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A hybrid discretization algorithm with guaranteed feasibility for the global solution of semi-infinite programs

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Abstract

A discretization-based algorithm for the global solution of semi-infinite programs (SIPs) is proposed, which is guaranteed to converge to a feasible, \(\varepsilon \)-optimal solution finitely under mild assumptions. The algorithm is based on the hybridization of two existing algorithms. The first algorithm (Mitsos in Optimization 60(10–11):1291–1308, 2011) is based on a restriction of the right-hand side of the constraints of a discretized SIP. The second algorithm (Tsoukalas and Rustem in Optim Lett 5(4):705–716, 2011) employs a discretized oracle problem and a binary search in the objective space. Hybridization of the approaches yields an algorithm, which leverages the strong convergence guarantees and the relatively tight upper bounding problem of the first approach while employing an oracle problem adapted from the second approach to generate cheap lower bounds and adaptive updates to the restriction of the first approach. These adaptive updates help in avoiding a dense population of the discretization. The hybrid algorithm is shown to be superior to its predecessors both theoretically and computationally. A proof of finite convergence is provided under weaker assumptions than the assumptions in the references. Numerical results from established SIP test cases are presented.

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Acknowledgements

We gratefully acknowledge the financial support from Deutsche Forschungsgemeinschaft (German Research Foundation) through the project “Optimization-based Control of Uncertain Systems” PAK 635/1; MA1188/34-1.

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Authors and Affiliations

  1. AVT Process Systems Engineering (SVT), RWTH Aachen University, Aachen, Germany

    Hatim Djelassi & Alexander Mitsos

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  1. Hatim Djelassi
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  2. Alexander Mitsos
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Correspondence to Alexander Mitsos.

Electronic supplementary material

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Supplementary material 1 (zip 24 KB)

Appendices

Appendix 1: Oracle algorithm

Here, the oracle algorithm is stated formally. As mentioned previously, Tsoukalas and Rustem [24] consider the exact solution of all subproblems. However, exact solutions cannot be provided for the investigated problems. Algorithm 3, is an adaptation of the original algorithm by Tsoukalas and Rustem [24] and considers the approximate solution of the subproblems according to Definition 1. Note that the convergence proof by Tsoukalas and Rustem [24] does not hold for Algorithm 3.

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Appendix 2: Numerical results

Table 2 Numerical results for Algorithm 1 for \(r^g = 1.2\)
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Table 3 Numerical results for Algorithm 1 for \(r^g = 2\)
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Table 4 Numerical results for Algorithm 2
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Table 5 Numerical results for Algorithm 3
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Djelassi, H., Mitsos, A. A hybrid discretization algorithm with guaranteed feasibility for the global solution of semi-infinite programs. J Glob Optim 68, 227–253 (2017). https://doi.org/10.1007/s10898-016-0476-7

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