Models and methods for integer programming under structure and uncertainty

Abstract

Integer Programming (IP) is a fundamental tool of modern decision-making; however, these problems are highly combinatorial and often require exponential time to solve. Such computational difficulty can also be enhanced by the specific algebraic structures of the solution space. Each combinatorial model has a unique structure, enabling the development of different methods to use these particular aspects. This thesis investigates two independent but complementary frontiers: exploiting mathematical structure in deterministic optimization and handling uncertainty in stochastic settings.

In the deterministic domain, we introduce prime programming, a subclass of integer programming whose decision variables are confined to prime numbers. Utilizing this algebraic structure, we develop a specific branch-and-bound method that employs number-theoretic branching rules designed for prime solutions. Next, we consider a quadratically constrained quadratic programming model characterized by nonconvex Euclidean norm constraints. Leveraging the geometric structure of these constraints, we derive a spatial branch-and-bound approach to solve the model more efficiently. This method is based on the intuition of replacing the Euclidean ball with a polyhedron that is iteratively refined to approximate the ball more closely. This approach provides a global optimality guarantee for k-hyperplane clustering problems where traditional heuristic methods frequently fail. We further provide a practical application of mixed-integer quadratic programming to rationalize the Household Waste Recycling Centre (HWRC) network in Hampshire, balancing fiscal efficiency with equitable public access.

In the stochastic domain, this thesis opposes traditional risk-averse paradigms by developing risk-seeking optimization models with probabilistic constraints. In this framework, constraint violations are viewed as lucky scenarios, allowing for a higher number of violations rather than requiring the constraints to be satisfied at a near-certain level. We treat these probabilistic constraints as a union of events and provide an alternative solution methodology; specifically, approximation models based on integer programming that provide valid lower and upper bounds. Finally, a chance-constrained study of the European energy market is provided to quantify the economic value of wind-solar complementarity.

This work collectively illustrates that specialized algorithms for specific frameworks (algebraic, geometric, or probabilistic-constrained) enhance the solvability of complicated optimization problems in mathematics, machine learning, and public policy.

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Identifiers

ORCID for Bismark Singh: orcid.org/0000-0002-6943-657X

ORCID for Joerg Fliege: orcid.org/0000-0002-4459-5419

ORCID for Stefano Coniglio: orcid.org/0000-0001-9568-4385

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