ENME 722
Equilibrium Programming in Engineering
Approximate Schedule and Homeworks (also on ELMS)
======OVERVIEW OF RECORDED LECTURES,
QUIZZES, HOMEWORKS, EXAM, PROJECT======
NOTE:
- "Due Date": To review the recorded lecture before that date and take the associated quiz.
- Class time on the due date will be devoted to solving new problems in real-time to reinforce concepts, proving certain results in more detail, going over homeworks and other questions as needed.
- In the recorded lectures, ignore Spring 2020 announcements and references to homeworks and other items for that semester
==============================================
>>>>START OF MODULE 1.1<<<<
Lecture #1 (Module 1.1.1)
Due Date: September 2, 2025
Duration: 2 hours 26 minutes
Slides: 1-47 (approximately)
Topics: Introduction to course, preview of Wardrop traffic equilibrium problem, transportation problem linear program and generalization to spatial price equilibrium problem conditions, portfolio optimization quadratic program, convex functions, convex sets, positive semi-definite matrices, positive definite matrices, multiplicity of solutions.
- Discussion Section (class time): Overview of course, going through "Lecture 0", going through GAMS software
- Quiz: Quiz #1, due by September 9, 2025, 9:30am (to give you more time for this first lecture and quiz)
- Homework: HW #1 given out, due by 9:30am on September 16, 2025
Lecture #2 (Module 1.1.2)
Due Date: September 9, 2025
Duration: 2 hours 13 minutes
Slides: 48-80 (approximately)
Topics: Overview of course modules (focus on equilibrium problems), unit commitment problem in power markets (integer program), non-convex pooling problem, Weierstrass Theorem, main convex program theorem, Karush-Kuhn-Tucker (KKT) optimality conditions- statement + analysis with a sample problem and discussion, KKT first-order sufficiency theorem, KKT first-order necessity condition, two-circle problem, box problem, constraint qualifications (linearity, linear independence of the binding constraint gradients), two-variable (linear) mixed complementarity problem (MCP), three-variable (linear) MCP, how to code MCPs in GAMS.
Quiz: Quiz #2, due by September 9, 2025 9:30am
Homework: HW #1 due by 9:30am on September 16, 2025
Lecture #3 (Module 1.1.3)
Due Date: September 16, 2025
Duration: 1 hour 27 minutes
Slides: 81-93 (approximately)
Topics: Directional derivatives, big picture/bigger picture relative to optimization and equilibrium problems, KKT conditions as special case of an MCP, variational inequality problem (VI) and relationship to MCPs, Principle of Symmetry, VI generalizing nonnegative directional derivatives, duopoly problem (MCP vs. VI).
Quiz: Quiz #3, due by September 16, 2025, 9:30am
Homework: HW #2 given out, due by 9:30am on September 30, 2025
Lecture #4 (Module 1.1.4)
Due Date: September 23, 2025
Duration: 2 hours 6 minutes 16 seconds
Slides: 94-103 (approximately)
Topics: Principle of Symmetry, feasible set of directions definition + examples cones, Tangent cone definition + examples, inner product cone (linearized tangent cone), definition + examples, relationship between the feasible set of directions, the tangent cone and the inner product cone.
Quiz: Quiz #4, due by September 23, 2025, 9:30am
Homework: HW #2 due by 9:30am on September 30, 2025
Lecture #5 (Module 1.1.5)
Due Date: September 30, 2025
Duration: 1 hour 55 minutes 57 seconds
Slides: 103-128 (approximately)
Topics: Feasible set of directions , Tangent cone , inner product cone,relationship between the feasible set of directions, the tangent cone and the inner product cone (con't), first-order necessary conditions in terms of directional derivative relative to the feasible set of directions and the tangent cone, polar of a cone, Moreau's Decomposition Theorem, closed convex cones, polyhedral cones, polar of the inner product cone,steps leading up to the KKT conditions relative to cones.
