A Homotopy Framework for Constrained Multiobjective Optimization
We develop a homotopy continuation framework for computing Karush--Kuhn--Tucker (KKT) points of constrained multiobjective optimization problems. The proposed homotopy map continuously deforms an easily solvable system into the KKT system associated with the original multiobjective problem, yielding a deterministic continuation path. Under the stated regularity and reference-point cone avoidance assumptions, we establish the existence of a smooth, bounded continuation path whose set of accumulation points as the homotopy parameter approaches zero is nonempty and consists of KKT points. The framework is implemented using two numerical continuation algorithms including an enhanced pseudo-arclength predictor--corrector strategy with adaptive step-size control and branch-restarting mechanisms to improve robustness for challenging nonconvex and degenerate problems. Numerical experiments demonstrate that the proposed method remains effective even when initialized from nonfeasible points, indicating practical robustness beyond the theoretical guarantees. Comparative studies evaluate the proposed framework against classical scalarization approaches and the evolutionary algorithms. Experiments on benchmark problems, disconnected and nonconvex multiobjective problems, analytically solvable three-objective problems, and a multiobjective IEEE 5-bus optimal power flow problem demonstrate the ability of the framework to compute KKT points in a variety of constrained settings. The results also highlight the effects of initialization and the limitations of continuation-based methods in recovering complete Pareto-front coverage. Overall, the proposed framework provides a deterministic and interpretable approach to computing Pareto-critical (KKT) points in constrained multiobjective optimization.