Integral Resolvent and Proximal Mixtures

Minh N. Bùi () and Patrick L. Combettes ()
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Minh N. Bùi: University of Graz
Patrick L. Combettes: North Carolina State University

Journal of Optimization Theory and Applications, 2024, vol. 203, issue 3, No 10, 2328-2353

Abstract: Abstract Using the theory of Hilbert direct integrals, we introduce and study a monotonicity-preserving operation, termed the integral resolvent mixture. It combines arbitrary families of monotone operators acting on different spaces and linear operators. As a special case, we investigate the resolvent expectation, an operation which combines monotone operators in such a way that the resulting resolvent is the Lebesgue expectation of the individual resolvents. Along the same lines, we introduce an operation that mixes arbitrary families of convex functions defined on different spaces and linear operators to create a composite convex function. Such constructs have so far been limited to finite families of operators and functions. The subdifferential of the integral proximal mixture is shown to be the integral resolvent mixture of the individual subdifferentials. Applications to the relaxation of systems of composite monotone inclusions are presented.

Keywords: Hilbert direct integral; Integral proximal mixture; Integral resolvent mixture; Monotone operator; Proximal expectation; Resolvent expectation (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02466-7

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