Ethic Duality: A Homological Framework for Primal-Dual Problems
Dmitry Pasechnyuk-Vilensky, Martin Tak\'a\v{c}

TL;DR
This paper introduces a homological duality framework for primal-dual problems using derived functors, unifying various classical dualities and providing a systematic hierarchy of obstructions to exactness across multiple domains.
Contribution
It develops a novel homological duality framework based on Ext-groups that unifies primal-dual exactness criteria across diverse fields like optimization, graph theory, and dynamical systems.
Findings
Ext^1 vanishing characterizes primal-dual exactness in optimization.
Higher Ext-groups encode obstructions in various duality contexts.
Framework is stable under derived Morita equivalence and domain-independent.
Abstract
We develop a homological duality framework based on a contravariant functor with dualizing object . A morphism is called ethic when it satisfies the canonical double-dual compatibility . In the derived setting, the functor produces a graded family of Ext-groups that measure all failures of this compatibility. The first layer identifies primal-dual gaps, while higher provide a systematic hierarchy of derived obstructions to exactness. This formulation specializes uniformly across several classical domains. In linear and conic optimization, Farkas- and Slater-type exactness criteria correspond to the vanishing of , and integer duality gaps coincide with torsion Ext-classes. In graph theory, Kirchhoff- and Baker-Norine-type dualities arise as…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Topological and Geometric Data Analysis · Polynomial and algebraic computation
