‎let $x$ be a real normed  space, then  $c(subseteq x)$  is  functionally  convex  (briefly, $f$-convex), if  $t(c)subseteq bbb r $ is  convex for all bounded linear transformations $tin b(x,r)$; and $k(subseteq x)$  is  functionally   closed (briefly, $f$-closed), if  $t(k)subseteq bbb r $ is  closed  for all bounded linear transformations $tin b(x,r)$. we improve the    krein-milman theorem  ...

Motivated by recent increased interest in optimization algorithms for non-convex application to training deep neural networks and other problems data analysis, we give an overview of theoretical results on global performance guarantees optimization. We start with classical arguments showing that general could not be solved efficiently a reasonable time. Then list can find the minimizer exploiti...

We lower bound the complexity of finding $$\epsilon $$ -stationary points (with gradient norm at most ) using stochastic first-order methods. In a well-studied model where algorithms access smooth, potentially non-convex functions through queries to an unbiased oracle with bounded variance, we prove that (in worst case) any algorithm requires least ^{-4}$$ find point. The is tight, and establis...

In this work, we study the existence of non-trivial multiple solutions for a class of quasilinear elliptic systems equipped with concave-convex nonlinearities and critical growth terms in bounded domains. By using the variational method, especially Nehari manifold and Palais-Smale condition, we prove the existence and multiplicity results of positive solutions.