Benchmarking Metaheuristics for Mixed-Integer Optimization

Provable Speedups From Dynamic Population Sizes in Evolutionary Algorithms for Multiobjective Optimization

This paper introduces the bi-objective problem class CLIMB and analyzes the runtime of GSEMO and the widely used NSGA-II on this problem, and proves that GSEMO and NSGA-II-DYN, a version of NSGA-II with dynamic population sizes, can find the Pareto front of CLIMB in expected fitness evaluations.

Improving Metaheuristic E ffi ciency for Stochastic Optimization by Sequential Predictive Sampling

Quantifying the Explore-Exploit Trade-off in Metaheuristics via Normalised Positional Diversity

A continuous development of metaheuristics necessitates a deeper understanding of internal search mechanisms, specifically the transition between exploration and exploitation, which is often overlooked in favour of final fitness values. This study introduces a standardized framework based on a Normalized Positional Diversity Index (D*) to quantify optimizer behaviour. We demonstrate that D* is a geometric generalization of existing measures, such as the one proposed by Hussain et al. [1], by replacing stochastic, path-dependent historical maximums with a fixed global upper bound anchored to the search space geometry. This transformation eliminates post-hoc bias and provides a scale-independent diagnostic. From D*, additional metrics including Area Under the Curve (AUC) and crossover point (txover) are derived to identify exact phase transitions. We evaluate five optimizers (GWO, DE, PSO, AGOA, and EES) across CEC 2020 benchmarks. Results confirm that D* offers consistent diversity monitoring with lower Coefficients of Variation compared to existing measures. Code and supplementary materials are available at: [https://github.com/saiddddd/D-Star-Metric-Optimization].

Solution of the Binary Knapsack Problem Using the Secretary Bird Optimization Algorithm

Many nature-inspired metaheuristic algorithms proposed in the literature are initially designed to operate in continuous search spaces. However, real-world problems are not limited to continuous domains; they also include discrete, combinatorial, and binary problem types.In this study, the performance of the Secretary Bird Optimization Algorithm (SBOA), a recently introduced metaheuristic method, is investigated on the knapsack problem, which belongs to the class of binary optimization problems. Numerous transfer functions have been proposed in the literature to transform continuous search spaces into binary ones. In this work, S-shaped transfer functions are employed for this purpose.Through this transformation, SBOA—originally designed for continuous domains—is adapted to the binary search space and referred to as the Binary SBOA (BSBOA). The performance of BSBOA is evaluated on well-known benchmark knapsack problem instances. While BSBOA achieves optimal solutions on 10 small-scale benchmark problems, it produces near-optimal results for 18 large-scale binary knapsack instances.This study aims to address the SBOA binary optimization problem and evaluate its performance in such an environment.

MixDiBB: Distributed Black-Box Optimization for Mixed-Parameter Search Spaces

Many real-world black-box optimization problems involve a mix of continuous, integer, and categorical parameters, where gradients are unavailable and gradient-based methods are inapplicable. An alternative can be found within the family of Black-Box Optimization (BBO) methods, which includes broadly adopted approaches such as Genetic Algorithms and Evolution Strategies. The most successful implementations maintain full covariance information between parameters, yielding high sample-efficiency at the cost of quadratic performance, which restricts their applicability to low-dimensional applications. This paper presents Mixed-Parameter DiBB (MixDiBB), a highly parallelizable and distributable system that scales state-of-the-art BBO algorithms to high-dimensional problems by leveraging partial separability, which is commonly found in real-world problems. This is achieved by partitioning the parameter space based on inter-parameter relations, then applying a custom, potentially different optimization method to each partition. MixDiBB is particularly effective in applications with mixed parameter types, as it allows users to select the ideal algorithm for each parameter subset. Our results demonstrate high performance on standard benchmarks, but with three orders of magnitude more dimensions than typically showcased in the literature, and within shorter wall-clock time.

Efficient Alternative Mixed-Integer Non-Linear Programs and a Customized Genetic-Based Hybrid Metaheuristic for a Resource-Constrained Project-Scheduling Problem with a Flexible Network

This paper aims to present efficient alternative mixed-integer non-linear programming models and a customized hybrid metaheuristic, the Genetic-Based Algorithm (GBA), for a resource-constrained project-scheduling problem with a flexible network structure (RCPSP-FNS). We also consider the cost–time trade-off in the problem with a flexible network by using activity-duration compression. We present three approaches to solve the problem, including a mixed-integer non-linear program (MINLP) using binary variables representing activity completion times (MINLP1), an alternative mixed-integer non-linear program using integer variables representing activity-completion times (MINLP2) that has not presented before in RCPSP-FNS modeling, and the GBA. A total of 35 different problems are solved to examine the computational efficiency of the solution approaches. The MINLP1 and MINLP2 models are both solved by the GEKKO solver. The results indicate that solving the MINLP2 model can reach the optimal objective value obtained by solving the MINLP1 model in significantly less time. In addition, the proposed genetic-based algorithm can solve some large problems in a more efficient way in comparison to solving MINLP1 by using GEKKO. However, solving the MINLP2 model using GEKKO is the most efficient solution approach in comparison to both MINLP1 and the proposed genetic-based algorithm. MINLP2 can be solved to proven optimality (in much less time) for problems in which the MINLP1 model can, at most, reach near-optimal solutions.