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# A Primer to Mathematical Optimization

### Overview

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ABOUT THE COURSE: In this era of Machine Learning and Data Science, students often crave for learning the basic tools of optimization theory because machine learning ideas essentially exploit the power of Numerical Linear Algebra, Optimization, and Statistics. The primary aim of this course is to hand over a complete readymade package for beginners in mathematical optimization. 'Complete' in the sense of its mathematical orientation, geometrical explanation, problem-solution sheets, and tutorial sheets. After attending this course, students will get to know the standard methods, and basic and modern results in optimization. The concepts will be explained not only with mathematical rigor but also with geometrical essence so that students feel optimization is fun. The course covers mathematical foundation for optimization and basic techniques of unconstrained and constrained optimization problems.INTENDED AUDIENCE: Third Year Undergraduates of Mathematics / Computer Science / Electrical / Mechanical EngineeringPREREQUISITES: Calculus, Linear Algebra, Coordinate GeometryINDUSTRY SUPPORT: Control, Machine Learning

### Syllabus

Week 1:

Lecture 1.Introduction and history on the development of optimization theory
Section I: Tools from Linear Algebra
Lecture 2.Vector space, basis and dimension, subspace, inner products and norms, matrix norms, projection, eigen values and eigen vectors
Lecture 3.Definiteness of matrices: eigen value criterion and Sylvester criterion, quadratic forms and quadratic functions
Section II: Tools from Calculus of Several Variables

Lecture 4.Sets in Rn: balls, neighborhood, interior point, open sets, closed sets, bounded sets, compact sets, level sets, epigraphs
Lecture 5.Supremum and infimum of a set, sequence, subsequence, limsup and liminf
Week 2:

Lecture 6.Order of convergence of a sequence
Lecture 7.Limit, continuity, uniform continuity and Lipschitz continuity and their geometries
Lecture 8.Partial derivative, gradient, Hessian, directional derivative
Lecture 9.Differentiablity, hierarchy of ‘PDs, DD and diffentiability’
Lecture 10.Little oh and big Oh notations, Taylor’s theorem
Week 3:

Section III: Convex Sets and Convex Functions
Lecture 11.Convex sets, examples of commonly occurring convex sets, convexity preserving operations
Lecture 12.Convexity preserving operations continued, separation result (a point and a closed convex set)
Lecture 13.Theorems of alternatives: Farkas’ theorem
Lecture 14.Theorems of alternatives: Gordan’s theorem and other results
Lecture 15.Convex functions, their continuity, Lipschitz continuity and directional derivative
Week 4:

Lecture 16.Zeroth order and first order characterizations of convexity
Lecture 17.Second order characterization of convexity, convexity preserving operations
Section IV: Unconstrained Optimization Theory

Lecture 18.Local optima and its variants, saddle point, coercive functions, existence of optima: Weierstrass theorem
Lecture 19.Existence of optima: Theorem for coercive functions. Descent direction and its first order and second order characterizations
Lecture 20.First order and second order necessary and sufficient conditions for minima
Week 5:

Section V: Unconstrained Optimization Methods

Lecture 21.General structure of an optimization algorithm, global convergence, descent property, quadratic termination property. Introduction to line search: exact and inexact line searches
Lecture 22.Inexact line searches: Armijo-Goldstein, Wolfe-Powell, Armijo-backtracking
Lecture 23.
On global convergence of gradient descent methods using line search methods: for exact line search
Lecture 24.On global convergence of gradient descent methods using line search methods: for inexact line searches
Lecture 25.Where do general descent methods converge? —almost always to a local minimizer. Capture theorem.
Week 6:

Lecture 26.Scaling of variables
Lecture 27.Practical stopping criteria
Lecture 28.Steepest descent method, descent property, zigzagging, scaling effect, Barzilai and Borwin gradient method
Lecture 29.Global convergence of steepest descent method and order of convergence. Scaling effect and dimension independency.
Lecture 30.Newton method and its local convergence.

Week 7:

Lecture 31.Dimension dependency of Newton method. Scaling effect.
Lecture 32.Levenberg-Marquardt and other modified Newton methods
Lecture 33.Quasi-Newton methods - quasi-Newton equation, general quasi-Newton algorithm, variable metric method, general comparison of Newton and quasi-Newton methods
Lecture 34.Global convergence of quasi-Newton methods for uniformly convex functions
Lecture 35.Superlinear convergence of quasi-Newton methods
Week 8:

Lecture 36.Basic conjugate direction method
Lecture 37.Convergence rate of the basic CG method
Lecture 38.Trust region methods: Cauchy point, dog leg and double dog leg methods
Section VI. Constraint Optimization Theory

Lecture 39.A revisit to Lagrange multipliers method
Lecture 40.
Cones for constraint optimization: cone, dual cone, cone of feasible directions, linearizing cone, cone of descent directions
Week 9:

Lecture 41.Cones for constraint optimization: tangent cone. Geometric optimality conditions.
Lecture 42.First order FJ and KKT necessary optimality conditions
Lecture 43.First order KKT sufficient optimality condition
Lecture 44.Second order KKT necessary and sufficient optimality conditions
Lecture 45.Constraint qualification
Week 10:

Section VII: Lagrangian Duality Theory

Lecture 46.Lagrangian duality: how does one discover duality? Geometric interpretation of duality.
Lecture 47.Results on dual function: dual problem is a convex optimization problem, differentiability of the dual function
Lecture 48.Weak duality theorem. Duality gap. Equal primal and dual objective values imply optimal.
Lecture 49.Strong duality theorem. Convex optimization problem and SCQ implies strong duality.
Lecture 50.Saddle point optimality and absence of duality gap. Relationship between saddle point criteria and KKT conditions
Week 11:

Section VIII: Linearly Constrained Nonlinear Optimization AlgorithmsLecture 51.Quadratic programming methods – Active set method
Lecture 52.Quadratic programming method – Interior point method
Section IX: Nonlinearly Constrained Nonlinear Optimization Algorithms

Lecture 55.Penalty method – Exterior penalty method
Lecture 56.Penalty method – Interior penalty method
Week 12:

Lecture 57.Penalty method – Augmented Lagrangian method
Lecture 58.Sequential quadratic programming – Lagrange-Newton method
Lecture 59.Sequential quadratic programming – Reduced Hessian matrix method
Lecture 60.Trust region method
Lecture 61.Null space method

### Taught by

Prof. Debdas Ghosh

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