Quiz: Quiz #5, due by September 30, 2025, 9:30am
Homework: HW #3 given out, due by 9:30am on October 14, 2025
>>>>START OF MODULE 1.2<<<<
Lecture #6 (Module 1.2.1)
Due Date: October 7, 2025
Duration: 2 hours 0 minutes 8 seconds
Slides: 1-48 (approximately)
Topics: Big picture of equilibrium problems (linear programming and complementary slackness as an example), variational inequality re-expressed as a linear program/or with constraint qualifications if you know the solution already (symbolic importance), overview of Module 1.2 (single and multi-level equilibrium problems, examples of important equilibrium problems), game theory, two-person zero-sum games (pure strategies and mixed strategies), two-person bimatrix games (pure strategies and mixed strategies), Stone-Paper-Scissors game, Prisoner's Dilemma game, Battle-of-the Sexes game (parts 1 and 2), Peace-War game, notions of equilibria in a game (equilibria in dominant actions, Nash equilibria), Pareto efficient equilibrium solution, linear programming primal and dual problems relating to two-person, mixed, zero-sum games, need a linear complementarity problem for a mixed bimatrix game, 1-finger/2-finger mixed two-person, zero-sum game, graphical solution.
Quiz: Quiz #6, due by October 7, 2025, 9:30am
Homework: HW #3 due by 9:30am on October 21, 2025
Lecture #7 (Module 1.2.2)--
NOTE:
1. NO CLASS DUE TO FALL BREAK
2. QUIZ #7 IS STILL DUE BUT SHIFTED TO OCTOBER 21
3. ALSO QUIZ #8 IS DUE ON OCTOBER 21
Due Date: October 14, 2025
Duration: 2 hours 3 minutes 47 seconds
Slides: 46-50 (approximately)
Topics: Minimax theorem related to saddle points (generalizing Strong Duality theorem from linear programming)+ proof for two-person, zero-sum games, see for example https://www.cantorsparadise.com/john-von-neumanns-minimax-theorem-70042d077bd9, Von Neumann and others, Lemke's (Lemke and Howson's) result to solved two-person bimatrix games via a certain linear complementarity problem (LCP) + proof, see for example https://www.jstor.org/stable/2946376#metadata_info_tab_contents, Braess' Paradox in transportation and other systems.
Quiz: Quiz #7, due by October 21, 2025, 9:30am
Homework: HW #3 due by 9:30am on October 21, 2025
Lecture #8 (Module 1.2.3)
Due Date: October 21, 2025
Duration: 1 hour 55 minutes 36 seconds
Slides: 49-71 (approximately)
Topics: Energy and transportation equilibrium models, duopoly in energy (or other) production- analysis, Wardrop traffic equilibrium problem formulation and results, P-matrix property, bisymmetric matrix, notions of monotonicity of vector-valued functions (monotone, strictly monotone, strongly monotone), Karamardian results for MCP (strictly monotone or strongly monotone functions).
Quiz: Quiz #8, due by October 21, 2025, 9:30am
Homework: HW #3 due by 9:30am on October 21, 2025
Homework: HW #4 given out, due by 9:30am on October 28, 2025
Lecture #9 (Module 1.2.4)
Due Date: October 28, 2025
Duration: 2 hours 8 minutes 45 seconds
Slides: 71-100 (approximately)
Topics: Producer duopoly MCP, P-matrices, Spatial Price Equilibrium Problem, linear program expressed as an MCP, Wardrop Traffic Equilibrium Problem, notions of monotonicity for vector-valued functions.
Quiz: Quiz #9, due by October 28, 2025, 9:30am
Homework: HW #4 given out, due by 9:30am on October 28, 2025
STUDY FOR EXAM #1 on November 4, 2025
Lecture #10 (Module 1.2.5)- first part
Due Date: November 4, 2025
Duration: 0 hours 58 minutes 50 seconds
Slides: 101-141 (approximately)
Topics: Stochastic MCPs, generalized Nash equilibria formulations/examples, theorems and as special cases of the quasi-variational inequality problem, pseudo-convex functions.
