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Optimality Analysis of a Class of Semi-infinite Programming Problems

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Abstract

In this paper, we consider a class of semi-infinite programming problems with a parameter. As the parameter increases, we prove that the optimal values decrease monotonically. Moreover, the limit of the sequence of optimal values exists as the parameter tends to infinity. In finding the limit, we decompose the original optimization problem into a series of subproblems. By calculating the maximum optimal values to the subproblems and applying a fixed-point theorem, we prove that the obtained maximum value is exactly the limit of the sequence of optimal values under certain conditions. As a result, the limit can be obtained efficiently by solving a series of simplified subproblems. Numerical examples are provided to verify the limit obtained by the proposed method.

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References

  1. Hettich, R., Kortanek, K.O.: Semi-infinite programming: theory, methods, and application. SIAM Rev. 35, 380–429 (1993)

    Article  MathSciNet  Google Scholar 

  2. López, M., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180, 491–518 (2007)

    Article  MathSciNet  Google Scholar 

  3. Shapiro, A.: Semi-infinite programming, duality, discretization and optimality conditions. Optimization 58, 133–161 (2009)

    Article  MathSciNet  Google Scholar 

  4. Feng, Z.G., Teo, K.L., Rehbock, V.: A smoothing approach for semi-infinite programming with projected new-type algorithm. J. Ind. Manag. Optim. 5, 141–151 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Shiu, T.J., Wu, S.Y.: Relaxed cutting plane method with convexification for solving nonlinear semi-infinite programming. Comput. Optim. Appl. 53(1), 91–113 (2012)

    Article  MathSciNet  Google Scholar 

  6. Wang, S.X., Yuan, Y.X.: Feasible method for convex semi-infinite programming. SIAM J. Optim. 25, 2537–2560 (2010)

    Article  Google Scholar 

  7. Zhang, L.P., Wu, S.Y., Lopez, M.A.: A new exchange method for convex semi-infinite programming. SIAM J. Optim. 20, 2959–2977 (2010)

    Article  MathSciNet  Google Scholar 

  8. Yiu, K.F.C., Yang, X.Q., Nordholm, S., Teo, K.L.: Near-field broadband beamformer design via multidimensional semi-infinite linear programming techniques. IEEE Trans. Speech Audio Process. 11(6), 725–732 (2003)

    Article  Google Scholar 

  9. Feng, Z.G., Yiu, K.F.C., Nordholm, S.: Performance limit of broadband beamformer designs in space and frequency. J. Optim. Theory Appl. 164, 316–341 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This paper is supported by the grant of National Natural Science Foundation of China (Nos. 11771064, 11991020, 11991023), the Natural Science Foundation of Chongqing (cstc2019jcyj-zdxmX0016), the grant of Guangdong Basic and Applied Basic Research Foundation (No. 2020A1515010463), the program for scientific research start-up funds of Guangdong Ocean University, the grant of Chongqing Normal University (No. 17XLB010). The fourth author is supported by RGC Grant PolyU. (152245/18E) and PolyU Grant ZZGS.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Guangdong Ocean University, Zhanjiang, Guangdong, People’s Republic of China

    Zhi Guo Feng

  2. School of Mathematical Sciences, Chongqing Normal University, Chongqing, People’s Republic of China

    Fei Chen & Lin Chen

  3. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, People’s Republic of China

    Lin Chen

  4. Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong, People’s Republic of China

    Ka Fai Cedric Yiu

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  1. Zhi Guo Feng
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  2. Fei Chen
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  3. Lin Chen
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  4. Ka Fai Cedric Yiu
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Correspondence to Lin Chen.

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Communicated by Marco Antonio López-Cerdá

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Feng, Z.G., Chen, F., Chen, L. et al. Optimality Analysis of a Class of Semi-infinite Programming Problems. J Optim Theory Appl 186, 398–411 (2020). https://doi.org/10.1007/s10957-020-01708-8

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  • DOI: https://doi.org/10.1007/s10957-020-01708-8

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