Volume 43 Issue 3
Mar.  2022
Turn off MathJax
Article Contents
SHI Xiaobo, GAO Ying. Properties of Quasiconvex Functions and Their Applications in Multiobjective Optimization Problems[J]. Applied Mathematics and Mechanics, 2022, 43(3): 322-329. doi: 10.21656/1000-0887.420275
Citation: SHI Xiaobo, GAO Ying. Properties of Quasiconvex Functions and Their Applications in Multiobjective Optimization Problems[J]. Applied Mathematics and Mechanics, 2022, 43(3): 322-329. doi: 10.21656/1000-0887.420275

Properties of Quasiconvex Functions and Their Applications in Multiobjective Optimization Problems

doi: 10.21656/1000-0887.420275
  • Received Date: 2021-09-09
  • Accepted Date: 2021-10-28
  • Rev Recd Date: 2021-09-23
  • Available Online: 2022-02-12
  • Publish Date: 2022-03-08
  • A new type of approximate subdifferential was proposed for quasiconvex functions. Their properties were studied, and the approximate subdifferential was applied to the characterization of approximate solutions to quasiconvex multiobjective optimization problems. Firstly, the existing approximate subdifferentials were improved to get a new approximate subdifferential of the quasiconvex function, and their relationships and properties were given. Then, the optimality conditions for approximate efficient solutions and approximate properly efficient solutions to quasiconvex multiobjective optimization problems were obtained by means of the new approximate subdifferential.

  • loading
  • [1]
    MANGASARIAN O L. Pseudo function[J]. Journal of the Society for Industrial and Applied Mathematics, 1965, 3(2): 23-32.
    [2]
    DUCA D I, LUPA L. On the E-epigraph of an E-convex function[J]. Journal of Optimization Theory and Applications, 2006, 129(2): 341-348. doi: 10.1007/s10957-006-9059-y
    [3]
    王海英, 符祖峰. D-η-E-半预不变凸映射和D-η-E-半预不变真拟凸映射[J]. 应用数学和力学, 2019, 40(3): 111-122. (WANG Haiying, FU Zufeng. D-η-E-semi-preinvex mapping and D-η-E-properly semi-prequasi-invex mapping[J]. Applied Mathematics and Mechanics, 2019, 40(3): 111-122.(in Chinese)
    [4]
    刘娟, 龙宪军. 非光滑多目标半无限规划问题的混合型对偶[J]. 应用数学和力学, 2021, 42(6): 595-601. (LIU Juan, LONG Xianjun. Mixed type duality for nonsmooth multiobjective semi-infinite programming problems[J]. Applied Mathematics and Mechanics, 2021, 42(6): 595-601.(in Chinese)
    [5]
    DE FINETTI B. Sulle stratificazioni convesse[J]. Annali di Matematica Pura ed Applicata, 1949, 30(1): 173-183. doi: 10.1007/BF02415006
    [6]
    FENCHEL W. Convex Cones, Sets, and Functions[M]. Princeton University, 1953.
    [7]
    杨新民. 拟凸函数的某些性质[J]. 工程数学学报, 1993, 10(1): 51-56. (YANG Xinmin. Some properties of quasiconvex functions[J]. Journal of Engineering Mathematics, 1993, 10(1): 51-56.(in Chinese)
    [8]
    YANG X M, LIU S Y. Technical note three kind of generalized convexity[J]. Journal of Optimization Theory and Applications, 1995, 86(2): 501-513. doi: 10.1007/BF02192092
    [9]
    杨新民, 戎卫东. 广义凸性及其应用[M]. 北京: 科学出版社, 2016.

    YANG Xinmin, RONG Weidong. Generalized Convexity and Its Application[M]. Beijing: Science Press, 2016. (in Chinese)
    [10]
    高岩. 非光滑优化[M]. 北京: 科学出版社, 2008.

    GAO Yan. Nonsmooth Optimization[M]. Beijing: Science Press, 2008. (in Chinese)
    [11]
    GREENBERG H P, PIERSKALLA W P. Quasi-conjugate functions and surrogate duality[J]. Cahiers du Centre d’Etude de Recherche Operationelle, 1973, 15: 437-448.
    [12]
    PENOT J P, ZALINESCU C. Elements of quasiconvex subdifferential calculus[J]. Journal of Convex Analysis, 2000, 7(7): 243-269.
    [13]
    GUTIÉRREZ D J M. Infragradients and directions of decrease[J]. Rev Real Acad Cienc Exact Fís Natur Madrid, 1984, 78(4): 523-532.
    [14]
    PLASRTIA F. Lower subdifferentiable functions and their minimization by cutting planes[J]. Journal of Optimization Theory and Applications, 1985, 46(1): 37-53. doi: 10.1007/BF00938758
    [15]
    PENOT J P. What is quasiconvex analysis?[J]. Optimization, 2000, 47(1/2): 35-110.
    [16]
    PENOT J P. Characterization of solution sets of quasiconvex programs[J]. Journal of Optimization Theory and Applications, 2003, 117(3): 627-636. doi: 10.1023/A:1023905907248
    [17]
    NGUYEN T H L, PENOT J P. Optimality conditions for quasiconvex programs[J]. SIAM Journal on Optimization, 2006, 17(2): 500-510. doi: 10.1137/040621843
    [18]
    SUZUKI S, KUROIWA D. Optimality conditions and the basic constraint qualification for quasiconvex programming[J]. Nonlinear Analysis Theory Methods and Applications, 2011, 74(4): 1279-1285. doi: 10.1016/j.na.2010.09.066
    [19]
    KHANH P Q, QUYEN H T, YAO J C. Optimality conditions under relaxed quasiconvexity assumptions using star and adjusted subdifferentials[J]. European Journal of Operational Research, 2011, 212(2): 235-241. doi: 10.1016/j.ejor.2011.01.024
    [20]
    陈瑞婷, 徐智会, 高英. 拟凸多目标优化问题近似解的最优性条件[J]. 运筹学学报, 2019, 23(1): 35-44. (CHEN Ruiting, XU Zhihui, GAO Ying. The optimality conditions of approximate solutions for quasiconvex multiobjective optimization problem[J]. Operations Research Transactions, 2019, 23(1): 35-44.(in Chinese)
    [21]
    岳瑞雪, 高英. 多目标优化问题(ε, $ \bar \varepsilon $)-拟近似真有效解的非线性标量化[J]. 数学的实践与认识, 2015, 45(21): 282-289. (YUE Ruixue, GAO Ying. Nonlinear scalarizations for (ε, $ \bar \varepsilon $)-approximate quasi solutions of multiobjective optimization problems[J]. Mathematics in Practice and Theory, 2015, 45(21): 282-289.(in Chinese)
  • 加载中

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Article Metrics

    Article views (606) PDF downloads(80) Cited by()
    Proportional views
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return