The minimizer of the sum of two strongly convex functions

The optimization problem concerning the determination of the minimizer for the sum of convex functions holds significant importance in the realm of distributed and decentralized optimization. In scenarios where full knowledge of the functions is not available, limiting information to individual mini...

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Bibliographic Details
Published in:Optimization Vol. 74; no. 16; pp. 4357 - 4397
Main Authors: Kuwaranancharoen, Kananart, Sundaram, Shreyas
Format: Journal Article
Language:English
Published: Philadelphia Taylor & Francis 10.12.2025
Taylor & Francis LLC
Subjects:
Approximation
Boundaries
Convex analysis
Convexity
decentralized optimization
fault-tolerant systems
Interiors
Optimization
quadratic functions
strongly convex functions
ISSN:0233-1934, 1029-4945
Online Access:Get full text
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Summary:The optimization problem concerning the determination of the minimizer for the sum of convex functions holds significant importance in the realm of distributed and decentralized optimization. In scenarios where full knowledge of the functions is not available, limiting information to individual minimizers and convexity parameters - either due to privacy concerns or the nature of solution analysis - necessitates an exploration of the region encompassing potential minimizers based solely on these known quantities. The characterization of this region becomes notably intricate when dealing with multivariate strongly convex functions compared to the univariate case. This paper contributes outer and inner approximations for the region harboring the minimizer of the sum of two strongly convex functions, given a constraint on the norm of the gradient at the minimizer of the sum. Notably, we explicitly delineate the boundaries and interiors of both the outer and inner approximations. Intriguingly, the boundaries as well as the interiors turn out to be identical. Furthermore, we establish that the boundary of the region containing potential minimizers aligns with that of the outer and inner approximations.
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ISSN:0233-1934
1029-4945
DOI:10.1080/02331934.2024.2402923