Doctor of Philosophy (Ph.D.)
Degree Granting Department
Industrial and Management Systems Engineering
Hadi Charkhgard, Ph.D.
Changhyun Kwon, Ph.D.
Alex Savachkin, Ph.D.
Nasir Ghani, Ph.D.
He Zhang, Ph.D.
Machine learning, Mixed integer programming, Multi-objective optimization, Nash bargaining solution, Optimization over the efficient set, Spatial conservation planning
This thesis presents a total of 3 groups of contributions related to multi-objective optimization. The first group includes the development of a new algorithm and an open-source user-friendly package for optimization over the efficient set for bi-objective mixed integer linear programs. The second group includes an application of a special case of optimization over the efficient on conservation planning problems modeled with modern portfolio theory. Finally, the third group presents a machine learning framework to enhance criterion space search algorithms for multi-objective binary linear programming.
In the first group of contributions, this thesis presents the first (criterion space search) algorithm for optimizing a linear function over the set of efficient solutions of bi-objective mixed integer linear programs. The proposed algorithm is developed based on the triangle splitting method (Boland et al.), which can find a full representation of the nondominated frontier of any bi-objective mixed integer linear program. The proposed algorithm is easy to understand and implement, and converges quickly to an optimal solution. An extensive computational study shows the efficacy of the algorithm. Is numerically shown in this thesis that the proposed algorithm can be used to quickly generate a provably high-quality approximate solution because it maintains a lower and an upper bound on the optimal value of the linear function at any point in time. Additionally, this thesis presents OOESAlgorithm.jl, a comprehensive julia package based on the proposed algorithm. The proposed package ex- tends the first implementation of the algorithm by adding two main features: (a) in addition to CPLEX, the package allows employing any single-objective solver supported by Math- ProgBase.jl, for example, GLPK, CPLEX, and SCIP; (b) the package supports execution
on multiple processors and is compatible with the JuMP modeling language. An extensive computational study shows the efficacy of the package and its features.
In the second group of contributions, this thesis presents a Nash bargaining solu- tion approach for spatial conservation planning problems modeled with modern portfolio theory. The proposed modern portfolio optimization formulation corresponds to a spatial conservation planning problem involving two conflicting objectives: maximizing return and minimizing risk. A Nash bargaining solution approach is presented in this thesis to directly compute a desirable Pareto-optimal (nondominated) solution for the proposed bi-objective optimization formulation in natural resource management problems. Numerical examples in this thesis show that to directly compute a Nash bargaining solution, a Binary Quadratically Constrained Quadratic Program (BQCQP) can be solved. This thesis also shows that the proposed approach (implementable with commercial solvers such as CPLEX) can effectively solve the proposed BQCQP for much larger problems than previous approaches published in the ecological literature. The new approach expands considerably the applicability of such optimization methods to address real spatial conservation planning problems.
In the third group of contributions, this thesis investigates the possibility of improving the performance of multi-objective optimization solution approaches using machine learning techniques. Specifically, this thesis focus on multi-objective binary linear programs and employs one of the most effective and recently developed criterion space search algorithms, the so-called KSA, during our study. This algorithm computes all nondominated points of a problem with p objectives by searching on a projected criterion space, i.e., a (p − 1)- dimensional criterion space. This thesis presents an effective and fast learning approach to identify on which projected space the KSA should work, and also presents several generic features that can be used in machine learning techniques for identifying the best-projected space. Finally, a bi-objective optimization-based heuristic for selecting the best subset of the features to overcome the issue of overfitting in learning is presented. Through an extensive computational study, the performance of the proposed learning approach is tested.
Scholar Commons Citation
Sierra Altamiranda, Alvaro Miguel, "Algorithms for Multi-Objective Mixed Integer Programming Problems" (2019). Graduate Theses and Dissertations.