Abstract

We perturb the (Minty-Browder monotone) gradients of some convex functions ff, defined on convex open subsets of the Euclidean space with suitable linear transformations AA and provide sufficient conditions on ff and AA in order to obtain the η\eta-monotonicity of the perturbation ∇f+A\nabla f+A for some η∈(−1,0)\eta\in (-1,0). These conditions are given in terms of the magnitude of the Hessian operators associated to ff and the spectra of the operators (A+A∗)/2(A+A^*)/2 and A∗AA^*A and they make the Minty-Browder monotonicity of the perturbation ∇f+A\nabla f+A to fail. While the symmetric perturbations are only considered as instructive examples in the Minty-Browder monotone case, we mostly rely, in the even dimensional context, on linear transformations which can be represented as linear combinations between the identity operator and skew-symmetric isometries. Note that the skew-symmetric isometries of R2m\mathbb{R}^{2m} are complex structures on R2m\mathbb{R}^{2m}. In the bidimensional case the rotations do admit such representations and the perturbations of the gradients of convex functions by rotations were treated before by D. Marian, I. R. Peter, and C. Pintea, A class of generalized monotone operators, J. Math. Anal. Appl. 421 (2015) 1827–1843. We also obtain some global injectivity results for suitable linear perturbations of some gradients of convex functions.

Author information

Contact details are reproduced from the original publication and may be historical.

Cornel Pintea

Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Str. Kogălniceanu 1, 400084 Cluj-Napoca, Romania

cpintea@math.ubbcluj.ro

Tiberiu Trif

Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Str. Kogălniceanu 1, 400084 Cluj-Napoca, Romania

ttrif@math.ubbcluj.ro

Keywords

  • Minty-Browder monotone operator
  • η-monotone operator
  • h-monotone operator

Mathematics Subject Classification

47H05, 15A18, 15B57, 47A12

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Suggested citation

C. Pintea, T. Trif. “The Monotonicity of Perturbed Gradients of Convex Functions.” Journal of Convex Analysis 24 (2017), No. 2, 525–545. https://doi.org/10.68381/jca24035

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press LLC.