Argomenc dl curs
- Nonlinear Optimization Modelling: Formulaic Structure of Nonlinear Optimization Models, Fundamental Models in Production Planning, Support Vector Machine, Energy Capacity Planning, Portfolio/Inventory Optimization, Facility Location, Engineering Design (Geometric Optimization), Regression, Optimal Control, and Robotics Motion Planning
- Mathematical Preliminaries and Topological Aspects of Nonlinear Optimization: Vector/Matrix Norms and Nonlinear Multivariable Function Approximation
- Optimality Conditions for Unconstrained Optimization Models: Formulaic Structure of Unconstrained Optimization Models, First/Second Order Analysis of Optimality, and Necessary and Sufficient Optimality Conditions
- Least Squares Models: Data Fitting, Noise Cancellation, and Circle Fitting
- First Order Algorithms for Unconstrained Optimization: Line Search, Gradient Method with Convergence Analysis, Gauss–Newton Method for Nonlinear Least Squares Models, and a Gradient-Based Fixed-Point Iteration Method for Solving the Fermat–Weber Problem
- Second Order Algorithms for Unconstrained Optimization: Newton’s Method with Convergence Analysis
- Convex Sets: Definition, Convex Balls in Various Norms, Algebraic Operations Preserving Convexity, Convex Hulls, Convex Cones, Conic Hulls, and Topological Properties of Convex Sets
- Convex Functions: Definition, Jensen’s Inequality, First/Second Order Analysis of Convexity, Global Optimality, Well-Known Convex Functions, Operations Preserving Functional Convexity, Level Sets and Epigraphs, and Quasi-Convex Functions
- Convex Optimization: Formulaic Structure, Global Optimality, Convex Quadratic Models, Chebyshev Center of a Set of Points, Analysis of the Markowitz Portfolio Optimization Model, Orthogonal Projection Methods, Analysis of Linear Classification Models, and Convex Form of the Trust Region Subproblem
- Optimization Over Convex Sets: Stationarity and Optimality Conditions, Gradient Projection Method with Convergence Analysis, Sparsity Constrained Optimization Models, and Iterative Hard-Thresholding Method
- Linearly Constrained Nonlinear Optimization Models: Formulaic Structure, Karush–Kuhn–Tucker (KKT) Conditions, Lagrangian Function, Orthogonal Projection onto Half-Spaces, and Orthogonal Regression
- KKT Conditions for Equality/Inequality Constrained Nonlinear Optimization Models: Feasible Descent Directions, Fritz–John Conditions, KKT Conditions, Sufficiency of KKT Conditions for Convex Optimization, Analysis of Constrained Least Squares Models, Second Order Optimality Conditions, and Total Least Squares Models
- Duality Theory in Nonlinear Optimization: Dual Model Definition, Weak and Strong Duality, Duality Gap and Optimality Bounds, Dual Models of Well-Known Nonlinear Optimization Problems, and Regularization and Denoising
- Complementary Topics: Penalty and Barrier Methods for Nonlinear Optimization, and Alternating Direction Method of Multipliers (ADMM) Algorithms
Modalité de ensegnament
Lectures: The course is delivered using lecture slides, which are regularly uploaded to Microsoft Teams. Additional explanations and derivations are provided on the board to clarify concepts and complement the slide material.
Exercises: Topic-oriented exercises are solved, primarily on the board, during class to reinforce students' understanding and help them internalize the theoretical concepts.
Software Laboratory: To illustrate the practical implementation and performance of optimization algorithms, main algorithms are implemented and tested in the MATLAB environment during dedicated laboratory sessions.