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Linear Equation

Author

Listed:
  • Bruce C. Dieffenbach

    (Independent author)

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 ∞.

Suggested Citation

  • Bruce C. Dieffenbach, 2026. "Linear Equation," Contributions to Economics,, Springer.
    • Handle: RePEc:spr:conchp:978-3-032-21396-9_12
    DOI: 10.1007/978-3-032-21396-9_12
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