K e y w o r d s F u z z y convexity; Fuzzy criterion sets, Pareto-optimal decision, Multiple objective optimization. 1. I N T R O D U C T I O N Since most of practical decision problems are fuzzy and approximate, fuzzy decision making becomes one of the most impor tan t practical approaches. However, the result ing problems are frequently complicated and difficult to solve. One effective way to...

Abstract We present a generalization of the notion neighborliness to non-polyhedral convex cones. Although definition is available in case literature, it fairly restrictive as requires all low-dimensional faces be polyhedral. Our approach more flexible and includes, for example, cone positive-semidefinite matrices special (this not neighborly general). term our Terracini convexity due its conce...

The seminal work by Edmonds [9] and Lovász [39] shows the strong connection between submodular functions and convex functions. Submodular functions have tight modular lower bounds, and a subdifferential structure [16] in a manner akin to convex functions. They also admit polynomial time algorithms for minimization and satisfy the Fenchel duality theorem [18] and the Discrete Seperation Theorem ...

In “High-Dimensional Learning Under Approximate Sparsity with Applications to Nonsmooth Estimation and Regularized Neural Networks,” Liu, Ye, Lee study a model fitting problem where there are much fewer data than dimensions. Of their particular focus the scenarios commonly imposed sparsity assumption is relaxed, usual condition of restricted strong convexity absent. The results show that genera...

This paper presents a modification of the q-BFGS method for nonlinear unconstrained optimization problems. For this modification, we use simple symmetric positive definite matrix and propose new q-quasi-Newton equation, which is close to ordinary equation in limiting case. uses only first order q-derivatives build an approximate q-Hessian over number iterations. The q-Armijo-Wolfe line search c...

Ricci curvature bounds in Riemannian geometry are known to be equivalent to the weak convexity (convexity along at least one geodesic between any two points) of certain functionals in the space of probability measures. We prove that the weak convexity can be reinforced into strong (usual) convexity, thus solving a question left open in [4].