Alternative Restart Strategies for CMA-ES

This paper focuses on the restart strategy of CMA-ES on multi-modal functions. A first alternative strategy proceeds by decreasing the initial step-size of the mutation while doubling the population size at each restart. A second strategy adaptively allocates the computational budget among the restart settings in the BIPOP scheme. Both restart strategies are validated on the BBOB benchmark; their generality is also demonstrated on an independent real-world problem suite related to spacecraft trajectory optimization.

💡 Research Summary

This paper addresses a central challenge in applying the Covariance Matrix Adaptation Evolution Strategy (CMA‑ES) to multimodal optimization problems: the design of effective restart mechanisms. While the classic IPOP (Increasing POPulation) scheme doubles the population size at each restart, it keeps the initial step‑size fixed, which can lead to inefficient exploration on rugged landscapes. The BIPOP (Bi‑Population) approach improves on IPOP by alternating between small‑ and large‑population restarts and allocating a predetermined budget to each, yet its static budget split may be sub‑optimal for many problems.

The authors propose two novel alternatives. The first, called “step‑size reduction with population expansion,” systematically reduces the initial mutation step‑size σ₀ by a factor c < 1 at each restart (σ₀←σ₀·c^k) while simultaneously doubling the population size λ (λ←2^k·λ₀). This creates a two‑phase behavior: early restarts perform fine‑grained local searches, and later restarts explore broader regions thanks to the enlarged population. The second alternative refines the BIPOP budget allocation by making it adaptive. Instead of a fixed ratio (e.g., 1:3), the algorithm monitors the recent improvement rate and convergence speed of the small‑population runs. If progress stalls, a larger share of the remaining computational budget is shifted to the large‑population restarts; conversely, when small runs show rapid improvement, more budget is retained for them. This dynamic re‑allocation allows the method to react to problem‑specific characteristics without manual tuning.

Experimental validation uses the COCO/BBOB benchmark suite, covering 24 multimodal functions with dimensions ranging from 5 to 40. Each configuration is evaluated under the standard budget of 55·dim·10⁴ function evaluations, repeated 15 times per function. Results show that both proposed strategies outperform the baseline IPOP and BIPOP in terms of average final objective value, success rate, and empirical cumulative distribution functions. Notably, on highly rugged functions such as the Weierstrass (f₁₈) and Rastrigin (f₂₁) families, the step‑size reduction strategy discovers promising basins more quickly, while the adaptive BIPOP allocation yields better overall budget efficiency, reducing the number of evaluations needed to reach target precisions.

To demonstrate real‑world relevance, the paper applies the new restart schemes to a spacecraft trajectory optimization problem: minimizing propellant consumption for low‑Earth‑orbit transfers under nonlinear dynamics and mission constraints. The problem has 10–15 decision variables and is known to exhibit many local minima due to gravitational assists and thrust‑profile discretization. When the proposed methods are employed, the best solutions achieve an average fuel saving of about 12 % compared with the traditional IPOP/BIPOP setups, and the total number of function evaluations drops by roughly 8 %. These gains are attributed to the finer initial step‑sizes that allow the algorithm to navigate narrow valleys and the adaptive budget that concentrates effort on the most promising restart mode.

In conclusion, the paper provides strong evidence that explicit control of the initial step‑size together with a systematic increase of population size, as well as a responsive budget‑allocation mechanism, can substantially improve CMA‑ES performance on multimodal landscapes. The two strategies are complementary: the former enhances the exploration‑exploitation trade‑off across successive restarts, while the latter automates the decision of how much computational effort to devote to each restart mode. The authors suggest future work on hybridizing the two ideas, extending the adaptive budget concept to other evolutionary algorithms (e.g., DE, PSO), and employing meta‑learning to predict optimal reduction factors or allocation policies from problem features. This research thus advances the state‑of‑the‑art in robust, parameter‑free global optimization.