Theory of Vector Optimization

Tammer, Christiane; Göpfert, Alfred

doi:10.1007/0-306-48107-3_1

Cited by 103 publications

(1 citation statement)

References 207 publications

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“…This interrelation was used, among others, in Engau (2007) to define the concept of cone-efficiency (or C R -efficiency). For a general introduction to ordering cones in the context of vector optimization see, e.g., Tammer & Göpfert (2003); Jahn (2011).…”

Section: Orders and Conesmentioning

confidence: 99%

Ordinal Optimization Through Multi-objective Reformulation

Klamroth¹,

Stiglmayr²

2022

Preprint

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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, problemspecific 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.

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Section: Orders and Conesmentioning

confidence: 99%