非凸映射的度量正则性和非凸优化
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摘要
利用泛函分析、变分分析和非光滑分析的方法和技术研究闭集值映射的度量正则性,度量次正则性,calmness,数学规划问题及向量优化问题,受到人们的密切关注,被广泛地应用于许多领域.本文在其他研究者的基础上,我们主要获得了如下四个方面成果：
一、给出一个单值映射g(·)与一个多值映射Q(·)之和M(·)=g(·)+Ω(·)的Clarke切锥,法锥及Bouligand切锥通过Ω(·)的相应切锥及法锥描述的表达式,讨论了M的subsmoothness及L-subsmoothness;给出(σ,δ)-subsmooth闭集值映射具有度量正则性的若干等价条件,把Ursescu-Robinson定理及Zheng和Ng的有关结果推广到更一般的非凸情形.利用正规对偶映射,给出(σ,δ)-subsmooth闭集值映射具有度量次正则性的充分与必要条件和point-based条件,改进并推广了Zheng和Ng等的有关结论.
二、研究了不等式和光滑等式约束的优化问题的sharp minimum与strong KKT条件、weak sharp minimum与quasi-strong KKT和sub-quasi-strong KKT条件的关系,并在Lipschitz条件及quasi-subsmooth条件下,获得了该类优化问题满足各种strong KKT条件的等价描述,把Zheng和Ng关于复合凸情形的有关结论推广到更一般的非凸情形.
三、获得了有限维空间上的Ck、1向量值函数的七+1阶Clarke次微分形式的泰勒公式.在(k+1)阶次微分正定的假设下,给出此类函数的高阶Clarke次微分形式的pareto及弱Pareto解的充分条件.在一般Banach空间中,我们定义了Ck·1函数的Michel-Penot高阶次微分,讨论了它的基本性质,并给出高阶次微分形式的Taylor中值公式,利用其研究了一般Banach空间上Ck、1映射的Michel-Penot同阶次微分优化条件.
四、在Banach空问中,我们证明,当目标集值映射F的图Gr(F)是有限个广义多面体之并且值集F(Γ)与序锥C的和F(Γ)+C是凸的或者序锥C是具有非空内部的多面体时,其弱pareto解集和弱Pareto最优值集也是有限个广义多面体之并.当目标集值映射的图Gr(F)是有限个凸多面体之并且值集F(Γ)与序锥C的和F(Γ)+C是凸的时,若C在Y关于F下的分解有限维子空间中的投影是闭的,则其pareto解集和最优值集也是有限个凸多面体之并.在去掉目标集值映射关于序锥是凸的假设下,序锥是广义凸多面体时,我们证明,其弱Pareto解集和弱最优值集也是有限个广义多面体之并.最后,我们证明,当目标映射关于序锥C是凸的时,其Pareto解集和pareto最优值集是道路连通的.
By techniques and methods of functional analysis, variational analysis and non-smooth analysis, many people have focused on such problems as metric regularity, metric subregularity and calmness of solutions for closed multifunctions, mathe-matic programming and vector optimization problems. They are widely used in many fields. Upon other researchers'work, in this dissertation, we mainly obtained four aspects of results below:
1. We build representations of the Clarke tangent cone, the Clarke normal cone and the Bouligand tangent cone for the multifunction M(·)= g(·)+Ω(·) by the corresponding ones for the multifunctionΩ(·). We also discuss sub smoothness and L-subsmoothness for M. We build some equivalent conditions of metric regularity for a (σ,δ)-subsmooth multifunction, in which we generalize Ursescu-Robinson theorem and Zheng and Ng's corresponding results to the more general nonconvex case. Finally, by the normal dual mapping, we provide some sufficient and necessary conditions and point-based conditions for metric subregularity of a (σ,δ)-subsmooth multifunction, improving and generalizing the corresponding results of zheng, Ng and others.
2. We research the relations between the sharp minimum and the strong KKT, between weak sharp minima and the quasi-strong KKT and the sub-quasi-strong KKT for the optimal problems with the constraint conditions of inequalities and equalities. Under the Lipschitzian and quasi-subsmooth conditions, We obtain equivalent conditions for the above strong KKTs. Our results generalize the results of Zheng and Ng in the convex-composite case to the more general nonconvex case.
3. We obtain the Taylor formula of Ck,l vector functions in the form of the (k+1)th order Clarke subdifferential. Under the assumption of positive definiteness of the (k+1)th order subdifferential, we provide sufficient conditions of Pareto and weak Pareto solutions for this kind of problems in the term of high order Clarke subdifferential. In Banach spaces, we define the Michel-Penot high order subdifferential for a Ck.1 function, give some properties of it and a new Taylor formula. Using our new Taylor formula, we obtain some high order subdifferential sufficient optimal conditions for Ck.1 functions in Banach spaces.
