Constrained Multi-Objective Optimization

The samyama-optimization crate (included in the open-source Community Edition) provides full support for multi-objective optimization, including NSGA-II and MOTLBO with the Constrained Dominance Principle for handling complex real-world constraints.

Note: All 22 metaheuristic solvers—including the multi-objective solvers NSGA-II and MOTLBO—are available in the OSS edition. The Enterprise edition adds GPU-accelerated constraint evaluation for large-scale problems.

The Reality of Constraints

In academic problems, objectives like “Minimize Cost” and “Maximize Quality” are often explored in a vacuum. In industry, these objectives must be solved while adhering to hard physical or regulatory constraints:

• Supply Chain: Minimize lead time AND maximize profit, but total warehouse volume cannot exceed 5,000m³.
• Energy: Maximize grid stability AND minimize carbon output, but no single plant can operate at >95% capacity for more than 4 hours.

Constrained Dominance Principle

The samyama-optimization crate implements this principle in the NSGA-II and MOTLBO solvers. Instead of a simple “penalty” approach (which often struggles to find feasible solutions in tight spaces), the selection logic follows a strict hierarchy:

graph TD
    Compare["Compare Solution A vs B"] --> FeasCheck{"Both<br>Feasible?"}

    FeasCheck -- "Yes" --> Pareto["Standard Pareto<br>Dominance"]
    FeasCheck -- "No" --> MixCheck{"One Feasible,<br>One Not?"}

    MixCheck -- "Yes" --> FeasWins["Feasible Solution<br>Always Wins"]
    MixCheck -- "No (both infeasible)" --> Violation["Lower Total<br>Constraint Violation Wins"]

    Pareto --> Select["Selected for<br>Next Generation"]
    FeasWins --> Select
    Violation --> Select

1. Feasibility First: A solution that satisfies all constraints is always preferred over one that violates any constraint.
2. Comparative Violation: Between two infeasible solutions, the one with the lower total constraint violation is preferred.
3. Standard Dominance: Between two feasible solutions, standard Pareto dominance rules apply.

Defining Constraints in Cypher

The algo.or.solve procedure allows for explicit constraint definitions:

CALL algo.or.solve({
  algorithm: 'NSGA2',
  label: 'Generator',
  objectives: ['cost', 'emissions'],
  constraints: [
    { property: 'load', max: 500.0 },
    { property: 'temperature', max: 100.0 }
  ],
  population_size: 100
})
YIELD pareto_front

This advanced logic ensures that the “Pareto Front” returned by the solver contains solutions that are not only optimal but also physically executable, making Samyama a powerful tool for industrial decision-making.