Linear Equation

Bruce C. Dieffenbach ()
Additional contact information
Bruce C. Dieffenbach: Independent author

Chapter 12 in Conjugate Duality in Economic Analysis, 2026, pp 99-102 from Springer

Abstract: Abstract Using perturbation duality, we calculate the general formula for the solution set to a linear equation. We express the solution as the optimum value of a convex optimization. We find the reciprocal dual, and calculate the value function. Its subdifferential is the solution set. To calculate the solution set contrasts the traditional approach, which is not a calculation. Instead, one conjectures the solution and then verifies it via the fundamental theorem of linear algebra. Consider the linear equation$${\textit {{\sf { {A}}}}}\boldsymbol {x}=\boldsymbol {y},$$ in which A:X→Y. A convex optimization that models the solution is$$\inf _{\boldsymbol {x}}\delta _{{\textit {{\sf { {A}}}}}\boldsymbol {x}=\boldsymbol {y}}.$$ If there is a solution, then the optimum value is zero. If there is no solution, then the infimum is ∞.

Date: 2026
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:conchp:978-3-032-21396-9_12

Ordering information: This item can be ordered from
http://www.springer.com/9783032213969

DOI: 10.1007/978-3-032-21396-9_12

Access Statistics for this chapter

More chapters in Contributions to Economics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().