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.
Introduction, Examples, Solving Optimization Problems, Least-Squares, Linear Programming, Convex Optimizations, How To Solve?, Course Goals
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, 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, 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, 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, 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, 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, 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, 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 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, Maximum Likelihood Estimation, Examples, Logistic Regression, (Binary) Hypothesis Testing, Scalarization, Experiment Design, D-Optimal Design
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.), 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.), 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, 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, 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.), 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, 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.), 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