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Efficiency Duality

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  • Bruce C. Dieffenbach

    (Independent author)

Abstract

We seek conditions for the feasible set to contain an efficient vector or a weakly efficient vector. For a closed, convex cone K in a Euclidean Jordan algebra, the “efficiency alternative” states that exactly one of the following has a solution: 1Some x≫0 belongs toK; 2Some x*≻0 belongs to K∘. The efficiency alternative leads to both efficiency duality and weak-efficiency duality. Consider a nonempty, closed, convex feasible set in a Euclidean Jordan algebra. “Efficiency duality” establishes that three properties are equivalent.EfficiencyThe feasible set contains an efficient vector;Horizon ConeThe horizon cone contains no vector greater than zero;Barrier ConeThe barrier cone contains a positive vector. “Weak-efficiency duality” establishes that three properties are equivalent.Weak EfficiencyThe feasible set contains a weakly efficient vector;Horizon ConeThe horizon cone contains no positive vector;Barrier ConeThe barrier cone contains a vector greater than zero. Invoke the Ekeland variational principle to prove these statements for the barrier cone. The chapter applies these relationships to activity analysis.

Suggested Citation

  • Bruce C. Dieffenbach, 2026. "Efficiency Duality," Contributions to Economics,, Springer.
    • Handle: RePEc:spr:conchp:978-3-032-21396-9_36
    DOI: 10.1007/978-3-032-21396-9_36
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