Definition and Characterization of Geoffrion Proper Efficiency for Real Vector Optimization with Infinitely Many Criteria

Abstract

The concept and characterization of proper efficiency is of significant theoretical and computational interest, in multiobjective optimization and decision-making, to prevent solutions with unbounded marginal rates of substitution. In this paper, we propose a slight modification to the original definition in the sense of Geoffrion, which maintains the common characterizations of properly efficient points as solutions to weighted sums or series and augmented or modified weighted Tchebycheff norms, also if the number of objective functions is countably infinite. We give new proofs and counterexamples which demonstrate that such results become invalid for infinitely many criteria with respect to the original definition, in general, and we address the motivation and practical relevance of our findings for possible applications in stochastic optimization and decision-making under uncertainty.

Appendix

For completeness, we state the slightly generalized lemma of the alternative that is used in the proof of Theorem 3.2, and that we extended to an arbitrary (and possible infinite) number of convex functions.

Lemma 1

(Mangasarian [23, page 65) Let $X$ be a nonempty, convex set, and let $g_i:X \rightarrow \mathbb {R}^{}$ be convex functions for all $i \in I$, such that $\sup \{g_i(x):i \in I\} < \infty$ for every $x \in X$. Then either (i) there exists $x \in X$ such that $g_i(x) < 0$ for all $i \in I$, or (ii) there exists $\lambda \in \Lambda$ such that $\sum _{i \in I}\lambda _ig_i(x) \ge 0$ for all $x \in X$, but never both.

Proof

First, if (i) is true, then there exists $x \in X$ such that $g_i(x) < 0$ for all $i \in I$. It follows that $\sum _{i \in I}\lambda _ig_i(x) < 0$ if $\lambda \in V_\ge$ and $\lambda \ne 0$, and thus (ii) cannot be true at the same time. Hence, it only remains to show that whenever (i) is false, then (ii) must be true. Define $S(x) := \left\{ s \in V:s_i > g_i(x) \text { for all } i \in I\right\}$ and $S := \bigcup _{x \in X} S(x)$. For each $x \in X$, the set $S(x)$ is nonempty and convex because $X$ and all $g_i$ are convex, and thus $S$ is nonempty and convex as well. Now, if (i) is false, then $0 \notin S$ so that $S$ and the singleton $T = \{0\}$ are disjoint. Hence, by the Hahn–Banach Separation Theorem for convex sets, it follows that there exists a continuous map $\ell :V \rightarrow \mathbb {R}^{}$ such that $\ell (s) = \sum _{i \in I}\lambda _is_i > 0 \text { for all } s \in S$, for some (not yet necessarily non-negative or summable) $\lambda \in V$. It remains to show that the $\lambda _i$ are indeed non-negative and can be chosen summable to 1. For this, note that $S$ is unbounded above so that each $s_i$ can be chosen arbitrarily large both individually by itself, and collectively for all $i \in I$. The former implies that $\lambda _i \ge 0$ for all $i \in I$, and $\lambda \ne 0$ because $\lambda (s) > 0$. The latter implies that $\lambda$ must be summable, and thus, without any loss of generality, we can choose $\sum _{i \in I}\lambda _i = 1$. $\square$