非凸多目标优化模型的一类鲁棒逼近最优性条件
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摘要
通过引入一类非凸多目标不确定优化问题,借助鲁棒优化方法,先建立了该不确定多目标优化问题的鲁棒对应模型;再借助标量化方法和广义次微分性质,刻画了该不确定多目标优化问题的鲁棒拟逼近有效解的最优性条件,推广和改进了相关文献的结论.
A class of nonconvex multi-objective optimization problems were introduced with data uncertainty. Then, with the robust optimization approach, the robust counterpart model for the uncertain multi-objective optimization problem was built. Moreover, with the scalarization method and the generalized subdifferential properties, the optimality conditions were characterized for robust quasi approximate efficient solutions to the uncertain multi-objective optimization problem. The work generalizes and improves some results in the recent literatures.
A class of nonconvex multi-objective optimization problems were introduced with data uncertainty. Then, with the robust optimization approach, the robust counterpart model for the uncertain multi-objective optimization problem was built. Moreover, with the scalarization method and the generalized subdifferential properties, the optimality conditions were characterized for robust quasi approximate efficient solutions to the uncertain multi-objective optimization problem. The work generalizes and improves some results in the recent literatures.
引文
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[4] 赵勇,彭再云,张石生.向量优化问题有效点集的稳定性[J].应用数学和力学,2013,34(6):643-650.(ZHAO Yong,PENG Zaiyun,ZHANG Shisheng.Stability of the sets of effective points of vector valued optimization problems[J].Applied Mathematics and Mechanics,2013,34(6):643-650.(in Chinese))
[5] 陈望,周志昂.基于改进集的带约束集值向量均衡问题的最优性条件[J].应用数学和力学,2018,39(10):1189-1197.(CHEN Wang,ZHOU Zhiang.Optimality conditions for set-valued vector equilibrium problems with constraints involving improvement sets[J].Applied Mathematics and Mechanics,2018,39(10):1189-1197.(in Chinese))
[6] BEN-TAL A,GHAOUI L E,NEMIROVSKI A.Robust Optimization[M].Princeton:Princeton University Press,2009.
[7] LEE G M,LEE J H.On nonsmooth optimality theorems for robust multiobjective optimization problems[J].Journal of Nonlinear and Convex Analysis,2015,16(10):2039-2052.
[8] EHRGOTT M,IDE J,SCH?BEL A.Minmax robustness for multi-objective optimization problems[J].European Journal of Operational Research,2014,239(1):17-31.
[9] SUN X K,PENG Z Y,GUO X L.Some characterizations of robust optimal solutions for uncertain convex optimization problems[J].Optimization Letters,2016,10(7):1463-1478.
[10] FAKHAR M,MAHYARINIA M R,ZAFARANI J.On nonsmooth robust multiobjective optimization under generalized convexity with applications to portfolio optimization[J].European Journal of Operational Research,2018,265(1):39-48.
[11] LORIDAN P.ε-solutions in vector minimization problems[J].Journal of Optimization Theory and Application,1984,43(2):265-276.
[12] SON T Q,STRODIOT J J,NGUYEN V H.ε-optimality and ε-Lagrangian duality for a nonconvex programming problem with an infinite number of constraints[J].Journal of Optimization Theory and Application,2009,141(2):389-409.
[13] SON T Q,KIM D S.ε-mixed type duality for nonconvex multiobjective programs with an infinite number of constraints[J].Journal of Global Optimization,2013,57(2):447-465.
[14] SUN X K,LI X B,LONG X J,et al.On robust approximate optimal solutions for uncertain convex optimization and applications to multi-objective optimization[J].Pacific Journal of Optimization,2017,13(4):621-643.
[15] CLARKE F H.Optimization and Nonsmooth Analysis[M].New York:John Wiley & Sons,1983.
[16] MIFFLIN R.Semismooth and semiconvex functions in constrained optimization[J].SIAM Journal on Control and Optimization,1977,15(6):959-972.
[2] BOT R I,GRAD S M,WANKA G.Duality in Vector Optimization[M].Berlin:Springer-Verlag,2009.
[3] 彭再云,李科科,张石生.D-η-E-半预不变凸映射与向量优化[J].应用数学和力学,2014,35(9):1020-1032.(PENG Zaiyun,LI Keke,ZHANG Shisheng.D-η-E-semipreinvex vector mappings and vector optimization[J].Applied Mathematics and Mechanics,2014,35(9):1020-1032.(in Chinese))
[4] 赵勇,彭再云,张石生.向量优化问题有效点集的稳定性[J].应用数学和力学,2013,34(6):643-650.(ZHAO Yong,PENG Zaiyun,ZHANG Shisheng.Stability of the sets of effective points of vector valued optimization problems[J].Applied Mathematics and Mechanics,2013,34(6):643-650.(in Chinese))
[5] 陈望,周志昂.基于改进集的带约束集值向量均衡问题的最优性条件[J].应用数学和力学,2018,39(10):1189-1197.(CHEN Wang,ZHOU Zhiang.Optimality conditions for set-valued vector equilibrium problems with constraints involving improvement sets[J].Applied Mathematics and Mechanics,2018,39(10):1189-1197.(in Chinese))
[6] BEN-TAL A,GHAOUI L E,NEMIROVSKI A.Robust Optimization[M].Princeton:Princeton University Press,2009.
[7] LEE G M,LEE J H.On nonsmooth optimality theorems for robust multiobjective optimization problems[J].Journal of Nonlinear and Convex Analysis,2015,16(10):2039-2052.
[8] EHRGOTT M,IDE J,SCH?BEL A.Minmax robustness for multi-objective optimization problems[J].European Journal of Operational Research,2014,239(1):17-31.
[9] SUN X K,PENG Z Y,GUO X L.Some characterizations of robust optimal solutions for uncertain convex optimization problems[J].Optimization Letters,2016,10(7):1463-1478.
[10] FAKHAR M,MAHYARINIA M R,ZAFARANI J.On nonsmooth robust multiobjective optimization under generalized convexity with applications to portfolio optimization[J].European Journal of Operational Research,2018,265(1):39-48.
[11] LORIDAN P.ε-solutions in vector minimization problems[J].Journal of Optimization Theory and Application,1984,43(2):265-276.
[12] SON T Q,STRODIOT J J,NGUYEN V H.ε-optimality and ε-Lagrangian duality for a nonconvex programming problem with an infinite number of constraints[J].Journal of Optimization Theory and Application,2009,141(2):389-409.
[13] SON T Q,KIM D S.ε-mixed type duality for nonconvex multiobjective programs with an infinite number of constraints[J].Journal of Global Optimization,2013,57(2):447-465.
[14] SUN X K,LI X B,LONG X J,et al.On robust approximate optimal solutions for uncertain convex optimization and applications to multi-objective optimization[J].Pacific Journal of Optimization,2017,13(4):621-643.
[15] CLARKE F H.Optimization and Nonsmooth Analysis[M].New York:John Wiley & Sons,1983.
[16] MIFFLIN R.Semismooth and semiconvex functions in constrained optimization[J].SIAM Journal on Control and Optimization,1977,15(6):959-972.