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Global Solution of Semi-infinite Programs with Existence Constraints

Author

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  • Hatim Djelassi

    (RWTH Aachen University)

  • Alexander Mitsos

    (RWTH Aachen University)

Abstract

We consider what we term existence-constrained semi-infinite programs. They contain a finite number of (upper-level) variables, a regular objective, and semi-infinite existence constraints. These constraints assert that for all (medial-level) variable values from a set of infinite cardinality, there must exist (lower-level) variable values from a second set that satisfy an inequality. Existence-constrained semi-infinite programs are a generalization of regular semi-infinite programs, possess three rather than two levels, and are found in a number of applications. Building on our previous work on the global solution of semi-infinite programs (Djelassi and Mitsos in J Glob Optim 68(2):227–253, 2017), we propose (for the first time) an algorithm for the global solution of existence-constrained semi-infinite programs absent any convexity or concavity assumptions. The algorithm is guaranteed to terminate with a globally optimal solution with guaranteed feasibility under assumptions that are similar to the ones made in the regular semi-infinite case. In particular, it is assumed that host sets are compact, defining functions are continuous, an appropriate global nonlinear programming subsolver is used, and that there exists a Slater point with respect to the semi-infinite existence constraints. A proof of finite termination is provided. Numerical results are provided for the solution of an adjustable robust design problem from the chemical engineering literature.

Suggested Citation

  • Hatim Djelassi & Alexander Mitsos, 2021. "Global Solution of Semi-infinite Programs with Existence Constraints," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 863-881, March.
    • Handle: RePEc:spr:joptap:v:188:y:2021:i:3:d:10.1007_s10957-021-01813-2
    DOI: 10.1007/s10957-021-01813-2
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    References listed on IDEAS

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    1. E. Polak & J. O. Royset, 2003. "Algorithms for Finite and Semi-Infinite Min–Max–Min Problems Using Adaptive Smoothing Techniques," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 421-457, December.
    2. Hatim Djelassi & Alexander Mitsos, 2017. "A hybrid discretization algorithm with guaranteed feasibility for the global solution of semi-infinite programs," Journal of Global Optimization, Springer, vol. 68(2), pages 227-253, June.
    3. Stein, Oliver, 2012. "How to solve a semi-infinite optimization problem," European Journal of Operational Research, Elsevier, vol. 223(2), pages 312-320.
    4. Lopez, Marco & Still, Georg, 2007. "Semi-infinite programming," European Journal of Operational Research, Elsevier, vol. 180(2), pages 491-518, July.
    5. Hatim Djelassi & Moll Glass & Alexander Mitsos, 2019. "Discretization-based algorithms for generalized semi-infinite and bilevel programs with coupling equality constraints," Journal of Global Optimization, Springer, vol. 75(2), pages 341-392, October.
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    Cited by:

    1. Daniel Jungen & Hatim Djelassi & Alexander Mitsos, 2022. "Adaptive discretization-based algorithms for semi-infinite programs with unbounded variables," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(1), pages 83-112, August.

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