One remark to Ekeland's variational principle

$(varphi_1, varphi_2)$-variational principle

In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $left(varphi_1, varphi_2right)$-convex function $g, $ with arbitrarily small norm,  such that $f + g $ attains its strong minimum on $X. $ This result extends some of the  well-known varitional principles as that of Ekeland [On the variational principle,  J. Ma...

Variational Principle

Variational principle for probabilistic learning Yet another justification More simplification of updates for mean-field family Examples Dirichlet Process Mixture On minimization of divergence measures Energy minimization justifications Variational learning with exponential family Mean parametrization and marginal polytopes Convex dualities The log-partition function and conjugate duality Belie...

A Remark on the Lower Semicontinuity Assumption in the Ekeland Variational Principle

What happens to the conclusion of the Ekeland variational principle (briefly, EVP) if a considered function f : X → R ∪ {+∞} is lower semicontinuous not on a whole metric space X but only on its domain? We provide a straightforward proof showing that it still holds but only for varying in some interval ]0, β − infX f [, where β is a quantity expressing quantitatively the violation of the lower ...

Variational Principle for

We show that some measures suuering from the so-called Renormalization Group pathologies satisfy a variational principle and that the corresponding limit of the pressure, with boundary conditions in a set of measure 1, is proportional to the pressure of the Ising model.

An extended variational principle