Lecture #10 (Module 1.2.6)-second part
Due Date: November 4, 2025
Duration: 1 hour 1 minute 43 seconds
Slides: 142-166 (approximately)
Topics: Algorithms for solving LCPs: pivotal methods (i.e., Lemke's method), iterative methods (i.e., splitting methods.) comparison of these two types of algorithms. examples of these algorithms, comparison with the Simplex Method and Lemke's method, selected theoretical results for these choices of algorithms
Quiz: Quiz #10 (on both the first and second parts), due by November 4, 2025, 9:30am
>>>IN-CLASS EXAM #1: 9:30-12:00 (in-person students, to be arranged for online students)
Homework: HW #5 given out, due by 9:30am on November 18, 2025
>>>>START OF MODULE 2.1<<<<
Lecture #11 (Module 2.1 )-- first part
Due Date: November 11, 2025
Duration: 0 hours 18 minutes 45 seconds
Slides: 1-17 (approximately)
Topics: Overview of Module 2.1 five parts, small mixed-integer nonlinear program (MINLP) examples, complementarity conditions, disjunctive constraints/big M method applied to a lower-level problem combined with an upper level expressed as an instance of an MINLP.
Lecture #11 (Module 2.2)-second part
Due Date: November 11, 2025
Duration: 0 hours 49 minutes 15 seconds
Slides: 1- 46
Topics: Two-level equilibrium problems expressed as:mathematical program with equilibrium constraints (MPECs) vs mathematical programs with complementarity constraints (MPCCs), examples of two-level equilibrium problems, several small examples of MPECs, power market example, triopoly example with 1 leader as the top player, a duopoly as the bottom-level problem, disjunctive constraints/big M approach vs. SOS1 approach vs. regularization approach.
Quiz: Quiz #11 (on both the first and second parts), due by November 11, 2025, 9:30am
Homework: HW #5 given out, due by 9:30am on November 18, 2025
Lecture #12 (Module 2.3)- first part
Due Date: November 18, 2025
Duration: 1 hour 5 minutes 47 seconds
Slides: 1- 61
Topics: Benders decomposition for linear programs, complicating variables (Benders, outer approximation) vs. complicating constraints (Dantzig-Wolfe, inner approximation), examples, theoretical results, subgradients and the subdifferential, optimal-value function for a minimization LP and changing the right-hand side, Lagrange multiplier as subgradient of the optimal-value function, LP basics review, example of solving a LP using Benders describing the
Lecture #12 (Module 2.4)-second part
Due Date: November 18, 2025
Duration: 1 hour 27 minutes 42 seconds
Slides: 1- 79
Topics: Nonlinear duality theory and connection to the generalized Benders method, nonlinear duality theory and game theory connection, review of linear programming duality theory: weak and strong duality theorems, nonlinear duality expressed as "game" between two players Peter and Harriet (zero-sum game), weak duality for nonlinear programs, saddle points, strong duality for nonlinear programs, relationship to the MinMax result, Lagrangian duality, primal and dual functions, duality gap, examples, Wolfe dual
Quiz: Quiz #12, due by November 18, 2025, 9:30am
Homework: HW #5 due by 9:30am on November 18, 2025
Lecture #13 (Module 2.5)
Due Date: November 25, 2025
Duration: 1 hour 20 minutes 47 seconds
Slides: 1- 79
Topics: Generalized Benders method, standard primal and dual problems, sample problem, GAMS code for the sample problem, master problem, subproblem, theoretical results.
Quiz: Quiz #13, due by November 25, 2025, 9:30am
Due Date: December 2, 2025
Open class for help with projects and/or review for exam
Due Date: December 9, 2025
Project Presentations and reports due
>>>IN-CLASS EXAM #2<<<
In-person students: This will be decided by the registar's office.
Online students: Some time on the same day as the in-person students, to be arranged with a proctor.
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