 No Duality GapBruce C. Dieffenbach ()
Chapter 2 in Conjugate Duality in Economic Analysis, 2026, pp 19-23 from Springer Abstract: Abstract We present the relative and polyhedral conditions for no duality gap. Consider the primal$$V\left ( \boldsymbol {y}\right ) :=\inf _{\boldsymbol {x}}F\left ( \boldsymbol {x},\boldsymbol {y}\right )$$ Its dual is$$V^{\ast \ast }\left ( \boldsymbol {y}\right ) =\sup _{\boldsymbol {y}^{\ast }}\left [ \left \langle \boldsymbol {y}^{\ast },\boldsymbol {y}\right \rangle -V^{\ast }\left ( \boldsymbol {y}^{\ast }\right ) \right ] \!.$$ For the perturbation function F to be proper, closed, and convex is not sufficient to ensure that the value function satisfies conjugate duality: there might be a duality gap, $$V\left ( \boldsymbol {y}\right ) >V^{\ast \ast }\left ( \boldsymbol {y}\right )$$. The relative and polyhedral conditions both imply that there is no duality gap. Furthermore, if the optimum value is finite, then it is attained by a dual solution. The relative condition for no duality gap is that there is some x such that $$\left ( \boldsymbol {x},\boldsymbol {y}\right )$$ is in the relative interior of dom F. The polyhedral condition for no duality gap is that F is polyhedral and there is some x such that $$\left ( \boldsymbol {x},\boldsymbol {y}\right )$$ is in dom F. Date: 2026 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:conchp:978-3-032-21396-9_2 Ordering information: This item can be ordered from DOI: 10.1007/978-3-032-21396-9_2 Access Statistics for this chapter More chapters in Contributions to Economics from Springer |
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