Related papers: Visualization of the {\epsilon}-Subdifferential of…
The paper is devoted to the study, characterizations, and applications of variational convexity of functions, the property that has been recently introduced by Rockafellar together with its strong counterpart. First we show that these…
We introduce a compositional framework for convex analysis based on the notion of convex bifunction of Rockafellar. This framework is well-suited to graphical reasoning, and exhibits rich dualities such as the Legendre-Fenchel transform,…
This work provides formulae for the $\epsilon$-subdifferential of integral functions in the framework of complete $\sigma$-finite measure spaces and locally convex spaces. In this work we present here new formulae for this…
Set-functions appear in many areas of computer science and applied mathematics, such as machine learning, computer vision, operations research or electrical networks. Among these set-functions, submodular functions play an important role,…
Functions with uniform sublevel sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used in multicriteria optimization, decision theory, mathematical…
We investigate inexact proximity operators for weakly convex functions. To this aim, we derive sum rules for proximal {\epsilon}-subdifferentials, by incorporating the moduli of weak convexity of the functions into the respective formulas.…
Separability of multivariate functions alleviates the difficulty in finding a minimum or maximum value of a function such that an optimal solution can be searched by solving several disjoint problems with lower dimensionalities. In most of…
In this paper, we introduce a new concept, namely $\epsilon$-arithmetics, for real vectors of any fixed dimension. The basic idea is to use vectors of rational values (called rational vectors) to approximate vectors of real values of the…
It is shown that a piecewise linear function can be represented as a Max-Min polynomial of its linear components.
The paper addresses the study and applications of a broad class of extended-real-valued functions, known as optimal value or marginal functions, which are frequently appeared in variational analysis, parametric optimization, and a variety…
This paper develops a correspondence relating convex hulls of fractional functions with those of polynomial functions over the same domain. Using this result, we develop a number of new reformulations and relaxations for fractional…
Submodular Functions are a special class of set functions, which generalize several information-theoretic quantities such as entropy and mutual information [1]. Submodular functions have subgradients and subdifferentials [2] and admit…
A key idea in convex optimization theory is to use well-structured affine functions to approximate general functions, leading to impactful developments in conjugate functions and convex duality theory. This raises the question: what are the…
We present a new method for proving Correa-Jofr\'e-Thibault theorem that monotonicity of subdifferential implies convexity of the function. This new method is based on barrier functions. Barrier functions help overcome some of the main…
Assume that f is a strict convex function with a unique minimum in R^n. We divide the vector of n-variables to d groups of vector subvariables with d at least two. We assume that we can find the partial minimum of f with respect to each…
A linear functional of an object from a convex symmetric set can be optimally estimated, in a worst-case sense, by a linear functional of observations made on the object. This well-known fact is extended here to a nonlinear setting: other…
We present two approximate versions of the proximal subgradient method for minimizing the sum of two convex functions (not necessarily differentiable). The algorithms involve, at each iteration, inexact evaluations of the proximal operator…
This paper demonstrates that the space of piecewise smooth functions can be well approximated by the space of functions defined by a set of simple (non-linear) operations on smooth uniform splines. The examples include bivariate functions…
Concerning bivariate least squares linear regression, the classical approach pursued for functional models in earlier attempts is reviewed using a new formalism in terms of deviation (matrix) traces. Within the framework of classical error…
We consider unsupervised classification by means of a latent multinomial variable which categorizes a scalar response into one of L components of a mixture model. This process can be thought as a hierarchical model with first level…