Abstract
We propose new descent methods for unconstrained multiobjective optimization problems, where each objective function can be written as the sum of a continuously differentiable function and a proper convex but not necessarily differentiable one. The methods extend the well-known proximal gradient algorithms for scalar-valued nonlinear optimization, which are shown to be efficient for particular problems. Here, we consider two types of algorithms: with and without line searches. Under mild assumptions, we prove that each accumulation point of the sequence generated by these algorithms, if exists, is Pareto stationary. Moreover, we present their applications in constrained multiobjective optimization and robust multiobjective optimization, which is a problem that considers uncertainties. In particular, for the robust case, we show that the subproblems of the proximal gradient algorithms can be seen as quadratic programming, second-order cone programming, or semidefinite programming problems. Considering these cases, we also carry out some numerical experiments, showing the validity of the proposed methods.
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Notes
We denote \(A \succeq (\succ ) O\) when A is positive semidefinite (positive definite). Also, \(A \succeq (\succ ) B\) if and only if \(A - B \succeq (\succ ) O\).
Here, \(\dim \) denotes dimension of a space and \(\ker \) means kernel of a matrix.
References
Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95(1), 1–52 (2001)
Beck, A., Eldar, Y.: Strong duality in nonconvex quadratic optimization with two quadratic constraints. SIAM J. Optim. 17(3), 844–860 (2006)
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)
Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23(4), 769–805 (1998)
Berge, C., Patterson, E.M.: Topological Spaces. Dover Publications, Edinburgh (1963)
Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)
Bertsekas, D.P.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)
Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15(4), 953–970 (2005)
Chen, G., Teboulle, M.: Convergence analysis of a proximal-like minimization algorithm using Bregman functions. SIAM J. Optim. 3(3), 538–543 (1993)
Cruz Neto, J.X., Silva, G.J.P., Ferreira, O.P., Lopes, J.O.: A subgradient method for multiobjective optimization. Comput. Optim. Appl. 54(3), 461–472 (2013)
Ehrgott, M., Ide, J., Schöbel, A.: Minmax robustness for multi-objective optimization problems. Eur. J. Oper. Res. 239(1), 17–31 (2014)
Fliege, J., Graña Drummond, L.M., Svaiter, B.F.: Newton’s method for multiobjective optimization. SIAM J. Optim. 20(2), 602–626 (2009)
Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51(3), 479–494 (2000)
Fliege, J., Werner, R.: Robust multiobjective optimization and applications in portfolio optimization. Eur. J. Oper. Res. 234(2), 422–433 (2014)
Fukuda, E.H., Graña Drummond, L.M.: A survey on multiobjective descent methods. Pesquisa Operacional 34(3), 585–620 (2014)
Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22(3), 618–630 (1968)
Graña Drummond, L.M., Iusem, A.N.: A projected gradient method for vector optimization problems. Comput. Optim. Appl. 28(1), 5–29 (2004)
Hogan, W.: Point-to-set maps in mathematical programming. SIAM Rev. 15(3), 591–603 (1973)
Morishita, M.: A descent method for robust multiobjective optimization in the presence of implemention errors. Master’s thesis, Kyoto University (2016)
Saul, G., Thomas, S.: The computational algorithm for the parametric objective function. Naval Res. Logist. Q. 2(12), 39–45 (1955)
Sturm, J.F.: Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11(1–4), 625–653 (1999)
Tseng, P.: Approximation accuracy, gradient methods, and error bound for structured convex optimization. Math. Program. 125(2), 263–295 (2010)
Zadeh, L.: Optimality and non-scalar-valued performance criteria. IEEE Trans. Autom. Control 8(1), 59–60 (1963)
Acknowledgements
This work was supported by the Kyoto University Foundation, and the Grant-in-Aid for Scientific Research (C) (17K00032) from Japan Society for the Promotion of Science. We are also grateful to the anonymous referees for their useful comments.
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Tanabe, H., Fukuda, E.H. & Yamashita, N. Proximal gradient methods for multiobjective optimization and their applications. Comput Optim Appl 72, 339–361 (2019). https://doi.org/10.1007/s10589-018-0043-x
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DOI: https://doi.org/10.1007/s10589-018-0043-x