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On the convergence analysis of a proximal gradient method for multiobjective optimization

  • Xiaopeng Zhao [2] ; Debdas Ghosh [3] ; Xiaolong Qin [1] ; Christiane Tammer [4] ; Jen-Chih Yao [5]
    1. [1] Zhejiang Normal University

      Zhejiang Normal University

      China

    2. [2] chool of Mathematical Sciences, Tiangong University, Tianjin, China
    3. [3] Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, India
    4. [4] Department of Mathematics, Martin-Luther-University Halle-Wittenberg, Germany
    5. [5] Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung, Taiwan
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  • Localización: Top, ISSN-e 1863-8279, ISSN 1134-5764, Vol. 33, Nº. 1, 2025, págs. 102-132
  • Idioma: inglés
  • Enlaces
  • Resumen

    • We propose a proximal gradient method for unconstrained nondifferentiable multiobjective optimization problems with the objective function being the sum of a proper lower semicontinuous convex function and a continuously differentiable function. We have shown under appropriate assumptions that each accumulation point of the sequence generated by the algorithm is Pareto stationary. Further, when imposing convexity on the smooth component of the objective function, the convergence of the generated iterative sequence to a weak Pareto optimal point of the problem is established. Meanwhile, the convergence rate of the proposed method is analyzed when the smooth component function in the objective function is non-convex (O(1/k)), convex (O(1/k)), and strongly convex (O(rk) for some r∈(0,1)), respectively, here k is the number of iterations. The performance of the proposed method on a few test problems with an ℓ1-norm function and with the indicator function is provided.

