InteractiveMap of Operations Research
Back to Optimization Core
TheoryAdvanced02.17
Optimization Core

Polyhedral Theory & Cutting Planes

Understand the geometry that makes integer optimization effective.

Overview

Polyhedral Theory & Cutting Planes focuses on understand the geometry that makes integer optimization effective. In the map of OR, it connects Facets, Valid inequalities, Relaxations to decisions that must be modeled, solved, explained, and revised as evidence changes.

Polyhedral theory studies facets, valid inequalities, relaxations, and extended formulations; cutting planes turn that geometry into stronger MIP algorithms. The practical use case is clearest in MIP solvers, Routing, Scheduling, Network design, where the method helps turn constraints and tradeoffs into a decision artifact someone can inspect.

Core ideas

Facets

Facets is a core checkpoint for Polyhedral Theory & Cutting Planes: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

Valid inequalities

Valid inequalities is a core checkpoint for Polyhedral Theory & Cutting Planes: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

Relaxations

Relaxations is a core checkpoint for Polyhedral Theory & Cutting Planes: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

Extended formulations

Extended formulations is a core checkpoint for Polyhedral Theory & Cutting Planes: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

Branch-and-cut

Branch-and-cut is a core checkpoint for Polyhedral Theory & Cutting Planes: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

How to use it

  1. 1Start with MIP solvers: write the decision, time horizon, actors, and objective in operational language.
  2. 2Translate the problem into Facets, Valid inequalities, and Relaxations; define units and data sources for each one.
  3. 3Build a small instance of Polyhedral Theory & Cutting Planes that can be solved or simulated by hand inspection before using full production data.
  4. 4Compare the recommendation against a baseline policy, not just against mathematical optimality.
  5. 5Document assumptions, sensitivity results, and the conditions under which the recommendation should be revisited.

Applications

MIP solversRoutingSchedulingNetwork design
  • MIP solvers: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.
  • Routing: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.
  • Scheduling: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.
  • Network design: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.

Common pitfalls

  • Applying Polyhedral Theory & Cutting Planes because the label sounds appropriate while leaving the actual decision boundary vague.
  • Treating Facets as a technical detail instead of a modeling choice that affects the recommendation.
  • Reporting one answer without showing sensitivity to demand, capacity, costs, or behavioral assumptions.
  • Ignoring implementation details such as data quality, explainability, ownership, and how users will override bad recommendations.

Resources