A Functional Analysis-Based Optimization Framework for Large-Scale Industrial Systems
Keywords:
Functional analysis, operator theory, hilbert space, variational inequality, optimization framework, industrial systems.Abstract
This paper presents a new optimization framework based on functional analysis for large-scale industrial system optimization, using operator theory, Hilbert space representations, and variational inequality formulations. In the proposed framework, the industrial decision space is represented as a Hilbert space, in which the complicated interactions among multiple system elements are captured by an abstract operator mapping. Through the mathematical properties of monotone and strongly monotone operators, the existence and uniqueness of the optimal operation solution can be guaranteed under proper mathematical assumptions. A projected functional operator iteration algorithm is also presented, whose convergence property is mathematically validated through contraction mapping. This framework is numerically evaluated with a large-scale industrial resource allocation problem and is benchmarked with traditional optimization approaches, such as gradient optimization, Alternating Direction Method of Multipliers (ADMM), and the classical projection method. Numerical results reveal the superior performance of this new optimization framework in terms of convergence stability, optimization error, constraint violation, and computational efficiency.
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