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Algorithms for vector optimization problems

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  1. Tez No: 402956
  2. Yazar: FİRDEVS ULUS
  3. Danışmanlar: PROF. BIRGIT RUDLOFF
  4. Tez Türü: Doktora
  5. Konular: Mühendislik Bilimleri, İşletme, Engineering Sciences, Business Administration
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 2015
  8. Dil: İngilizce
  9. Üniversite: Princeton University
  10. Enstitü: Yurtdışı Enstitü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 112

Özet

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Özet (Çeviri)

This dissertation studies algorithms to solve linear and convex vector (multi-objective) optimization problems. A parametric simplex algorithm for solving linear vector optimization problems (LVOPs) and two approximation algorithms for solving convex vector optimization problems (CVOPs) are provided. The algorithms work for any number of objectives and for general polyhedral ordering cones. The parametric simplex algorithm can be seen as a variant of the multi-objective simplex (Evans-Steuer) algorithm [17]. Di erent from it, this algorithm works in the parameter space and does not aim to nd the set of all ecient solutions. Instead, it nds a subset of ecient solutions that is enough to generate the whole frontier and corresponds to a solution concept introduced in [34]. In that sense, this algorithm can also be seen as a generalization of the parametric self-dual simplex algorithm, which originally is designed for solving single objective linear optimization problems, and is modi ed to solve two objective bounded LVOPs with the positive orthant as the ordering cone in [43] by Ruszczynski and Vanderbei. The algorithm proposed here works for any dimension, any solid pointed polyhedral ordering cone and for bounded as well as unbounded problems. Numerical results are provided to compare the proposed algorithm with an objective space based LVOP algorithm (Benson's algorithm in [22]), which also provides a solution in the sense of [34], and with the Evans-Steuer algorithm [17]. The results show that for nondegenerate problems the proposed algorithm outperforms Benson's algorithm and is on par with Evan-Steuer algorithm. For highly degenerate problems Benson's algorithm [22] excels the simplex-type algorithms as expected by being an objective space algorithm; however, the parametric simplex algorithm performs much better than Evans-Steuer algorithm. For solving convex vector optimization problems (CVOPs), two approximation algorithms are provided. Both algorithms solve the CVOP and its geometric dual problem simultaneously. The rst algorithm is an extension of Benson's outer approximation algorithm, and the second one is a dual variant of it. Both algorithms provide an inner as well as an outer approximation of the (upper respectively lower) image. Only one scalar convex program has to be solved in each iteration. We allow objective and constraint functions that are not necessarily di erentiable, allow solid pointed polyhedral ordering cones, and relate the approximations to an appropriate -solution concept. Some illustrative examples and also numerical results are provided for the approximation algorithms.

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