Related papers: On the convexity of piecewise-defined functions
A multi-convex optimization problem is one in which the variables can be partitioned into sets over which the problem is convex when the other variables are fixed. Multi-convex problems are generally solved approximately using variations on…
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…
We provide three new proofs of the strong concavity of the dual function of some convex optimization problems. For problems with nonlinear constraints, we show that the the assumption of strong convexity of the objective cannot be weakened…
A general condition determining the optimal performance of a complex system has not yet been found and the possibility of its existence is unknown. To contribute in this direction, an optimization algorithm as a complex system is presented.…
The condition number of a differentiable convex function, namely the ratio of its smoothness to strong convexity constants, is closely tied to fundamental properties of the function. In particular, the condition number of a quadratic convex…
We give a necessary and sufficient condition for strict convexity of the rate function of a random vector in $R^d$. This condition is always satisfied when the random vector has finite Laplace transform. We also completely describe the…
In information theory, some optimization problems result in convex optimization problems on strictly convex functionals of probability densities. In this note, we study these problems and show conditions of minimizers and the uniqueness of…
A class of real functions, which is the generalization of a family of convex functions, is introduced; in this connection, we have defined $X$-convex, strictly $X$-convex, quasi-$X$-convex, strictly quasi-$X$-convex, and semi-strictly…
We provide a sufficient condition for an invertible (locally strongly) convex vector-valued function on $\mathbb{R}^N$ to have a (locally strongly) convex inverse. We show under suitable conditions that if the gradient of each component of…
We consider spline estimates which preserve prescribed piecewise convex properties of the unknown function. A robust version of the penalized likelihood is given and shown to correspond to a variable halfwidth kernel smoother where the…
We provide new necessary and sufficient conditons for ensuring strong quasiconvexity in the nonsmooth case and, as a consequence, we provide a proof for the differentiable case. Furthermore, we improve the quadratic growth property for…
A piecewise linear function can be described in different forms: as an arbitrarily nested expression of $\min$- and $\max$-functions, as a difference of two convex piecewise linear functions, or as a linear combination of maxima of…
Optimal uncertainty quantification (OUQ) is a framework for numerical extreme-case analysis of stochastic systems with imperfect knowledge of the underlying probability distribution. This paper presents sufficient conditions under which an…
Given a function $f$ defined on a nonempty and convex subset of the $d$-dimensional Euclidean space, we prove that if $f$ is bounded from below and it satisfies a convexity-type functional inequality with infinite convex combinations, then…
For convex univalent functions we give instances where the sharp bound for various coefficient functionals are identical to those for the corresponding bound for the inverse function. We give instances where the sharp bounds differ and also…
In the article the necessary and sufficient conditions for a representation of Lipschitz function of two variables as a difference of two convex functions are formulated. An algorithm of this representation is given. The outcome of this…
Convexity, though extremely important in mathematical programming, has not drawn enough attention in the field of dynamic programming. This paper gives conditions for verifying convexity of the cost-to-go functions, and introduces an…
It is known that, in finite dimensions, the support function of a compact convex set with non empty interior is differentiable excepting the origin if and only if the set is strictly convex. In this paper we realize a thorough study of the…
We define convexity canonically in the setting of monoids. We show that many classical results from convex analysis hold for functions defined on such groups and semigroups, rather than only on vector spaces. Some examples and…
The property of isotonicity of a continuous convex function defined on the entire space or only on the positive cone is characterized via subdifferentials. Numerous examples illustrating the obtained results are included.