introduction to course logistics and organization. Key concepts include:

  • Understanding subgradients and subdifferentials.
  • Application of basic inequalities and rules in subgradient calculus.
  • Exploring pointwise supremum and weak rules.
  • Minimization and composition related to subgradients.

  Students will also learn about quasigradients and their applications in optimization problems.

• Recap: Subgradients

  Stephen Boyd

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  This module revisits the concept of subgradients, building on the foundational knowledge established previously. It covers:

  • Optimality conditions for both unconstrained and constrained problems.
  • Application of the subgradient method on various optimization problems, including piecewise linear minimization.
  • Directional derivatives and their relationship with the subdifferential.
  • Step size rules and convergence results for the subgradient method.

  Students will also examine examples illustrating the application of these methods.

• Convergence Proof, Stopping Criterion

  Stephen Boyd

  Playing

  This module focuses on convergence proofs and stopping criteria in optimization, providing a rigorous framework for evaluating the efficiency of optimization algorithms. Key topics include:

  • Proof of convergence for subgradient methods.
  • Establishing stopping criteria based on convergence results.
  • Techniques for locating points within the intersection of convex sets using alternating projections.
  • Examples highlighting the application of subgradient methods in constrained environments, such as least L₁-norm problems.
• Project Subgradient For Dual Problem

  Stephen Boyd

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  This module introduces the project subgradient method specifically tailored for dual problems. Students will learn about:

  • The subgradient of the negative dual function.
  • Examples illustrating the application of subgradient methods in constrained optimization.
  • The stochastic subgradient method, including handling noisy unbiased subgradients.
  • Convergence results and proofs related to stochastic programming.

  This module emphasizes practical applications of these methods in optimization problems.

• Stochastic Programming

  Stephen Boyd

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  This module focuses on stochastic programming, examining its variations and the expected value of convex functions. Key content includes:

  • Understanding expected values through examples like piecewise linear functions.
  • Application of on-line learning and adaptive signal processing techniques.
  • Cutting-plane methods, including the use of cutting-plane oracles.
  • Exploring feasibility problems and inequality-constrained scenarios.

  Students will also delve into specific cutting-plane methods and convergence of algorithms.

• Addendum: Hit-And-Run CG Algorithm

  Stephen Boyd

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  This addendum explores the Hit-And-Run Cutting-Plane (CG) Algorithm and its applications in optimization. Topics covered include:

  • Maximum volume ellipsoid and Chebyshev center methods.
  • Analytic center cutting-plane methods and their extensions.
  • Dropping constraints and constructing cutting-planes effectively.
  • Properties of the infeasible start Newton method algorithm.

  This module emphasizes constructing and utilizing cutting-planes in optimization problems.

• Example: Piecewise Linear Minimization

  Stephen Boyd

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  Focusing on piecewise linear minimization, this module provides practical examples to reinforce learning. Key concepts include:

  • Application of the Analytic Center Cutting-Plane Method (ACCPM) with constraint dropping.
  • Using the ellipsoid algorithm for minimizing convex functions.
  • Updating ellipsoids and understanding stopping criteria.
  • Interpreting results from the basic ellipsoid algorithm.

  Students will engage in hands-on examples that illustrate these concepts.

• Recap: Ellipsoid Method

  Stephen Boyd

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  This recap module consolidates knowledge of the ellipsoid method, emphasizing improvements and understanding convergence. Topics include:

  • Proofs of convergence and complexity analysis.
  • Deep cut techniques in the ellipsoid method.
  • Applications of primal and dual decomposition in handling nondifferentiable convex optimization problems.
  • Examples illustrating dual decomposition applications.

  Students will gain a comprehensive understanding of the methods for managing complex optimization scenarios.

• Comments: Latex Typesetting Style

  Stephen Boyd

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  This module discusses decomposition methods, their structures, and applications in optimization. Key points include:

  • Understanding primal and dual decomposition with and without constraints.
  • General decomposition structures and their applications.
  • Finding feasible iterates and interpreting results.
  • Complex examples illustrating decomposition methods.

  Students will learn about various applications of decomposition in different optimization scenarios.

