Homotopy for Multi-Objective Optimization
Student: Olaoluwa (or Ola) Ogunleye
Advisor: Dr. Guangmin Yao
PhD Committee members: Dr. Jianhua Zhang, Dr. James Greene, Dr. Kathleen Kavanagh, Dr. Emmanuel Asante-Asamani
Monday, May 5th, 2025, 2:00 pm, Snell 177
Zoom Link: Olaoluwa’s Proposal Defense
Abstract
This dissertation presents a homotopy-based framework for solving multi-objective optimization problems by transforming them into a parametrized single-objective formulation using a weighted sum approach. We introduce a novel homotopy map associated with the Karush-Kuhn-Tucker (KKT) conditions and demonstrate that, under a Weak Normal Cone Condition (WNCC), a relaxed variant of classical constraint qualifications. This map retains key properties of continuity and convergence, thereby extending classical assumptions commonly required in the literature.
The main contribution lies in proving the existence, boundedness, and global convergence of solution trajectories generated by the proposed homotopy continuation method. This is achieved through tools from differential geometry and topology, including Sard’s theorem and the classification of one-dimensional manifolds with boundary. We show that the algorithm leads to smooth solution paths that converge to Pareto-optimal KKT points, even in the absence of strict normal cone conditions.
To improve numerical stability and computational efficiency, we incorporate adaptive step size control and predictor-corrector schemes based on Newton-type methods. Computational experiments further validate the theoretical results, illustrating the method’s robustness in locating efficient solutions across diverse initializations.
Future work will enhance the homotopy formulation to allow dynamic evolution of the weight vector λ, improving exploration of the Pareto front. Comparative analyses between Newton and Quasi-Newton approaches, as well as investigations into solver behavior under different initializations, will further refine computational performance. Applications of this framework span diverse fields, including energy systems, Smart Grid Scheduling, and machine learning hyperparameter tuning, each requiring principled trade-offs between competing objectives.