# Linear Systems and Optimization | Convex Optimization I

Instructor: Boyd, Stephen

## Course Meetings: 19

Lecture 1    View Now >

1 hr 21 min

• Topics: Introduction, Examples, Solving Optimization Problems, Least-Squares, Linear Programming, Convex Optimizations, How To Solve?, Course Goals
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Lecture 2    View Now >

1 hr 17 min

• Topics: 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
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Lecture 3    View Now >

1 hr 17 min

• Topics: 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
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Lecture 4    View Now >

1 hr 14 min

• Topics: 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)
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Lecture 5    View Now >

1 hr 16 min

• Topics: 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
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Lecture 6    View Now >

1 hr 9 min

• Topics: (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)
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Lecture 7    View Now >

1 hr 15 min

• Topics: 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
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Lecture 8    View Now >

1 hr 16 min

• Topics: 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
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Lecture 9    View Now >

1 hr 17 min

• Topics: 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
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Lecture 10    View Now >

1 hr 18 min

• Topics: 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
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Lecture 11    View Now >

1 hr 17 min

• Topics: Statistical Estimation, Maximum Likelihood Estimation, Examples, Logistic Regression, (Binary) Hypothesis Testing, Scalarization, Experiment Design, D-Optimal Design
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Lecture 12    View Now >

1 hr 16 min

• Topics: 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
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Lecture 13    View Now >

1 hr 15 min

• Topics: 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
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Lecture 14    View Now >

1 hr 10 min

• Topics: 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
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Lecture 15    View Now >

1 hr 17 min

• Topics: 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
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Lecture 16    View Now >

1 hr 14 min

• Topics: 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
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Lecture 17    View Now >

1 hr 19 min

• Topics: 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
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Lecture 18    View Now >

1 hr 17 min

• Topics: 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
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Lecture 19    View Now >

1 hr 15 min

• Topics: 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
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*Segments of this lecture have been edited out due to copyright restrictions.