A usufel lemma for Lagrange multiplier rules in infinite dimension

TL;DR

This paper establishes conditions on dual Banach space sequences to prevent weak* convergence to zero, ensuring nontrivial Lagrange multipliers in infinite-dimensional optimization problems like Pontryagin Principles.

Contribution

It introduces a usable lemma providing conditions that guarantee nontrivial Lagrange multipliers in infinite-dimensional optimization.

Findings

Sequences satisfying the lemma do not converge to zero in the weak* topology.

The lemma aids in ensuring nontrivial multipliers in infinite horizon Pontryagin problems.

Provides a practical criterion for analyzing dual sequences in Banach spaces.

Abstract

We give some reasonable and usable conditions on a sequence of norm one in a dual banach space under which the sequence does not converges to the origin in the $w^*$-topology. These requirements help to ensure that the Lagrange multipliers are nontrivial, when we are interested for example on the infinite dimensional infinite-horizon Pontryagin Principles for discrete-time problems.