A MEASURE-BASED METHODOLOGICAL FRAMEWORK FOR CONTINUOUS MAXIMUM COVERAGE: GEOMETRIC FORMULATION AND HYBRID SWARM OPTIMIZATION

Sergiy Yakovlev, Sergiy Shekhovtsov, Yehor Havryliuk

Abstract


Today, technical systems that rely on spatial monitoring, autonomous decision-making, and distributed sensing are widely used in industry, energy, environmental protection, telecommunications, and security. Many modern infrastructures, such as sensor networks, robotic platforms, surveillance systems, and environmental monitoring grids, require reliable geometric coverage of a given territory under operational constraints. Achieving high-quality coverage in the presence of forbidden zones, irregular boundaries, and heterogeneous service requirements represents an important scientific and practical challenge. The goal of this study is to develop a measure-based methodological framework for the formulation and computational treatment of continuous maximum coverage problems involving geometric objects of arbitrary shapes. To achieve this goal, the paper addresses the formalization of continuous coverage problems through set measures and configuration parameters, the construction of scalable methods for evaluating coverage areas with controllable accuracy, and the integration of global and local optimization mechanisms as computational instruments within the proposed approach, rather than as objects of algorithmic comparison. The subject of the research concerns methodological principles of hybrid swarm-based optimization applied to high-dimensional geometric configurations of covering objects. The proposed framework combines concepts from computational geometry, inclusion–exclusion formulas, Monte Carlo sampling, and population-based optimization, which are interpreted as geometric motion models in a continuous parameter space. The study provides several important methodological results. A general nonlinear optimization formulation of the continuous maximum coverage problem based on set measures is constructed, and computational schemes for estimating coverage areas under different accuracy–complexity trade-offs are introduced. Several families of hybrid swarm-based methods are adapted, interpreted within a unified geometric framework, and provided with evidence-based guidelines for parameter selection and hybridization configuration. The reported numerical results demonstrate the practical applicability of the proposed framework and illustrate typical effects of hybridization on coverage quality under a fixed computational budget. The scientific novelty of the work consists of three contributions. A geometry-agnostic measure-based formulation is developed that simultaneously accommodates arbitrary object shapes, non-convex target regions, forbidden zones of general form, and continuous rotation parameters within a single unified framework, without relying on shape-specific predicates or problem-dependent geometric decompositions. A unified geometric reinterpretation of four swarm intelligence paradigms is established, in which algorithmic update rules are interpreted as coordinated rigid motions in the continuous configuration space, providing a principled geometry-driven basis for algorithm selection depending on the structural properties of the feasible domain. A structured hybridization scheme is formulated for integrating global stochastic search with deterministic local refinement applied to the penalized measure-based coverage functional, with evidence-based guidelines for activation frequency, stopping criteria, and gradient approximation under non-smooth geometric objectives. The proposed framework provides a flexible and extensible methodological foundation for coverage modeling in engineering, energy, and environmental monitoring systems where geometric complexity, forbidden zones, and placement constraints are unavoidable

Keywords


mathematical model, continuous maximum coverage, measure-based methodology, swarm intelligence, memetic optimization, computational geometry, geometric modelling

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DOI: https://doi.org/10.32620/reks.2026.2.01

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