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Course

# Convex Optimization I

Stanford University

Concentrates on recognizing and solving convex optimization problems that arise in engineering. Topics include: Convex sets, functions, and optimization problems. Basics of convex analysis. Least-squares, linear and quadratic programs, semidefinite programming, minimax, extremal volume, and other problems. Optimality conditions, duality theory, theorems of alternative, and applications. Interiorpoint methods. Applications to signal processing, control, digital and analog circuit design, computational geometry, statistics, and mechanical engineering. Prerequisites: Good knowledge of linear algebra. Exposure to numerical computing, optimization, and application fields helpful but not required; the engineering applications will be kept basic and simple.

Home > Mathematics > Calculus > Convex Optimization I Lectures:
• ### Introduction to Convex Optimization I

01:20:33Stephen Boyd

Introduction, Examples, Solving Optimization Problems, Least-Squares, Linear Programming, Convex Optimizations, How To Solve?, Course Goals

• ### Guest Lecturer: Jacob Mattingley

01:16:51Stephen Boyd

Guest Lecturer: Jacob Mattingley, Logistics, Agenda, Convex Set, Convex Cone, Polyhedra, Positive Semidefinite Cone, Operations That Preserve Convexity, Intersection, Affine Function, Generalized Inequalities, Minimum And Minimal Elements, Supporting Hyperlane Theorem, Minimum And Minimal Elements Via Dual Inequalities

• ### Logistics

01:17:14Stephen Boyd

Logistics, Convex Functions, Examples, Restriction Of A Convex Function To A Line, First-Order Condition, Examples (FOC And SOC), Epigraph And Sublevel Set, Jensen's Inequality, Operations That Preserve Convexity, Pointwise Maximum, Pointwise Maximum, Composition With Scalar Functions, Vector Composition

• ### Vector Composition

01:13:38Stephen Boyd

Vector Composition, Perspective, The Conjugate Function, Quasiconvex Functions, Examples, Properties (Of Quasiconvex Functions), Log-Concave And Log-Convex Functions, Properties (Of Log-Concave And Log-Convex Functions), Examples (Of Log-Concave And Log-Convex Functions)

• ### Optimal And Locally Optimal Points

01:16:10Stephen Boyd

Optimal And Locally Optimal Points, Feasibility Problem, Convex Optimization Problem, Local And Global Optima, Optimality Criterion For Differentiable F0, Equivalent Convex Problems, Quasiconvex Optimization, Problem Families, Linear Program

• ### (Generalized) Linear-Fractional Program

01:09:20Stephen Boyd

(Generalized) Linear-Fractional Program, Quadratic Program (QP), Quadratically Constrained Quadratic Program (QCQP), Second-Order Cone Programming, Robust Linear Programming, Geometric Programming, Example (Design Of Cantilever Beam), GP Examples (Minimizing Spectral Radius Of Nonnegative Matrix)

• ### Generalized Inequality Constraints

01:14:38Stephen Boyd

Generalized Inequality Constraints, Semidefinite Program (SDP), LP And SOCP As SDP, Eigenvalue Minimization, Matrix Norm Minimization, Vector Optimization, Optimal And Pareto Optimal Points, Multicriterion Optimization, Risk Return Trade-Off In Portfolio Optimization, Scalarization, Scalarization For Multicriterion Problems

• ### Lagrangian

01:16:30Stephen Boyd

Lagrangian, Lagrange Dual Function, Least-Norm Solution Of Linear Equations, Standard Form LP, Two-Way Partitioning, Dual Problem, Weak And Strong Duality, Slater's Constraint Qualification, Inequality Form LP, Quadratic Program, Complementary Slackness

• ### Complementary Slackness

01:16:35Stephen Boyd

Complementary Slackness, Karush-Kuhn-Tucker (KKT) Conditions, KKT Conditions For Convex Problem, Perturbation And Sensitivity Analysis, Global Sensitivity Result, Local Sensitivity, Duality And Problem Reformulations, Introducing New Variables And Equality Constraints, Implicit Constraints, Semidefinite Program

• ### Applications Section of Course

01:17:55Stephen Boyd

Applications Section Of The Course, Norm Approximation, Penalty Function Approximation, Least-Norm Problems, Regularized Approximation, Scalarized Problem, Signal Reconstruction, Robust Approximation, Stochastic Robust LS, Worst-Case Robust LS

• ### Statistical Estimation

01:17:03Stephen Boyd

Statistical Estimation, Maximum Likelihood Estimation, Examples, Logistic Regression, (Binary) Hypothesis Testing, Scalarization, Experiment Design, D-Optimal Design

• ### Continue On Experiment Design

01:16:05Stephen Boyd

Continue On Experiment Design, Geometric Problems, Minimum Volume Ellipsoid Around A Set, Maximum Volume Inscribed Ellipsoid, Efficiency Of Ellipsoidal Approximations, Centering, Analytic Center Of A Set Of Inequalities, Linear Discrimination

• ### Linear Discrimination (Cont.)

01:15:17Stephen Boyd

Linear Discrimination (Cont.), Robust Linear Discrimination, Approximate Linear Separation Of Non-Separable Sets, Support Vector Classifier, Nonlinear Discrimination, Placement And Facility Location, Numerical Linear Algebra Background, Matrix Structure And Algorithm Complexity, Linear Equations That Are Easy To Solve, The Factor-Solve Method For Solving Ax = B, LU Factorization

• ### LU Factorization (Cont.)

01:10:12Stephen Boyd

LU Factorization (Cont.), Sparse LU Factorization, Cholesky Factorization, Sparse Cholesky Factorization, LDLT Factorization, Equations With Structured Sub-Blocks, Dominant Terms In Flop Count, Structured Matrix Plus Low Rank Term

• ### Algorithm Section of The Course

01:16:45Stephen Boyd

Algorithm Section Of The Course, Unconstrained Minimization, Initial Point And Sublevel Set, Strong Convexity And Implications, Descent Methods, Gradient Descent Method, Steepest Descent Method, Newton Step, Newton's Method, Classical Convergence Analysis, Examples

• ### Continue on Unconstrained Minimization

01:13:59Stephen Boyd

Continue On Unconstrained Minimization, Self-Concordance, Convergence Analysis For Self-Concordant Functions, Implementation, Example Of Dense Newton System With Structure, Equality Constrained Minimization, Eliminating Equality Constraints, Newton Step, Newton's Method With Equality Constraints

• ### Newton's Method (Cont.)

01:19:25Stephen Boyd

Newton's Method (Cont.), Newton Step At Infeasible Points, Solving KKT Systems, Equality Constrained Analytic Centering, Complexity Per Iteration Of Three Methods Is Identical, Network Flow Optimization, Analytic Center Of Linear Matrix Inequality, Interior-Point Methods, Logarithmic Barrier

• ### Logarithmic Barrier

01:16:53Stephen Boyd

Logarithmic Barrier, Central Path, Dual Points On Central Path, Interpretation Via KKT Conditions, Force Field Interpretation, Barrier Method, Convergence Analysis, Examples, Feasibility And Phase I Methods

• ### Interior-Point Methods (Cont.)

01:15:02Stephen Boyd

Interior-Point Methods (Cont.), Example, Barrier Method (Review), Complexity Analysis Via Self-Concordance, Total Number Of Newton Iterations, Generalized Inequalities, Logarithmic Barrier And Central Path, Barrier Method, Course Conclusion, Further Topics