Determination of convex functions via subgradients of minimal norm. (English) Zbl 1482.26018
The paper shows that when the norm of the minimal subgradients of two convex proper lower semicontinuous functions bounded from below and defined on a Hilbert space coincide, then these functions coincide up to a constant. Under additional boundary conditions, the same result is achieved for continuous functions defined on open convex domains. Consequently, a positive answer to the conjecture posed in [T.Z. Boulmezaoud et al., SIAM J. Optim. 28, No. 3, 2049–2066 (2018; Zbl 1404.26016)] is provided.
Reviewer: Sorin-Mihai Grad (Paris)
MSC:
| 26B25 | Convexity of real functions of several variables, generalizations |
| 34A60 | Ordinary differential inclusions |
| 34C12 | Monotone systems involving ordinary differential equations |
| 34G20 | Nonlinear differential equations in abstract spaces |
| 49J40 | Variational inequalities |
| 90C56 | Derivative-free methods and methods using generalized derivatives |
Keywords:
subdifferential determination; subgradient flows; Moreau-Yosida approximation; Dirichlet boundary condition; slopeCitations:
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