MultiSQP-GS: a sequential quadratic programming algorithm via gradient sampling for nonsmooth constrained multiobjective optimization

In this paper, we propose a method for solving constrained nonsmooth multiobjective optimization problems which is based on a Sequential Quadratic Programming (SQP) type approach and the Gradient Sampling (GS) technique. We consider the multiobjective problems with noncovex and nonsmooth objective a...

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Vydané v:	Computational optimization and applications Ročník 89; číslo 3; s. 729 - 767
Hlavní autori:	Rashidi, Mehri, Soleimani-damaneh, Majid
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.12.2024
Predmet:	Convex and Discrete Geometry Management Science Mathematics Mathematics and Statistics Operations Research Operations Research/Decision Theory Optimization Statistics Gradient Sampling (GS) Nonconvex optimization 49J52 Multiobjective programming 90C29 Nonsmooth optimization 90C26 Exact penalization Sequential Quadratic Programming (SQP) Constrained optimization
ISSN:	0926-6003, 1573-2894
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Abstract	In this paper, we propose a method for solving constrained nonsmooth multiobjective optimization problems which is based on a Sequential Quadratic Programming (SQP) type approach and the Gradient Sampling (GS) technique. We consider the multiobjective problems with noncovex and nonsmooth objective and constraint functions. The problem functions are assumed to be locally Lipschitz. Such problems arise in important applications, many having (weak) Pareto solutions at points of nondifferentiability of the problem functions. In our algorithm, a penalty function is applied to regularize the constraints, GS is employed to overcome the subdifferential calculation burden and make the search direction computation effective in nonsmooth regions, and SQP is used for getting a local linearization. We prove the global convergence properties of our algorithm to the stationary points which approximate (weak) Pareto front. Furthermore, we illustrate the ability and efficiency of the proposed method via a MATLAB implementation on several tests problems and compare it with some existing algorithms.
AbstractList	In this paper, we propose a method for solving constrained nonsmooth multiobjective optimization problems which is based on a Sequential Quadratic Programming (SQP) type approach and the Gradient Sampling (GS) technique. We consider the multiobjective problems with noncovex and nonsmooth objective and constraint functions. The problem functions are assumed to be locally Lipschitz. Such problems arise in important applications, many having (weak) Pareto solutions at points of nondifferentiability of the problem functions. In our algorithm, a penalty function is applied to regularize the constraints, GS is employed to overcome the subdifferential calculation burden and make the search direction computation effective in nonsmooth regions, and SQP is used for getting a local linearization. We prove the global convergence properties of our algorithm to the stationary points which approximate (weak) Pareto front. Furthermore, we illustrate the ability and efficiency of the proposed method via a MATLAB implementation on several tests problems and compare it with some existing algorithms.
Author	Rashidi, Mehri Soleimani-damaneh, Majid
Author_xml	– sequence: 1 givenname: Mehri surname: Rashidi fullname: Rashidi, Mehri organization: Faculty of Mathematics and Computer Science, Amirkabir University of Technology – sequence: 2 givenname: Majid orcidid: 0000-0002-5913-1035 surname: Soleimani-damaneh fullname: Soleimani-damaneh, Majid email: m.soleimani.d@ut.ac.ir, soleimani_d@yahoo.com organization: School of Mathematics, Statistics, and Computer Science, College of Science, University of Tehran
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References_xml	– reference: EhrgottMMulticriteria Optimization2005BerlinSpringer – reference: Jin, Y., Olhofer, M., Sendhoff, B.: Dynamic weighted aggregation for evolutionary multiobjective opti- mization: Why does it work and how? In: Proceedings of the genetic and evolutionary computation conference, 1042–1049 (2001) – reference: Lukšan, L., Vlček., J.: Test problems for nonsmooth unconstrained and linearly constrained optimization. Technical report, No. 798, Academy of Sciences of the Czech Republic (2000) – reference: BurkeJVLewisASOvertonMLA robust gradient sampling algorithm for nonsmooth, nonconvex optimizationSIAM J. Control. Optim.2005157517792142859 – reference: NocedalJWrightSJNumerical Optimization20062BerlinSpringer-Verlag – reference: RockafellarRTWetsRJBVariational Analysis1998BerlinSpringer – reference: Mäkelä, M., Karmitsa, N., Wilppu, O.: Multiobjective proximal bundle