Ordinal Optimization Through Multi-objective Reformulation

TL;DR

This paper introduces a linear transformation that converts ordinal combinatorial optimization problems into standard multi-objective problems, enabling the use of existing solution methods like dynamic programming.

Contribution

It presents a bijective linear transformation that preserves problem properties, bridging ordinal and multi-objective optimization for combinatorial problems.

Findings

Transformation preserves problem properties

Dynamic programming applies to ordinal shortest path

Extension to mixed ordinal and real-valued objectives

Abstract

We analyze combinatorial optimization problems with ordinal, i.e., non-additive, objective functions that assign categories (like good, medium and bad) rather than cost coefficients to the elements of feasible solutions. We review different optimality concepts for ordinal optimization problems and discuss their similarities and differences. We then focus on two prevalent optimality concepts that are shown to be equivalent. Our main result is a bijective linear transformation that transforms ordinal optimization problems to associated standard multi-objective optimization problems with binary cost coefficients. Since this transformation preserves all properties of the underlying problem, problem-specific solution methods remain applicable. A prominent example is dynamic programming and Bellman's principle of optimality, that can be applied, e.g., to ordinal shortest path and ordinal knapsack problems. We extend our results to multi-objective optimization problems that combine ordinal and real-valued objective functions.