Abstract
Multilevel Stackelberg problems are nested optimization problems which reply optimally to hierarchical decisions of subproblems. These kind of problems are common in hierarchical decision making systems and are known to be NP-hard. In this paper, a systematic evolutionary algorithm has been proposed for such types of problems. A unique feature of the algorithm is that it is not affected by the nature of the objective and constraint functions involved in the problem as long as the problem has a solution. The convergence proof of the proposed algorithm is given for special problems containing non-convex and non-differentiable functions. Moreover, a new concept of \((\varepsilon ,\delta )\)-approximation for Stackelberg solutions is defined. Using this definition comparison of approximate Stackelberg solutions has been studied in this work. The numerical results on various problems demonstrated that the proposed algorithm is very much promising to multilevel Stackelberg problems with bounded constraints, and it can be used as a benchmark for a comparison of approximate results by other algorithms.
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Notes
Systematic sampling is a statistical method of selecting sample members from a larger population according to a random starting point and a fixed, periodic interval Wikipedia (2013).
\(|X_{ji}|\) stands for a number of VCPs on \(X_{j}\).
In MATLAB \(floor\left( \frac{a_{j}}{2}\right) \) displays the greatest integer which is less than \(\frac{a_{j}}{2}\).
\(VCPs^{*}\) of any decision space contains all representatives/ VCPs of the decision space and offsprings of the best representatives of the decision space.
\(\beta (x,\mu _{1}) :=\) an open ball of radius \(\mu _{1}\) about the point \(x\).
\(\beta ((x_{j}',\dots ,x_{k}'),\mu _{j}) :=\) an open ball of radius \(\mu _{j}\) about the point \((x_{j}',\dots ,x_{k}')\).
\(\Vert x\Vert :=\) Euclidean Norm of \(x=(x_{1},\dots ,x_{k})\in R^{n}\), i.e \(\Vert x\Vert =\left( \sum _{i=1}^{i=n}x_{i}^{2}\right) ^{\frac{1}{2}}\).
In this and the next examples, log stands for the natural logarithm; and the floor and ceiling (ceil) functions are functions that map a real number to the largest previous or the smallest following integer, respectively.
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This work is in part supported by the International Science Program (ISP) of Sweden, a research project at the Department of Mathematics, Addis Ababa University.
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Woldemariam, A.T., Kassa, S.M. Systematic evolutionary algorithm for general multilevel Stackelberg problems with bounded decision variables (SEAMSP). Ann Oper Res 229, 771–790 (2015). https://doi.org/10.1007/s10479-015-1842-4
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DOI: https://doi.org/10.1007/s10479-015-1842-4