Matheuristics

Matheuristics is a term used in operations research and computer science to describe a class of algorithms that combine heuristics and metaheuristics with mathematical programming techniques. These methods are designed to solve complex optimization problems, particularly those that are computationally intractable for exact methods within a reasonable time frame. The core idea is to leverage the strengths of both approaches. Mathematical programming provides a rigorous framework for problem formulation and the use of optimization tools like linear programming or integer programming solvers, which can generate feasible solutions or bounds. Heuristics, on the other hand, employ problem-specific rules or shortcuts to quickly find good, though not necessarily optimal, solutions. Metaheuristics offer more general strategies for exploring the solution space and escaping local optima, guiding the search process.

By integrating these elements, matheuristics aim to achieve a better balance between solution quality and computational effort compared to pure heuristic or exact methods. For instance, a matheuristic might use a mathematical programming solver to generate an initial solution or to improve a solution found by a heuristic. Alternatively, it could use heuristic rules to guide the search within a metaheuristic framework, or to identify promising subproblems for a mathematical solver. This hybrid approach has proven effective for a wide range of combinatorial optimization problems, including scheduling, routing, and resource allocation, offering practical solutions for real-world applications where finding the absolute optimum is often secondary to obtaining a high-quality solution quickly.