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A Primer to Mathematical Optimization

Prof. Debdas Ghosh · IIT(BHU) Varanasi

Start with lecture 1 At NPTEL

About

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 Engineering

Prerequisites: Calculus, Linear Algebra, Coordinate Geometry

Lectures

Week 01

  1. Lecture 01: Introduction and Histroy on the Development of Optimization Theory YouTube
  2. Lecture 02: Tools from Linear Algebra YouTube
  3. Lecture 03: Tools from Linear Algebra YouTube

Week 02

  1. Lecture 04 : Tools for calculus of several variables YouTube
  2. Lecture 05 : Tools from calculus of several variables YouTube

Week 03

  1. Lecture 6: Order of Convergence YouTube
  2. Lecture 7: Continuity YouTube

Week 04

  1. Lecture 8 & 9 : Partial and Directional Derivatives and Differentiability YouTube

Week 05

  1. Lecture - 10 : Taylor's Theorem YouTube
  2. Lecture - 11 : Convex Sets & Convexity Preserving Operations YouTube
  3. Lecture - 12 : Separation Result YouTube

Week 06

  1. Lecture - 13 & 14 : Theorems of Alternatives YouTube
  2. Lecture - 15 : Convex Functions YouTube
  3. Lecture -16 : Properties and Zeroth Order Characterization of Convex Functions YouTube
  4. Lecture - 17 : First-Order and Second-Order Characterization of Convex Functions YouTube
  5. Lecture - 18 : Convexity Preserving Operations YouTube
  6. Lecture - 19 : Optimality and Coerciveness YouTube

Week 07

  1. Lecture 20 part 1: First Order Optimality Condition YouTube
  2. Lecture 20 part 2: Second Order Optimality Conditions YouTube
  3. Lecture 21: General Structure of Unconstrained Optimization Algorithms YouTube
  4. Lecture 22: Inexact Line Search YouTube
  5. Lecture 23 & 24: Global Convergence of Descent Methods YouTube
  6. Lecture 25: Where do Descent Methods Converge? YouTube

Week 08

  1. Lecture - 26 : Scaling Of Variables YouTube
  2. Lecture - 27 : Practical Stopping Criteria YouTube
  3. Lecture - 28 & 29 : Steepest Descent Method YouTube
  4. Lecture - 30, 31 & 32 : Newton's Method YouTube

Week 09

  1. Lecture - 33, 34 & 35 YouTube
  2. Lecture - 36 & 37 YouTube
  3. Lecture - 38 Part 1 YouTube
  4. Lecture - 38 Part 2 YouTube

Week 10

  1. Lecture - 39 : A Revisit to Lagrange Multipliears Method YouTube
  2. Lecture - 40 : Special Cones for Contrained Optimization YouTube
  3. Lecture - 41 : Tangent Cone YouTube
  4. Lecture - 42 & 43 : First-Order KKT Optimality Conditions YouTube
  5. Lecture 44 : Second-Order KKT Optimality Conditions YouTube
  6. Lecture 45 : Constraint Qualifications YouTube

Week 11

  1. Lecture 46, 47, 48, 49 & 50 : Lagrangian Duality Theory YouTube
  2. Lecture 51, 52 & 53 : Methods for Linearly Constrained Problems YouTube

Week 12

  1. Lecture 54 : Interior-Point Method for QPP YouTube
  2. Lecture 55 : Penalty Methods YouTube
  3. Lecture 56 : Sequential Quadratic Programming Method YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT(BHU) Varanasi
Department
Mathematics
Teacher
Prof. Debdas Ghosh
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Kanpur.
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