method for nonsmooth optimization. TUCS Technical Report, No. 1120, Turku Optimization Group (2014) – reference: MengKWLiMHYaoWFYangXQLipschitz-like property relative to a set and the generalized Mordukhovich criterionMath. Program.20211894554894306081 – reference: Beck, A.: Introduction to Nonlinear Optimization. MOS-SIAM Series in Optimization (2014) – reference: ChanWLHuangLRNgKFOn generalized second-order derivatives and Taylor expansions in nonsmooth optimizationSIAM J. Control. Optim.19943235916111269984 – reference: MukaiHAlgorithms for multicriterion optimizationIEEE Trans. Autom. Control1980252177186567375 – reference: KiwielKCConvergence of the gradient sampling algorithm for nonsmooth nonconvex optimizationSIAM J. Optim.2007183793882338443 – reference: CustodioALMadeiraJFVazAIVicenteLNDirect multisearch for multiobjective optimizationSIAM J. Optim.201121110911402837565 – reference: DebKPratapAMeyarivanTZitzlerEThieleLDebKCoello CoelloCACorneDConstrained test problems for multi-objective evolutionary optimizationEvolutionary Multi-Criterion Optimization2001BerlinSpringer284298 – reference: DebKThieleLLaumannsMZitzlerEScalable Test Problems for Evolutionary Multiobjective Optimization2005LondonSpringer – reference: FukudaEHDrummondLGInexact projected gradient method for vector optimizationComput. Optim. Appl.2013544734933029305 – reference: BonnelHIusemANSvaiterBFProximal methods in vector optimizationSIAM J. Optim.20051549539702178482 – reference: DrummondLGA Projected gradient method for vector optimization problemsComput. Optim. Appl.2004285292049673 – reference: CocchiGLapucciMMansuetoPPareto front approximation through a multi-objective augmented Lagrangian methodEURO J. Comput. Optim.202191000084397740 – reference: Knowles, J., Thiele, L., Zitzler, E.: A tutorial on the performance assessment of stochastic multiobjective optimizers. TIK Report 214, ETH Zurich (2006) – reference: HaaralaMMiettinenKMäkeläMMNew limited memory bundle method for large-scale nonsmooth optimizationOptim. Methods Softw.20041966736922102221 – reference: KiwielKCMethods of Descent for Nondifferentiable Optimization1985BerlinSpringer-Verlag – reference: MiettinenKNonlinear Multiobjective Optimization1999New YorkSpringer – reference: Andrei, N.: Modern Numerical Nonlinear Optimization. Springer (2022) – reference: ByrdRHLopez-CalvaGNocedalJA line search exact penalty method using steering rulesMath. Program.201213339732921091 – reference: DolanEDMoréJJBenchmarking optimization software with performance profilesMath. Program.2002912012131875515 – reference: EichfelderGAdaptive Scalarization Methods in Multiobjective Optimization2008BerlinSpringer – reference: BandyopadhyaySPalSKArunaBMultiobjective GAs, quantitative indices, and pattern classificationIEEE Trans. Syst. Man Cybern Part B (Cybern)200434520882099 – reference: ClarkeFHFunctional Analysis, Calculus of Variations and Optimal Control2013LondonSpringer – reference: FukudaEHDrummondLGRauppFMAn external penalty-type method for multicriteriaTOP2016244935133509465 – reference: MordukhovichBSVariational Analysis and Generalized Differentiation I: Basic Theory2013ChamSpringer – reference: MäkeläMNeittaanmakiPNonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control1992SingaporeWorld Scientific – reference: Mäkelä, M., Montonen, O.: New multiobjective proximal bundle method with scaled improvement function. In: Bagirov, A., Gaudioso, M., Karmitsa, N., Mäkelä, M., Taheri, S. (eds). Numerical Nonsmooth Optimization. Springer (2020) – reference: MorovatiVPourkarimiLExtension of Zoutendijk method for solving constrained multiobjective optimization problemsEur. J. Oper. Res.201927344573863513 – reference: AubinJ-PLipschitz behavior of solutions to convex minimization problemsMath. Oper. Res.1984987111736641 – reference: CocchiGLapucciMAn augmented Lagrangian algorithm for multi-objective optimizationComput. Optim. Appl.202077129564123858 – reference: CurtisFEQueXAn adaptive gradient sampling algorithm for non-smooth optimizationOptim. Methods Softw.2013286130213243175468 – reference: FliegeJSvaiterBFSteepest descent methods for multicriteria optimizationMath. Methods Oper. 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Snippet	In this paper, we propose a method for solving constrained nonsmooth multiobjective optimization problems which is based on a Sequential Quadratic Programming...
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SubjectTerms	Convex and Discrete Geometry Management Science Mathematics Mathematics and Statistics Operations Research Operations Research/Decision Theory Optimization Statistics
Title	MultiSQP-GS: a sequential quadratic programming algorithm via gradient sampling for nonsmooth constrained multiobjective optimization
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