Avoiding Redundant Restarts in Multimodal Global Optimization

TL;DR

The paper addresses wasted evaluations in CMA-ES due to duplicate restarts in multimodal landscapes by formalizing a redundancy measure based on basin visitation and introducing a repelling restart mechanism. It combines Hill-Valley-based basins, tabu-region repulsion, and covariance-aware sampling to deter converging to previously found optima, and evaluates redundancy on BBOB and CEC'13 benchmarks, showing meaningful reductions in redundant restarts on multimodal problems. The results reveal landscape-dependent benefits: notable redundancy reductions on multimodal benchmarks, with potential trade-offs on well-structured global landscapes. The work suggests practical pathways to more budget-efficient black-box global optimization and outlines avenues for refining region shapes, fitness-informed rejection criteria, and smarter restart strategies.

Abstract

Naïve restarts of global optimization solvers when operating on multimodal search landscapes may resemble the Coupon's Collector Problem, with a potential to waste significant function evaluations budget on revisiting the same basins of attractions. In this paper, we assess the degree to which such ``duplicate restarts'' occur on standard multimodal benchmark functions, which defines the \textit{redundancy potential} of each particular landscape. We then propose a repelling mechanism to avoid such wasted restarts with the CMA-ES and investigate its efficacy on test cases with high redundancy potential compared to the standard restart mechanism.

Figures (7)

• Figure 1: A single run of the CMA-ES with a simple u.a.r. restart strategy ($^0 = 2$), optimizing a modified version of the Himmelblau function ($f_{mh}$, see eq. \ref{['eq:modhim']}). Every restart is visualized as a line, which shows the trace followed by the CMA-ES, where the circle shows the start of the trace and the star shows the finally obtained solution. The restart that converges to the global optimum is shown in green, and restarts converging to a new local optimum are shown in yellow. The traces in red show restarts that converge to local optima, which have already been found during previous restarts (yellow).
• Figure 2: Boxplot showing the fraction of the total budget spent by restarts converging to a previously visited basin of attraction, the RRF (Equation \ref{['eq:potential']}). All 24 objective functions from the BBOB benchmark are aggregated over all dimensions and instances for each tested restart strategy.
• Figure 3: The average Relative Redundancy Factor over all instances for the CMA-ES using the 'RESTART' strategy for the BBOB functions with any redundant restarts. The grid shows the RRF (Eq. \ref{['eq:potential']}) per dimension and function individually.
• Figure 4: Fraction of budget which could be saved by avoiding convergence to redundant regions of the search space in subsequent restarts (as defined in Equation \ref{['eq:redundant']}), for 16 functions from the CEC 2013 benchmark, aggregated over all instances for runs with the CMA-ES using different restart strategies.
• Figure 5: Distribution of redundant function evaluation over the BBOB functions where redundant restarts were found in Section \ref{['sec:redundancy']}, i.e. $f_3, f_3, f_{15}, f_{19}, f_{21}, f_{22}, f_{23}$ and $f_{24}$, separated by problem dimensionality. The CMA-ES with the 'RESTART' strategy is compared to the RR-CMA-ES, with different coverage factors $c$.