A Solution Concept for Convex Vector Optimization Problems based on a User-defined Region of Interest

Abstract

This work addresses arbitrary convex vector optimization problems, which constitute a general framework for multi-criteria decision-making in diverse real-world applications. Due to their complexity, such problems are typically tackled using polyhedral approximation. Existing solution concepts rely on additional assumptions, such as boundedness, polyhedrality of the ordering cone, or existence of interior points in the ordering cone, and typically focus on absolute error measures. We introduce a solution concept based on the homogenization of the upper image that employs relative error measures and avoids additional structural assumptions. Although minimality is not explicitly required, a form of approximate minimality is implicitly ensured. The concept is straightforward, requiring only a single precision parameter and, owing to relative errors, remains robust under scaling of the target functions. Homogenization also eliminates the need for the binary distinction between points far from the origin and directions which can lead to numerical difficulties. Furthermore, in practice decision-makers often identify a region where preferred solutions are expected. Our concept supports both a global overview of the upper image and a refined local perspective within such a user-defined region of interest (RoI).We present a decision-making procedure enabling iterative refinement of this region and the associated preferences.

Figures (8)

• Figure 1: Homogenization of a convex set $Z$.
• Figure 2: Error bound $\alpha(r,\delta)$ within the region of interest depending on $r$ for $\delta = 0.1$. $R_\delta$ marks the radius for the region of validity for the error bound.
• Figure 3: Error bound $\alpha(r,\delta)$ on the distance of boundary points of a homogeneous $\delta$-approximation to the boundary of the upper image (see Theorem \ref{['absch']}). The bound is given as a function of the radius $r$ for fixed values of large $\delta$ on the left and smaller $\delta$ on the right. The plots illustrate how the approximation error grows with the size of the RoI, with smaller $\delta$ yielding tighter bounds. The vertical dashed lines indicate the bound of the region of validity $R_\delta$ for each $\delta$, beyond which the error bound is undefined. As an example, the point on the right graph highlights a feasible trade-off between accuracy and coverage that is a possible choice for the decision-maker when setting $\delta = 0.1$. It means that, in an approximate upper image that is a homogeneous $0.1$-approximation, any boundary point in the RoI with radius 5 is in at most 5.2632 units distance of a weakly $C-$minimal element (given feasibility).
• Figure 4: Bounded portion of the unbounded upper image in Example \ref{['example']}.
• Figure 5: Step 1 of the decision-making procedure. Locating the upper image and generating a rough impression using a high value for $\delta$ ($\delta = 0.9$) and an initial reference point ($p = (0,-20)^T$).

Theorems & Definitions (9)

• Definition 3.1
• Proposition 3.2
• Definition 3.3
• Theorem 3.4
• Theorem 3.5
• Remark 3.6
• Theorem 3.7
• Example 4.1
• Lemma 6.1