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Risk Measures in the Form of Infimal Convolution
The properties of risk measures in the form of infimal convolution are analyzed. The dual representation of such measures, their subdifferential,...
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Infimal Convolution and Duality in Problems with Third-Order Discrete and Differential Inclusions
This paper is concerned with the Mayer problem for third-order evolution differential inclusions; to this end, first we use auxiliary problems with...
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Superquantiles at Work: Machine Learning Applications and Efficient Subgradient Computation
R. Tyrell Rockafellar and his collaborators introduced, in a series of works, new regression modeling methods based on the notion of superquantile...
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Well-posedness and Subdifferentials of Optimal Value and Infimal Convolution
We show that well-posedness (namely approximative well-posedness) properties of optimization problems are very efficient tools in subdifferential...
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Convolution operators on weighted spaces of continuous functions and supremal convolution
The convolution of two weighted balls of measures is proved to be contained in a third weighted ball if and only if the supremal convolution of the...
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Duality Problems with Second-Order Polyhedral Discrete and Differential Inclusions
The present paper deals with the theory of duality for the Mayer problem given by second-order polyhedral discrete and differential inclusions....
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Equivalent norms
In this chapter we discuss the basic notion of an equivalent norm (Definition 2 above) in Section 3.1, and for finite-dimensional spaces, in Section...
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Convex Non-convex Variational Models
An important class of computational techniques to solve inverse problems in image processing relies on a variational approach: the optimal output is...
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Duality in the problems of optimal control described by Darboux-type differential inclusions
This paper is devoted to the optimization of the Mayer problem with hyperbolic differential inclusions of the Darboux type and duality. We use the...
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Extremal Points and Sparse Optimization for Generalized Kantorovich–Rubinstein Norms
A precise characterization of the extremal points of sublevel sets of nonsmooth penalties provides both detailed information about minimizers, and...
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On Duality in Second-Order Discrete and Differential Inclusions with Delay
The present paper studies the duality theory for the Mayer problem with second-order evolution differential inclusions with delay and state...
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Fenchel Conjugate and Further Topics in Subdifferentiation
Duality is one of the central themes of convex analysis and its applications. A large part of this chapter is devoted to Fenchel conjugates, their...
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Resolvent and Proximal Compositions
We introduce the resolvent composition, a monotonicity-preserving operation between a linear operator and a set-valued operator, as well as the...
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Game Theory and Its Applications in Imaging and Vision
It is very common to see many terms in a variational model from Imaging and Vision, each aiming to optimize some desirable measure. This is naturally...
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A Notion of Fenchel Conjugate for Set-Valued Map**s
In this paper, we present a novel concept of the Fenchel conjugate for set-valued map**s and investigate its properties in finite and infinite...