  • Referencias bibliográficas
    • Attouch H, Bolte J, Redont P, Soubeyran A (2010) Proximal alternating minimization and projection methods for nonconvex problems: an approach...
    • Bagchi U (1989) Simultaneous minimization of mean and variation of flow time and waiting time in single machine systems. Oper Res 37:118–125....
    • Baltar M, Abreu V, Ribeiro G, Bahiense L (2021) Multi-objective model for the problem of locating tows for incident servicing on expressways....
    • Beck A (2017) First-order methods in optimization. MOS-SIAM Ser. Optim., vol 25. SIAM, Philadelphia
    • Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J Imaging Sci 2:183–202. https://doi.org/10.1137/080716542
    • Bello-Cruz Y, Melo JG, Serra RVG (2022) A proximal gradient splitting method for solving convex vector optimization problem. Optimization...
    • Bento GC, Cruz Neto JX, López G, Soubeyran A, Souza JCO (2018) The proximal point method for locally Lipschitz functions in multiobjective...
    • Bolte J, Nguyen TP, Peypouquet J, Suter BW (2016) From error bounds to the complexity of first-order descent methods for convex functions....
    • Bonnel H, Iusem AN, Svaiter BF (2005) Proximal methods in vector optimization. SIAM J Optim 15:953– 970. https://doi.org/10.1137/S1052623403429093
    • Bo¸t RI, Grad SM (2018) Inertial forward-backward methods for solving vector optimization problems. Optimization 67:959–974. https://doi.org/10.1080/02331934.2018.1440553
    • Branke J, Deb K, Miettinen K, Slowinski R (2008) Multiobjective optimization: interactive and evolutionary approaches. Springer, Berlin
    • Burachik R, Graña Drummond LM, Iusem AN, Svaiter BF (1995) Full convergence of the steepest descent method with inexact line searches. Optimization...
    • Burachik RS, Kaya CY, Rizvi MM (2017) A new scalarization technique and new algorithms to generate Pareto fronts. SIAM J Optim 27:1010–1034....
    • Carrizo GA, Lotito PA, Maciel MC (2016) Trust region globalization strategy for the nonconvex unconstrained multiobjective optimization problem....
    • Chankong V, Haimes YY (1983) Multiobjective decision making. North-Holl and Publishing Co., New York
    • Chen GY, Huang XX, Yang XQ (2005) Vector optimization: set-valued and variational analysis. Springer, Berlin
    • Combettes PL, Wajs VR (2005) Signal recovery by proximal forward-backward splitting. Multiscale Model Simul 4:1168–1200. https://doi.org/10.1137/050626090
    • Eichfelder G (2008) Adaptive scalarization methods in multiobjective optimization. Springer, Berlin
    • Ermol’ev YM (1969) On the method of generalized stochastic gradients and quasi-Fejér sequences. Cybernetics 5:208–220. https://doi.org/10.1007/BF01071091
    • Eschenauer H, Koski J, Osyczka A (2012) Multicriteria design optimization: procedures and applications.
    • Fliege J, Svaiter BF (2000) Steepest descent methods for multicriteria optimization. Math Methods Oper Res 51:479–494. https://doi.org/10.1007/s001860000043
    • Fliege J, Graña Drummond LM, Svaiter BF (2009) Newton’s method for multiobjective optimization. SIAM J Optim 20:602–626. https://doi.org/10.1137/08071692X
    • Fliege J, Vaz AIF, Vicente LN (2019) Complexity of gradient descent for multiobjective optimization. Optim Methods Softw 34:949–959. https://doi.org/10.1080/10556788.2018.1510928
    • Göpfert A, Riahi H, Tammer C, Z˘alinescu C (2003) Variational methods in partially ordered spaces. Springer, Berlin
    • Graña Drummond LM, Iusem AN (2004) A projected gradient method for vector optimization problems. Comput Optim Appl 28:5–29. https://doi.org/10.1023/B:COAP.0000018877.86161.8b
    • Graña Drummond LM, Svaiter BF (2005) A steepest descent method for vector optimization. J Comput Appl Math 175:395–414. https://doi.org/10.1016/j.cam.2004.06.018
    • Graña Drummond LM, Maculan N, Svaiter BF (2008) On the choice of parameters for the weighting method in vector optimization. Math Program...
    • Hiriart Urruty J-B, Lemarèchal C (1993) Convex analysis and minimization algorithms. Springer, Berlin
    • Hu YH, Li C, Meng KW, Yang XQ (2021) Linear convergence of inexact descent method and inexact proximal gradient algorithms for lower-order...
    • Huband S, Hingston P, Barone L, While L (2006) A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans Evolut...
    • Iusem AN, Svaiter BF, Teboulle M (1994) Entropy-like proximal methods in convex programming. Math Oper Res 19:790–814. https://doi.org/10.2307/3690314
    • Jahn J (2011) Vector optimization: theory, applications, and extensions. Springer, New York
    • Kacem A, Dammak A (2021) Multi-objective scheduling on two dedicated processors. TOP 29:694–721. https://doi.org/10.1007/s11750-020-00588-5
    • Leschine TM, Wallenius H, Verdini WA (1992) Interactive multiobjective analysis and assimilative capacity-based ocean disposal decisions....
    • Luc DT (1989) Theory of vector optimization, lecture notes in economics and mathematical systems. Springer, Berlin
    • Lucambio Pérez LR, Prudente LF (2018) Nonlinear conjugate gradient methods for vector optimization. SIAM J Optim 28:2690–2720. https://doi.org/10.1137/17M1126588
    • Miettinen KM (2012) Nonlinear multiobjective optimization. Springer, Berlin
    • Mita K, Fukuda EH, Yamashita N (2019) Nonmonotone line searches for unconstrained multiobjective optimization problems. J Glob Optim 75:63–90....
    • Nesterov Y (2004) Introductory lectures on convex optimization. Kluwer Academic Publishers, Dordrecht
    • Papa Quiroz EA, Cruzado S (2022) An inexact scalarization proximal point method for multiobjective quasiconvex minimization. Ann Oper Res...
    • Parikh N, Boyd S (2014) Proximal algorithms. Found Trends Optim 1:127–239. https://doi.org/10.1561/ 2400000003
    • Pascoletti A, Serafini P (1984) Scalarizing vector optimization problems. J Optim Theory Appl 42:499–524. https://doi.org/10.1007/bf00934564
    • Rockafellar RT (1970) Convex analysis. Priceton University Press, Priceton
    • Tanabe H, Fukuda EH, Yamashita N (2019) Proximal gradient methods for multiobjective optimization and their applications. Comput Optim Appl...
    • Tanabe H, Fukuda EH, Yamashita N (2023a) Convergence rates analysis of a multiobjective proximal gradient method. Optim Lett 17:333–350. https://doi.org/10.1007/s11590-022-01877-7
    • Tanabe H, Fukuda EH, Yamashita N (2023b) An accelerated proximal gradient method for multiobjective optimization. Comput Optim Appl 86:421–455....
    • Tanabe H, Fukuda EH, Yamashita N (2023c) New merit functions for multiobjective optimization and their properties. Optimization. https://doi.org/10.1080/02331934.2023.2232794
    • Tapia M, Coello C (2007) Applications of multi-objective evolutionary algorithms in economics and finance: a survey. In: Proceedings of the...
    • Tibshirani R (1996) Regression shrinkage and selection via the Lasso. J R Stat Soc Ser B (Methodological) 58:267–288. https://doi.org/10.1111/j.2517-6161.1996.tb02080.x
    • Tseng P (2010) Approximation accuracy, gradient methods, and error bound for structured convex optimization. Math Program 125:263–295. https://doi.org/10.1007/s10107-010-0394-2
    • Wang JH, Hu YH, Yu CKW, Li C, Yang XQ (2019) Extended Newton methods for multiobjective optimization: majorizing function technique and convergence...
    • Wen B, Chen X, Pong TK (2017) Linear convergence of proximal gradient algorithm with extrapolation for a class of nonconvex nonsmooth minimization...
    • Zhao XP, Köbis MA, Yao YH, Yao JC (2021) A projected subgradient method for nondifferentiable quasiconvex multiobjective optimization problems....
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