Convexity and Lipschitz Behavior of Small Pseudospectra

Approximate Convexity and Submonotonicity in Locally Convex Spaces

Triangular norms and k-Lipschitz property

Inspired by an open problem of Alsina, Frank and Schweizer, k-Lipschitz t-norms are studied. The k-convexity of continuous monotone functions is introduced. Additive generators of k-Lipschitz tnorms are completely characterized by means of k-convexity. For a given k ∈ [1,∞[ the pointwise infimum A∗k of the class of all k-Lipschitz t-norms is introduced.

Local strong convexity and local Lipschitz continuity of the gradient of convex functions

Given a pair of convex conjugate functions f and f∗, we investigate the relationship between local Lipschitz continuity of ∇f and local strong convexity properties of f∗.

Optimization and Pseudospectra, with Applications to Robust Stability

The -pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of all complex matrices within a distance of A. We are interested in two aspects of “optimization and pseudospectra.” The first concerns maximizing the function “real part” over an -pseudospectrum of a fixed matrix: this defines a function known as the -pseudospectral abscissa of a matrix. We pres...

Beyond Convexity: Stochastic Quasi-Convex Optimization

Stochastic convex optimization is a basic and well studied primitive in machine learning. It is well known that convex and Lipschitz functions can be minimized efficiently using Stochastic Gradient Descent (SGD). The Normalized Gradient Descent (NGD) algorithm, is an adaptation of Gradient Descent, which updates according to the direction of the gradients, rather than the gradients themselves. ...