Abstract
In this paper, a new class of unified penalty functions are derived for the semi-infinite optimization problems, which include many penalty functions as special cases. They are proved to be exact in the sense that under Mangasarian–Fromovitz constraint qualification conditions, a local solution of penalty problem is a corresponding local solution of original problem when the penalty parameter is sufficiently large. Furthermore, global convergence properties are shown under some conditions. The paper is concluded with some numerical examples proving the applicability of our methods to PID controller design and linear-phase FIR digital filter design.
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References
Brayton R, Hachtel G, Sangiovanni-Vincentelli A (1981) A survey of optimization techniques for integrated-circuit design. Proc IEEE 69:1334–1362
Dam H, Teo K, Nordebo S, Cantoni A (2000) The dual parameterization approach to optimal least square FIR filter design subject to maximum error constraints. IEEE Trans Signal Process 48:2314–2320
Gonzaga C, Polak E, Trahan R (1980) An improved algorithm for optimization problems with functional inequality constraints. IEEE Trans Autom Control 25:49–54
Jennings L, Teo K (1990) A computational algorithm for functional inequality constrained optimization problems. Automatica 26:371–375
Jiang C, Lin Q, Yu C, Teo K, Duan G (2012) An exact penalty method for free terminal time optimal control problem with continuous inequality constraints. J Optim Theory Appl 154:30–53
Li D, Qi L, Tam J, Wu S (2004) A smoothing Newton method for semi-infinite programming. J Glob Optim 30:169–194
Lin Q, Loxton R, Teo K, Wu Y, Yu C (2014) A new exact penalty method for semi-infinite programming problems. J Comput Appl Math 261:271–286
Ling C, Qi L, Zhou G, Wu S (2006) Global convergence of a robust smoothing SQP method for semi-infinite programming. J Optim Theory Appl 129:147–164
Liu Q, Wang C, Yang X (2013) On the convergence of a smoothed penalty algorithm for semi-infinite programming. Math Methods Oper Res 78:203–220
Liu Q, Xu Y, Zhou Y (2019) A class of exact penalty functions and penalty algorithms for nonsmooth constrained optimization problem. J Glob Optim. https://doi.org/10.1007/s10898-019-00842-6
Ma C, Lee Y, Chan C, Wei Y (2017) Auction and contracting mechanisms for channel coordination with consideration of participants’ risk attitudes. J Ind Manag Optim 13:775–801
Nordebo S, Zang Z, Claesson I (2001) A semi-infinite quadratic programming algorithm with applications to array pattern synthesis. IEEE Trans Circuits Syst II 48:225–232
Panier E, Tits A (1989) A globally convergent algorithm with adaptively refined discretization for semi-infinite optimization problems arising in engineering design. IEEE Trans Autom Control 34:903–908
Qi L, Shapiro A, Ling C (2005) Differentiability and semismoothness properties of integral functions and their applications. Math Program 102:223–248
Qi L, Ling C, Tong X, Zhou G (2009) A smoothing projected Newton-type algorithm for semi-infinite programming. Comput Optim Appl 42:1–30
Teo K, Rehbock V, Jennings L (1993) A new computational algorithm for functional inequality constrained optimization problems. Autom J IFAC 29:789–792
Wang L, Gui W, Teo K, Loxton R, Yang C (2009) Time delayed optimal control problems with multiple characteristic time points: computation and industrial applications. J Ind Manag Optim 5:705–718
Wang C, Ma C, Zhou J (2014) A new class of exact penalty functions and penalty algorithms. J Glob Optim 58:51–73
Yu C, Teo K, Zhang L, Bai Y (2010) A new exact penalty function method for continuous inequality constrained optimization problems. J Ind Manag Optim 6:895–910
Yu C, Teo K, Zhang L, Bai Y (2012) On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem. J Ind Manag Optim 8:485–491
Yu C, Teo K, Zhang L (2014) A new exact penalty function approach to semi-infinite programming problem. In: Rassias T, Floudas C, Butenko S (eds) Optimization in science and engineering. Springer, New York, pp 583–596
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The work in this paper was supported by the National Natural Science Foundation of China (11271233, 11271226).
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Ju, J., Liu, Q. Convergence properties of a class of exact penalty methods for semi-infinite optimization problems. Math Meth Oper Res 91, 383–403 (2020). https://doi.org/10.1007/s00186-019-00693-7
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DOI: https://doi.org/10.1007/s00186-019-00693-7