Proper Efficiency in Vector Optimization

• Proper efficiency is a refinement of Pareto minimality in vector optimization that excludes unstable efficient points via cone separation.
• The framework unifies diverse efficiency notions as instances of Q-minimality and uses a sublinear scalar function to certify unique minimizers.
• It operates in normed spaces without convexity or boundedness assumptions by leveraging the strict separation property (SSP) for scalar reformulations.

In nonconvex vector optimization, proper efficiency designates refinements of Pareto-type minimality intended to exclude pathological efficient points and to support scalar reformulations. A unifying treatment is given by recasting several proper efficient solution concepts as instances of $\mathcal{Q}$-minimality in a normed space and then deriving scalarization theorems from a cone separation principle called the strict separation property (SSP). In this framework, many proper and approximate proper efficient points are characterized by the existence of a sublinear scalar function of the form $f(x-x_0)+\alpha\|x-x_0\|$ for which the candidate point $x_0$ is the unique minimizer, and these results are obtained without convexity or boundedness assumptions on the feasible set (García-Castaño et al., 2024).

1. $\mathcal{Q}$-minimality as the organizing abstraction

Let $X$ be a normed space, let $\mathcal{Q}\subset X$ be an open cone, and let $A\subset X$. The central notion is that $x_0\in A$ is a $\mathcal{Q}$-minimal point of $A$ if and only if

$f(x-x_0)+\alpha\|x-x_0\|$0

This formulation expresses minimality as the nonexistence of feasible displacements from $f(x-x_0)+\alpha\|x-x_0\|$1 into the negative cone $f(x-x_0)+\alpha\|x-x_0\|$2 (García-Castaño et al., 2024).

The significance of this definition is structural rather than merely terminological. It allows proper efficiency notions that were historically introduced separately to be treated as instances of the same geometric pattern: once an appropriate open cone $f(x-x_0)+\alpha\|x-x_0\|$3 is identified, proper efficiency becomes $f(x-x_0)+\alpha\|x-x_0\|$4-minimality. This suggests that the essential difficulty lies in separating the translated feasible set $f(x-x_0)+\alpha\|x-x_0\|$5 from $f(x-x_0)+\alpha\|x-x_0\|$6, rather than in the idiosyncratic definitions of individual properness concepts.

The scalarization developed in this setting is built from sublinear functions

$f(x-x_0)+\alpha\|x-x_0\|$7

The role of this function is to turn a vector comparison problem into a scalar minimization problem while preserving the proper efficiency structure. A $f(x-x_0)+\alpha\|x-x_0\|$8-minimal point is then shown, under SSP, to be the unique minimizer of such a function on $f(x-x_0)+\alpha\|x-x_0\|$9 (García-Castaño et al., 2024).

2. Strict separation property and cone geometry

The geometric engine behind the scalarization theory is the strict separation property (SSP) for a pair of cones $x_0$0. Using

$x_0$1

SSP is defined by

$x_0$2

This is presented as a variation of an earlier separation property from Kasimbeyli, but in a strengthened and flexible form suited to scalarization in general normed spaces (García-Castaño et al., 2024).

A key theorem states that if $x_0$3 has SSP, then there exist $x_0$4 and $x_0$5 such that for all $x_0$6,

$x_0$7

and the scalar function $x_0$8 strictly separates the relevant cones. The refined form needed for optimization is

$x_0$9

for all nonzero $\mathcal{Q}$0 and all $\mathcal{Q}$1 (García-Castaño et al., 2024).

This separation mechanism is the technical reason the theory does not require convexity of the feasible set or boundedness assumptions. The separation is imposed at the level of cones associated with domination and feasible directions, not at the level of a convex optimization set. A plausible implication is that SSP functions as a replacement for several more restrictive geometric hypotheses that had previously been used to obtain scalarization results.

3. Proper efficient points unified by open cones

The framework treats nine notions of proper efficiency as $\mathcal{Q}$2-minimality for suitable cones $\mathcal{Q}$3. The paper identifies the following classes (García-Castaño et al., 2024):

• positive proper efficient
• Hurwicz proper efficient
• Benson proper efficient
• Hartley proper efficient
• Borwein proper efficient
• Henig global proper efficient
• Henig proper efficient
• super efficient
• tangentially Borwein proper efficient

This unification is formalized in Theorem 3.7, where each concept is linked to a specific open cone. Three representative examples are explicitly given. Positive proper efficiency uses

$\mathcal{Q}$4

Henig proper efficiency uses

$\mathcal{Q}$5

and tangentially Borwein proper efficiency uses

$\mathcal{Q}$6

The common pattern is that properness is encoded by choosing a cone that excludes precisely the directions considered inadmissible for the relevant notion (García-Castaño et al., 2024).

