Principles of Optimization - Lecture Notes and Videos

Math 364 (Spring 2019): Lecture Notes and Videos on Principles of Optimization

 Copyright: I (Bala Krishnamoorthy) hold the copyright for all lecture scribes/notes, documents, and other materials including videos posted on these course web pages. These materials might not be used for commercial purposes without my consent.
Scribes from all lectures so far (as a single big file)

Lec # Date Topic(s) Scribe Panopto
1 Jan   8 syllabus, optimization in calculus, the system $$A\mathbf{x} = \mathbf{b}$$, EROs, basic/non-basic variables, parametric vector form scribe video
2 Jan 10 matrix transpose, multiplication, linear independence (LI), rank, inverse, general form of Gauss-Jordan (GJ) method scribe video
3 Jan 15 GJ example, Homework 1, LP formulation, decision variables (d.v.'s), objective function, constraints, sign restrictions scribe video
4 Jan 17 second formulation of Jones LP , assumptions of LP, graphical solution, feasible region, optimal solution, convex set scribe video
5 Jan 22 extreme point/vertex of LP, cases of LP, alternative optimal solutions, infeasible LP, unbounded LP, a full 2D example scribe video
6 Jan 24 LP formulation problems: staffing, blending: average quality, inventory planning: unified flow balance constraints scribe video
7 Jan 29 computer leasing problem, hints on Homework 3 problems, introduction to AMPL, model for Farmer Jones LP scribe video
8 Jan 31 video lecture: more AMPL, Farmer Jones LP using # crops, inventory planning LP: all balance constraints in one go software video
9 Feb  5 LP standard form, slack/excess vars, basic solutions, basic feasible solution (bfs), corner point $$\equiv$$ bfs, optimal bfs scribe video
10 Feb  7 correspondence of bfs's and corner points, adjacent bfs, steps of simplex method for a max LP with $$\leq$$ constraints scribe video
11 Feb 12 simplex method: entering variable, min-ratio test, leaving variable, pivoting, optimal bfs, tableau simplex method scribe video
12 Feb 14 make-up: simplex for min-LP, describing all alternative optima, unbounded LP, big-M simplex, artificial variable scribe video
13 Feb 19 infeasibility in big-M simplex, handling unrestricted in sign (urs) vars: $$x_i \leftarrow x_i^+ - x_i^-$$, for $$x_i^+, x_i^- \geq 0$$, full example scribe video
14 Feb 21 review: problems from the Practice midterm scribe video
15 Feb 26 Midterm exam
16 Feb 28 more AMPL: Chukee toys LP; sensitivity analysis in graphical LP, changing objective function coefficient scribe video
17 Mar  5 changing coefficient of wheat, changing rhs of constraint $$b_i \rightarrow b_i + \Delta$$, shadow price, economic interpretation scribe video
18 Mar  7 simplex in matrix form, $$\mathbf{x}^T = [\mathbf{x}_B^T~\mathbf{x}_N^T ]$$, optimal $$\mathbf{x}_B = B^{-1}\mathbf{b},\, z^* = \mathbf{c}_B^T B^{-1}\mathbf{b}$$, optimal tableau given optimal basis scribe video
19 Mar 19 video: sensitivity analysis in matrix form: changing $$c_j$$ for non-basic $$x_j$$, reduced cost, changing $$c_j$$ for basic $$x_j$$ scribe video
20 Mar 21 $$b_i \rightarrow b_i + \Delta$$, new $$\mathbf{x}_B =$$ old $$\mathbf{x}_B + \Delta [B^{-1}]_{\cdot i}$$, shadow price, changing $$c_j$$ and $$A_{ij}$$ for nonbasic $$x_j$$ simultaneously scribe video
21 Mar 26 video: LP duality, normal constraints/variables, dual of normal max LP, primal-dual relations, motivation for the dual scribe video
22 Mar 28 economic interpretation of Farmer Jones and Gaseous LPs (Hw2), duality in matrix form, weak and stroing duality scribe video
23 Apr   2 dual theorem: $$\mathbf{y}^T = \mathbf{c}_B^T B^{-1}$$ is optimal for (D), read off optimal dual solution (Jones LP), shadow price$$_i = y_i$$, AMPL scribe video
24 Apr   4 hints on Homework 10 problems, complementary slackness conditions (CSCs), using CSCs to solve LPs scribe video
25 Apr   9 discussion on Project, integer programming (IP), IP formulation: basketball starting line-up, if-then constaints scribe video
26 Apr 11 fixed charge MIP, forcing constraint, general either-or constraint, OR vs XOR, modeling more than two options scribe video
27 Apr 16 more discussion on Project, general if-then statement, $$A \Rightarrow B ~\equiv~ \mbox{not} A ~\mbox{OR}~ B$$, warehouse location MIP scribe video
28 Apr 18 modeling general either-or and general if-then statements using extra binary variables scribe video
29 Apr 23 problems from Homework 8 and Homework 10, review of practice final exam scribe video
30 Apr 25 more review of practice final exam scribe video