One-parameter Filled Function Method for Non-convex Multi-objective Optimization Problems

Abstract: In this paper, a new one-parameter filled function approach is developed for nonlinear multi-objective optimization. Inspired by key filled function ideas from single-objective optimization, the proposed method is adapted to the multi-objective setting. It avoids scalarization weights and does not impose any prior preference ordering among objectives. A descent-based procedure is first applied to obtain a local weak efficient solution. An associated filled function is then constructed and used to derive another local weak efficient solution that improves upon the current one. Repeating these phases drives the search toward global weak efficiency and yields an approximation of the global Pareto front, including in non-convex problems where multiple local Pareto fronts may exist. Numerical experiments on a set of test problems demonstrate the effectiveness of the proposed approach relative to existing methods.

• The paper introduces a one-parameter filled function method that bypasses scalarization to recover global Pareto fronts in challenging nonconvex problems.
• It develops a rigorous algorithm with adaptive parameter scheduling and theoretical descent guarantees for navigating disconnected efficient sets.
• Numerical results demonstrate superior purity and hypervolume metrics alongside reduced function evaluations compared to NSGA-II and MOSQCQP.

One-parameter Filled Function Method for Non-convex Multi-objective Optimization

Introduction and Motivation

The paper introduces a new one-parameter filled function method (OMFFM) targeted at non-convex multi-objective optimization problems, motivated by the limitations of conventional scalarization techniques and heuristics. Scalarization, which transforms multi-objective problems into single-objective formulations via weighted aggregations or constraints, generally struggles in non-convex scenarios to recover the full Pareto front and places excessive dependency on parameter tuning. Metaheuristics and evolutionary algorithms, although flexible, lack theoretical convergence guarantees and are computationally demanding. Recent descent-based methods—extensions of classical single-objective descent—have only attained local Pareto-optimality in non-convex landscapes, leaving unaddressed issues of multiple distinct local Pareto fronts and premature convergence.

The filled function paradigm in global optimization modifies the objective landscape to facilitate escape from local minima; its extension to the multi-objective context for global Pareto front approximation is nontrivial, particularly due to the absence of a total order among objectives and disconnected efficient sets. The paper builds systematically upon refined one-parameter filled function techniques in single-objective global optimization, leveraging their stability and theoretical descent guarantees.

Theoretical Foundations

The filled function methodology is formalized via a one-parameter auxiliary function, $F_{,}$, defined componentwise for each objective. The main filled function class is

$$F_{j,,}(y) = -\|y-\|^2 + \varphi_\nu({j}(y) - {j}()), \quad j=1,\dots,m$$

with

$$\varphi_\nu(t) = \begin{cases} -\nu t^3 & t \ge 0 \ -\frac{1}{\nu} t^2 & t < 0 \end{cases}$$

where $\nu > 0$ is the sole tuning parameter. This construction ensures smoothness ($C^1$) and preserves theoretical descent properties for all objective components near superior points, facilitating escape from local Pareto traps. The filled function is constructed on the basis of three conditions guaranteeing that: (F1) the current point is a local weak efficient solution for $-F_{,}$; (F2) no critical point is admissible in the non-improving region; (F3) strict improvement is possible across disconnected Pareto fronts under appropriate parameter selection.

The proposed approach avoids scalarization weights, does not require any prior ordering among objectives, and admits an algorithmic scheme to dynamically traverse the Pareto landscape. The parameter $\nu$ is chosen according to a theoretically derived admissible interval, ensuring that descent directions exist and that iterates can successfully escape local non-dominated regions.

Algorithmic Framework

The OMFFM algorithm follows a multi-phase workflow:

1. Initialization: Multiple well-distributed initial points are generated and refined to satisfy necessary conditions. Ideal and nadir vectors for each objective are estimated via single-objective global descent.
2. Local Phase: Starting from a current point, a local multi-objective solver is applied to determine a local weak efficient solution, updating the archive of Pareto vectors.
3. Global Phase: For each candidate outside a defined neighborhood, the filled function is used to compute descent directions. The algorithm employs adaptive parameter scheduling, line search, and restart mechanisms, ensuring robust performance with low function evaluation cost. When strict multi-objective improvement is realized, the process returns to the local phase.

The filled function enables principled restart/escape strategies from local Pareto traps, efficiently facilitating traversal between disconnected Pareto fronts. Upon completion, non-dominated filtering yields approximations of both the conventional and filled function Pareto fronts.

Numerical Results and Performance Analysis

Comprehensive MATLAB implementation and benchmarking confirm OMFFM's efficacy across a broad suite of test problems, including high-dimensional and non-convex instances. The method is evaluated against deterministic MOSQCQP solvers and NSGA-II, utilizing purity, $\Delta$-spread, $\Gamma$-spread, hypervolume, and function evaluation indicators. OMFFM outperforms competitors on purity and hypervolume, demonstrating superior convergence to globally efficient solutions and robust behavior in complex landscapes. It achieves improved $\Gamma$-spread relative to MOSQCQP, although a trade-off in uniform approximation ($$F_{j,,}(y) = -\|y-\|^2 + \varphi_\nu({j}(y) - {j}()), \quad j=1,\dots,m$$0) is observed. Notably, OMFFM requires fewer function evaluations than NSGA-II, indicating practical computational efficiency.

Theoretical justification for the algorithm's behavior is linked directly to established filled function properties (theorems proven in the text). Empirical evidence supports OMFFM's scalability and reliability, especially in problems where standard scalarization or heuristics cannot capture intricate non-convex Pareto structures.

Practical and Theoretical Implications

OMFFM represents a rigorous shift from parameter-heavy, heuristic, or scalarization-based paradigms in multi-objective optimization. The method's ability to traverse disconnected Pareto fronts, combined with robust theoretical guarantees, positions it as a preferable approach for global weak efficient solution identification in non-convex settings. Practical implications include enhanced solution quality in engineering, portfolio, environmental, or medical applications where global trade-offs are essential.

From a theoretical perspective, OMFFM's one-parameter structure simplifies tuning, improves algorithmic stability, and extends classical filled function global optimization theory into the multi-objective field. The provided proofs and algorithm design resolve scenarios where objectives are not totally ordered, and efficient sets may be disconnected.

Open challenges include extension to constrained or nonsmooth multi-objective optimization, as noted by the authors. Furthermore, the approach could be adapted for uncertainty-robust optimization, multiobjective Bayesian methods, or derivative-free settings.

Conclusion

OMFFM stands as a theoretically sound, scalable, and computationally efficient method for global Pareto front approximation in non-convex multi-objective optimization. Its filled function-based auxiliary construction, principled parameter selection, and adaptive algorithmic workflow enable escape from local Pareto traps and traversal of disconnected efficient sets without scalarization or preference information. Empirical and theoretical results confirm superior coverage, convergence, and computational performance compared to established deterministic and evolutionary methods. Future research will focus on incorporating constraints, nonsmooth objectives, and uncertainty modeling to further extend OMFFM's applicability (2603.29338).