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MethodAdvanced02.18
Optimization Core

Complementarity & Equilibrium Models

Model markets, traffic, contact, and KKT systems with either-or conditions.

Overview

Complementarity & Equilibrium Models focuses on model markets, traffic, contact, and KKT systems with either-or conditions. In the map of OR, it connects LCP, MCP, Variational inequalities to decisions that must be modeled, solved, explained, and revised as evidence changes.

Complementarity models encode mutually exclusive slackness conditions and equilibrium relationships that arise in economics, engineering, games, and optimization optimality systems. The practical use case is clearest in Traffic assignment, Energy markets, Contact mechanics, Economic equilibria, where the method helps turn constraints and tradeoffs into a decision artifact someone can inspect.

Core ideas

LCP

LCP is a core checkpoint for Complementarity & Equilibrium Models: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

MCP

MCP is a core checkpoint for Complementarity & Equilibrium Models: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

Variational inequalities

Variational inequalities is a core checkpoint for Complementarity & Equilibrium Models: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

KKT systems

KKT systems is a core checkpoint for Complementarity & Equilibrium Models: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

Equilibrium

Equilibrium is a core checkpoint for Complementarity & Equilibrium Models: 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 Traffic assignment: write the decision, time horizon, actors, and objective in operational language.
  2. 2Translate the problem into LCP, MCP, and Variational inequalities; define units and data sources for each one.
  3. 3Build a small instance of Complementarity & Equilibrium Models 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

Traffic assignmentEnergy marketsContact mechanicsEconomic equilibria
  • Traffic assignment: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.
  • Energy markets: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.
  • Contact mechanics: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.
  • Economic equilibria: compare feasible policies, quantify the operating tradeoffs, and make the assumptions behind the recommendation visible.

Common pitfalls

  • Applying Complementarity & Equilibrium Models because the label sounds appropriate while leaving the actual decision boundary vague.
  • Treating LCP 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