Abstract
We propose an algorithm to generate inner and outer polyhedral approximations to the upper image of a bounded convex vector optimization problem. It is an outer approximation algorithm and is based on solving norm-minimizing scalarizations. Unlike Pascoletti–Serafini scalarization used in the literature for similar purposes, it does not involve a direction parameter. Therefore, the algorithm is free of direction-biasedness. We also propose a modification of the algorithm by introducing a suitable compact subset of the upper image, which helps in proving for the first time the finiteness of an algorithm for convex vector optimization. The computational performance of the algorithms is illustrated using some of the benchmark test problems, which shows promising results in comparison to a similar algorithm that is based on Pascoletti–Serafini scalarization.
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Notes
Alternatively, one may start with \({\mathcal {X}}_0=\emptyset \) in line 2 of Algorithm 1. This would decrease \(|{\mathcal {X}}|\) by J, the number of generating vectors of C.
Note that many solvers yield both primal and dual optimal solutions when called only for one of the problems.
Since the solution \(x^v\) found in line 9 of Algorithm 1 is a weak minimizer, it is also possible to update the set of weak minimizers right after line 9 (without checking the value of \(\Vert z^v\Vert \)) and subsequently ignore lines 17 and 18. This would yield a finite weak \(\epsilon \)-solution with an increased cardinality.
More precisely, in Algorithm 1, we apply \(\bar{\mathcal {P}}^{\text {out}}_{k+1}=\bar{\mathcal {P}}^{\text {out}}_{k}\cap \mathcal {H}_k\) in line 12, \(\bar{\mathcal {X}}_{k+1}=\bar{\mathcal {X}}_k\) in line 13, \(\bar{\mathcal {X}}_k\leftarrow \bar{\mathcal {X}}\cup \{x^v\}\) in line 18.
The same cone is used as a dual cone in [31, Example 9].
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Communicated by Matthias Ehrgott.
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This work was funded by TÜBİTAK (Scientific & Technological Research Council of Turkey), Project No. 118M479.
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Ararat, Ç., Ulus, F. & Umer, M. A Norm Minimization-Based Convex Vector Optimization Algorithm. J Optim Theory Appl 194, 681–712 (2022). https://doi.org/10.1007/s10957-022-02045-8
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DOI: https://doi.org/10.1007/s10957-022-02045-8
Keywords
- Convex vector optimization
- Multiobjective optimization
- Approximation algorithm
- Scalarization
- Norm minimization