Among these notions, the paper highlights several classical forms. A point $\mathcal{Q}$7 is Benson proper efficient if

$\mathcal{Q}$8

Equivalently, the cone generated by feasible directions together with the ordering cone does not intersect the negative cone except at $\mathcal{Q}$9. A point $X$0 is Henig proper efficient if there exists an open cone $X$1 dilating $X$2 such that

$X$3

The Henig condition is thus expressed through a strict interior enlargement of the ordering cone (García-Castaño et al., 2024).

The theory also includes approximate variants. For an approximating set $X$4 and $X$5, $X$6 is approximate Benson proper efficient if

$X$7

and approximate Henig proper efficient if there exists a cone $X$8 in an admissible family $X$9 such that

$\mathcal{Q}\subset X$0

These definitions introduce approximation through the perturbation set $\mathcal{Q}\subset X$1, but preserve the same cone-minimality logic (García-Castaño et al., 2024).

4. Sublinear scalarization and unique minimizers

The main scalarization theorem states that if $\mathcal{Q}\subset X$2 has SSP and $\mathcal{Q}\subset X$3, then there exists $\mathcal{Q}\subset X$4 such that

$\mathcal{Q}\subset X$5

is attained only at $\mathcal{Q}\subset X$6 (García-Castaño et al., 2024).

This result turns cone minimality into a scalar optimization certificate. The scalarization is sublinear because it combines a continuous linear functional with a norm term. The norm term is not an auxiliary penalty added for regularization in the modern algorithmic sense; it is part of the separating functional furnished by SSP. Its presence is what allows strict inequalities on cones and ultimately uniqueness of the minimizer.

The theorem yields a necessary condition for any proper efficient point that has been represented as a $\mathcal{Q}\subset X$7-minimal point under an SSP hypothesis. Because the same scalar form

$\mathcal{Q}\subset X$8

appears across all treated notions, the framework is not merely a collection of parallel scalarization results. It is a genuinely unified scalarization principle. This suggests that many distinctions among proper efficiency concepts are cone-theoretic rather than scalarization-theoretic.

5. Exact characterizations for Benson, Henig, and approximate proper efficiency

The Benson case is the strongest. Provided the relevant SSP condition holds,

$\mathcal{Q}\subset X$9

is attained only at $A\subset X$0 (García-Castaño et al., 2024). For Benson proper efficiency, scalarization is therefore both sufficient and necessary.

The paper derives further identifications under additional assumptions. If $A\subset X$1 is convex and $A\subset X$2 has a weakly compact base, then

$A\subset X$3

so the Benson scalarization also characterizes Henig global proper efficiency. If $A\subset X$4 is starshaped at $A\subset X$5, then Benson and tangentially Borwein proper efficiency coincide at $A\subset X$6, and the same scalarization applies there as well (García-Castaño et al., 2024).

For approximate Benson proper efficient points, the paper proves both a necessary scalarization condition and a sufficient scalarization condition, yielding the characterization

$A\subset X$7

is attained only at $A\subset X$8 (García-Castaño et al., 2024).

A similar characterization is obtained for approximate Henig proper efficient points, under an additional cone-approximation or separation-family condition. The approximate theory is important because it shows that the scalarization method is not confined to exact efficiency concepts. It extends to perturbative variants in which approximation is built into the feasible-direction geometry rather than treated as an external numerical tolerance.

6. Mathematical significance in nonconvex vector optimization

The principal significance of the framework is that it works in normed spaces, does not require convexity assumptions on the feasible set, does not require boundedness assumptions, and treats many proper efficiency notions through a single framework and a single sublinear scalar function (García-Castaño et al., 2024). These features distinguish it from scalarization theories that rely on convex feasible sets, compactness, or concept-specific constructions.

In the broader theory of vector optimization, proper efficiency is meant to exclude efficient points that are mathematically admissible but unstable or degenerate from the standpoint of trade-off analysis. The present framework advances that program by replacing disparate properness definitions with cone minimality and by replacing ad hoc scalarizations with the common template $A\subset X$9. Benson proper efficiency emerges as the central exact case, while Henig and approximate variants are incorporated through additional geometric hypotheses (García-Castaño et al., 2024).

A common misconception is that scalarization of proper efficiency is intrinsically tied to convex analysis. The results here show otherwise: the decisive ingredient is not convexity of the feasible set, but a suitable separation property for cones. Another plausible implication is that future work on nonconvex vector optimization may profit more from refining separation principles than from seeking increasingly specialized scalarizing functions.