interactive resource
lpviz
Explore feasible regions and compare several linear-programming algorithms in the browser.
Open resourceISE 5405 · Optimization I
Fall 2026
Linear-programming theory, geometry, algorithms, modeling, and computation.
- Instructor
- Jiaxiang (Jason) Li
- Meetings
- Tuesdays and Thursdays, 9:30–10:45 a.m.
- Location
- Durham Hall 261
course materials
Lectures
- lecture 0
Syllabus & Logistics
Course logistics, instructional team, materials, assessment, policies, key dates, and first-week setup.
- lecture 1
Introduction & Linear Optimization
Why optimization matters, LP language and modeling, graphical solution concepts, canonical forms, and piecewise-linear reformulations.
- lecture 2
Proof Techniques & LaTeX
Logical implication, proof techniques, counterexamples, mathematical writing, and a practical introduction to LaTeX.
- lecture 3
The Geometry of Linear Programming I
Convexity, polyhedra, extreme points, vertices, basic feasible solutions, and adjacency.
- lecture 4
The Geometry of Linear Programming II
Standard-form bases, degeneracy, existence and optimality of extreme points, convex hulls, and Fourier–Motzkin elimination.
- lecture 5
Simplex I: Basis Directions and Reduced Costs
Basis partitions, equality-preserving feasible directions, maximum feasible steps, reduced costs, and the basis optimality test.
- lecture 6
Simplex II: Cycling and Anticycling
Cycling, lexicographic pivoting, Bland’s rule, and finite termination, with a complete three-dimensional worked example.
- lecture 7
Simplex III: Two-Phase Simplex
Find an initial feasible basis with Phase I, remove artificial variables, restore the original objective, and complete the two-phase method.
- lecture 8
Simplex IV: Column Geometry and Efficiency
Column geometry and a geometric pivot, the textbook long-path example, Dantzig’s rule on the n-dimensional Klee–Minty family, the Hirsch counterexample, and an introduction to ellipsoid and interior-point methods.
- lecture 9
Duality I: Bounds and the Dual Problem
Build objective bounds from weighted constraints, form LP duals, and use weak duality to verify optimality certificates.
- lecture 10
Duality II: Optimality and Complementary Slackness
Connect strong duality, complementary slackness, and simplex reduced costs through complete numerical optimality certificates.
- lecture 11
Duality III: Prices and Certificates
Interpret dual prices, connect reduced costs to column geometry, and use row combinations to certify infeasibility.
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