Example: Piecewise Linear Minimization

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.

Course Lectures

• Basic Rules for Subgradient Calculus

  Stephen Boyd

  Play

  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

  Play

  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

  Play

  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

  Play

  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

  Play

  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

  Play

  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

  Playing

  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

  Play

  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

  Play

  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

  Play

  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

  Play

  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

  Play

  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

  Play

  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

  Play

  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

  Play

  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

  Play

  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

  Play

  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

  Play

  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.