Related papers: Minimax theorems in a fully non-convex setting
Min-max problems have broad applications in machine learning, including learning with non-decomposable loss and learning with robustness to data distribution. Convex-concave min-max problem is an active topic of research with efficient…
We give an example of a convex, finite and lower semicontinuous function whose subdifferential is everywhere empty. This is possible since the function is defined on an incomplete normed space. The function serves as a universal…
Estimation of convex functions finds broad applications in engineering and science, while convex shape constraint gives rise to numerous challenges in asymptotic performance analysis. This paper is devoted to minimax optimal estimation of…
A min-max formula is proved for the minimum of an integer-valued separable discrete convex function where the minimum is taken over the set of integral elements of a box total dual integral (box-TDI) polyhedron. One variant of the theorem…
We provide theory for the computation of convex envelopes of non-convex functionals including an l2-term, and use these to suggest a method for regularizing a more general set of problems. The applications are particularly aimed at…
This paper presents a necessary and sufficient condition for a real-valued function defined on an open and convex subset of a Banach space to be quasi-concave, and a sufficient condition for such a function to be strictly quasi-concave.…
Optimal estimation and inference for both the minimizer and minimum of a convex regression function under the white noise and nonparametric regression models are studied in a nonasymptotic local minimax framework, where the performance of a…
Detecting hidden convexity is one of the tools to address nonconvex minimization problems. After giving a formal definition of hidden convexity, we introduce the notion of conditional infimum, as it will prove instrumental in detecting…
A real valued function $f$ defined on a convex $K$ is anemconvex function iff it satisfies $$ f((x+y)/2) \le (f(x)+f(y))/2 + 1. $$ A thorough study of approximately convex functions is made. The principal results are a sharp universal upper…
Two different proofs for an inf-sup type representation formula (minimax formula) of the additive eigenvalues corresponding to first-order Hamilton-Jacobi equations are given for quasiconvex (level-set convex) Hamiltonians not necessarily…
Nearly convex sets play important roles in convex analysis, optimization and theory of monotone operators. We give a systematic study of nearly convex sets, and construct examples of subdifferentials of lower semicontinuous convex functions…
Integrally convex functions constitute a fundamental function class in discrete convex analysis, including M-convex functions, L-convex functions, and many others. This paper aims at a rather comprehensive survey of recent results on…
In this paper, we introduce new properties of the relative interior calculus for nearly convex sets, functions, and set-valued mappings. These properties are important for the development of duality theory in optimization. Then we…
We develop an approach for estimating models described via conditional moment restrictions, with a prototypical application being non-parametric instrumental variable regression. We introduce a min-max criterion function, under which the…
We present a new neural network to approximate convex functions. This network has the particularity to approximate the function with cuts and can be easily adapted to partial convexity. We give an universal approximation theorem in the full…
In this paper, we provide a number of subdifferential formulas for a class of nonconvex infimal convolutions in normed spaces. The formulas obtained unify several results on subdifferentials of the distance function and the minimal time…
In the recent paper \cite{Hos}, surjective isometries, not necessarily linear, $T: {\rm AC}(X,E) \longrightarrow {\rm AC}(Y,F)$ between vector-valued absolutely continuous functions on compact subsets $X$ and $Y$ of the real line, has been…
This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operator convex functions, and strongly operator convex functions. Strongly operator convex functions were previously treated in [3] and…
Discrete Fenchel duality is one of the central issues in discrete convex analysis. The Fenchel-type min-max theorem for a pair of integer-valued M-natural-convex functions generalizes the min-max formulas for polymatroid intersection and…
We consider the problem of testing a particular type of composite null hypothesis under a nonparametric multivariate regression model. For a given quadratic functional $Q$, the null hypothesis states that the regression function $f$…