On a conjecture with implications for multicriteria decision making
Anas Mifrani
TL;DR
Problem: whether every efficient decision in multicriteria optimization can be characterized as the unique maximizer of a continuous, strictly increasing and strictly concave real-valued function on the criterion space $Y$. Approach: construct an explicit counterexample with $p=2$, $X=[0,\infty)$, $f_1(x)=x^2$, $f_2(x)=-x^3$, and use KKT/subdifferential reasoning to show no such $v$ exists for the efficient but improperly efficient $x^ irc=0$. Findings: the result confirms Soland's conjecture in a strong form, demonstrating nonexistence of a suitable $v$ for certain efficient decisions, and highlighting that continuity and strict concavity are not required prerequisites. Implications: this limits the applicability of the value-function approach in multicriteria decision making and suggests caution when discarding efficient decisions based on concave value-function representability.
Abstract
We prove a conjecture by Richard Soland that given an efficient solution to a multicriteria optimization problem, there need not exist a continuous, strictly increasing and strictly concave criterion space function that attains its maximum at the vector of criteria values achieved by that solution.