Filled Function Methods for Global Optimization

Filled function methods constitute a class of deterministic global optimisation techniques designed to overcome the proliferation of local minima in complex, multimodal landscapes. By constructing an auxiliary “filled” function that raises the value of the objective within local basins, these methods guide a subsequent local search towards unexplored regions and potential global minimisers. The process alternates between two phases: identification of a local optimum on the original function and minimisation of the filled function to obtain a new starting point for the next local search. Classic filled functions often employ exponential or logarithmic terms, and require careful tuning of parameters to balance basin‐filling with numerical stability. Over the past decade, research has extended the approach to handle non‐differentiable or constrained problems, integer variables and high‐dimensional domains, while reducing sensitivity to initial conditions and parameter settings. Practical applications span engineering design, supply chain planning and combinatorial scheduling, where global solutions are essential to performance. Recent advances have delivered parameterless or single‐parameter constructions that are continuously differentiable and free of overflow issues, and have enhanced robustness through adaptive step sizes and smoothing techniques. Collectively, these developments affirm the enduring role of filled function methods in the global optimisation toolbox and underscore their capacity to deliver reliable solutions in challenging problem domains.

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One recent study introduced a parameterless filled function that avoids exponential and logarithmic components, yielding a continuously differentiable auxiliary surface. By eliminating the need for manual parameter adjustment, the method accelerates convergence on benchmark test suites and demonstrates stable performance across diverse starting points. The authors validated the approach on fourteen standard functions, reporting improved success rates compared with established filled function algorithms.

Another contribution proposed a one‐parameter filled function designed to maintain numerical stability while satisfying the formal requirements of basin‐filling. The single parameter governs the depth of the filled region and can be bounded to prevent overflow, thereby simplifying the tuning process. Computational experiments on well‐known test problems confirmed the efficiency of the method in locating global minimisers with fewer function evaluations and reduced sensitivity to the choice of initial guess.

A third line of work extended filled function ideas to constrained integer programming by constructing a nonparametric auxiliary function tailored to discrete decision variables under box constraints. The resulting global optimisation algorithm integrates a discrete steepest descent phase with filled function minimisation, effectively escaping suboptimal integer‐valued local minima. Numerical examples in combinatorial scheduling and resource allocation illustrate the method’s feasibility and enhanced global search capability.

Filled Function Methods for Global Optimization publication trend

The graph below shows the total number of articles in filled function methods for global optimization across all publications each year (not limited to Nature Index journals).

Technical terms

Filled function: An auxiliary function built to elevate values within local basins of the objective, thereby enabling escape from local minima in a subsequent minimisation phase.

Global minimizer: A point in the feasible domain at which the original objective function attains its lowest overall value.

Multimodal function: An objective function characterised by multiple local minima, which poses challenges for optimisation algorithms seeking the global minimum.

Local search: An iterative optimisation procedure that refines a candidate solution by exploring its immediate neighbourhood for improvements.

References

1. A New Filled Function Method with One Parameter for Global Optimization. Mathematical Problems in Engineering (2013).
2. A New Parameterless Filled Function Method for Global Optimization. Axioms (2022).
3. A New Nonparametric Filled Function Method for Integer Programming Problems with Constraints. Mathematics (2022).