• Decomposition Applications

  Stephen Boyd

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  This module focuses on various decomposition applications, particularly in rate control and network flow problems. Students will learn about:

  • Rate control setups and problems, including Lagrangian formulations.
  • Dual decomposition algorithms and generating feasible flows.
  • Convergence of primal and dual objectives in network problems.
  • Examples illustrating network flow setups and optimal flow strategies.

  Students will engage with practical applications to solidify their understanding.

• Sequential Convex Programming

  Stephen Boyd

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  This module delves into sequential convex programming (SCP) as a method for nonconvex optimization problems. Key topics include:

  • Basic principles of SCP and its applications.
  • Trust region methods and affine approximations.
  • Particle methods and fitting functions to data.
  • Example applications, including nonlinear optimal control and discretization.

  Students will gain insights into the progress and convergence of SCP methods.

• Recap: 'Difference Of Convex' Programming

  Stephen Boyd

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  This recap module revisits the 'Difference of Convex' programming, discussing its relevance and applications. Key points include:

  • Alternating convex optimization techniques.
  • Application in nonnegative matrix factorization.
  • Overview of conjugate gradient methods and their applications in linear equations.
  • Properties of Krylov sequences and their spectral analysis.

  Students will understand the significance of these concepts in optimization.

• Recap: Conjugate Gradient Method

  Stephen Boyd

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  This module emphasizes the conjugate gradient method, covering its properties and applications. Key content includes:

  • Overview of Krylov subspace and its properties.
  • Convergence rates and residual convergence analysis.
  • Efficient matrix-vector multiplication techniques.
  • Shifting and preconditioned conjugate gradient algorithms.

  Students will gain a comprehensive understanding of these methods and their applications in optimization.

• Methods (Truncated Newton Method)

  Stephen Boyd

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  This module covers truncated Newton methods, focusing on their applications and convergence properties. Key topics include:

  • Understanding convergence versus iterations and cumulative CG steps.
  • Truncated Newton interior-point methods and their applications.
  • Evaluating dual rate control problems and primal-dual search directions.
  • Solving convex-cardinality problems and sparse modeling techniques.

  Students will explore various applications, including portfolio investment and sparse signal reconstruction.

• Recap: Example: Minimum Cardinality Problem

  Stephen Boyd

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  This recap module focuses on the minimum cardinality problem, discussing its interpretation as a convex relaxation. Key points include:

  • Understanding weighted and asymmetric L₁ heuristics.
  • Applications in regressor selection and sparse signal reconstruction.
  • Exploring total variation reconstruction techniques.
  • Strategies for detecting changes in time series models.

  Students will gain insights into L₁-norm methods and their applications in various optimization contexts.

• Model Predictive Control

  Stephen Boyd

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  This module explores model predictive control (MPC) strategies and their applications in linear time-invariant systems. Key topics include:

  • Understanding greedy control and dynamic programming solutions.
  • Linear quadratic regulator considerations and finite horizon approximations.
  • Analyzing MPC performance and variations.
  • Application of MPC in optimal trajectories and supply chain management.

  Students will learn about the nuances of MPC and its optimization strategies in various contexts.

• Stochastic Model Predictive Control

  Stephen Boyd

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  This module focuses on stochastic model predictive control, exploring its applications in optimization. Key content includes:

  • Causal state-feedback control strategies.
  • Stochastic finite horizon control solutions via dynamic programming.
  • Understanding independent process noise in control systems.
  • Branch and bound methods for solving nonconvex optimization problems.

  Students will engage with sample trajectories and cost histograms to analyze system performance.

• Recap: Branch And Bound Methods, Basic Idea, Unconstrained, Nonconvex Minimization

  Stephen Boyd

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  This recap module emphasizes branch and bound methods in nonconvex minimization, exploring their fundamental concepts and applications. Key points include:

  • Understanding lower and upper bound functions in optimization.
  • Analyzing the branch and bound algorithm structure.
  • Pruning techniques and convergence analysis in optimization.
  • Exploring mixed Boolean-convex problems and their solution methods.

  Students will gain insights into global lower and upper bounds through various examples.