4. In Banach spaces, we prove that if the graph Gr(F) of the objective multifunction F is the union of finitely many generalized polyhedra and F(■)+C, the sum of the range F(■) and the order cone C, is convex or C is a polyhedron with nonempty interior, then its weak Pareto solution set and weak Pareto optimal value set are the unions of finitely many generalized polyhedra, respectively, and that if the graph of the objective multifunction F is the union of finitely many convex polyhedra, F(T)+Cis convex and the projection C2 of C, in finite dimensional subspace Y2 of Y with respect to F, is closed, then its Pareto solution set and Pareto optimal value set are the unions of finitely many convex polyhedra, respectively. Finally. dropping the assumption of the ordering cone C having a weakly base but requiring the objective multifunction being convex with respect to C, we prove that the Pareto solution set and Pareto optimal solution set of the objective multifunction are pathwise connected, respectively.
一、给出一个单值映射g(·)与一个多值映射Q(·)之和M(·)=g(·)+Ω(·)的Clarke切锥,法锥及Bouligand切锥通过Ω(·)的相应切锥及法锥描述的表达式,讨论了M的subsmoothness及L-subsmoothness;给出(σ,δ)-subsmooth闭集值映射具有度量正则性的若干等价条件,把Ursescu-Robinson定理及Zheng和Ng的有关结果推广到更一般的非凸情形.利用正规对偶映射,给出(σ,δ)-subsmooth闭集值映射具有度量次正则性的充分与必要条件和point-based条件,改进并推广了Zheng和Ng等的有关结论.
二、研究了不等式和光滑等式约束的优化问题的sharp minimum与strong KKT条件、weak sharp minimum与quasi-strong KKT和sub-quasi-strong KKT条件的关系,并在Lipschitz条件及quasi-subsmooth条件下,获得了该类优化问题满足各种strong KKT条件的等价描述,把Zheng和Ng关于复合凸情形的有关结论推广到更一般的非凸情形.
三、获得了有限维空间上的Ck、1向量值函数的七+1阶Clarke次微分形式的泰勒公式.在(k+1)阶次微分正定的假设下,给出此类函数的高阶Clarke次微分形式的pareto及弱Pareto解的充分条件.在一般Banach空间中,我们定义了Ck·1函数的Michel-Penot高阶次微分,讨论了它的基本性质,并给出高阶次微分形式的Taylor中值公式,利用其研究了一般Banach空间上Ck、1映射的Michel-Penot同阶次微分优化条件.
四、在Banach空问中,我们证明,当目标集值映射F的图Gr(F)是有限个广义多面体之并且值集F(Γ)与序锥C的和F(Γ)+C是凸的或者序锥C是具有非空内部的多面体时,其弱pareto解集和弱Pareto最优值集也是有限个广义多面体之并.当目标集值映射的图Gr(F)是有限个凸多面体之并且值集F(Γ)与序锥C的和F(Γ)+C是凸的时,若C在Y关于F下的分解有限维子空间中的投影是闭的,则其pareto解集和最优值集也是有限个凸多面体之并.在去掉目标集值映射关于序锥是凸的假设下,序锥是广义凸多面体时,我们证明,其弱Pareto解集和弱最优值集也是有限个广义多面体之并.最后,我们证明,当目标映射关于序锥C是凸的时,其Pareto解集和pareto最优值集是道路连通的.
By techniques and methods of functional analysis, variational analysis and non-smooth analysis, many people have focused on such problems as metric regularity, metric subregularity and calmness of solutions for closed multifunctions, mathe-matic programming and vector optimization problems. They are widely used in many fields. Upon other researchers'work, in this dissertation, we mainly obtained four aspects of results below:
1. We build representations of the Clarke tangent cone, the Clarke normal cone and the Bouligand tangent cone for the multifunction M(·)= g(·)+Ω(·) by the corresponding ones for the multifunctionΩ(·). We also discuss sub smoothness and L-subsmoothness for M. We build some equivalent conditions of metric regularity for a (σ,δ)-subsmooth multifunction, in which we generalize Ursescu-Robinson theorem and Zheng and Ng's corresponding results to the more general nonconvex case. Finally, by the normal dual mapping, we provide some sufficient and necessary conditions and point-based conditions for metric subregularity of a (σ,δ)-subsmooth multifunction, improving and generalizing the corresponding results of zheng, Ng and others.
2. We research the relations between the sharp minimum and the strong KKT, between weak sharp minima and the quasi-strong KKT and the sub-quasi-strong KKT for the optimal problems with the constraint conditions of inequalities and equalities. Under the Lipschitzian and quasi-subsmooth conditions, We obtain equivalent conditions for the above strong KKTs. Our results generalize the results of Zheng and Ng in the convex-composite case to the more general nonconvex case.
3. We obtain the Taylor formula of Ck,l vector functions in the form of the (k+1)th order Clarke subdifferential. Under the assumption of positive definiteness of the (k+1)th order subdifferential, we provide sufficient conditions of Pareto and weak Pareto solutions for this kind of problems in the term of high order Clarke subdifferential. In Banach spaces, we define the Michel-Penot high order subdifferential for a Ck.1 function, give some properties of it and a new Taylor formula. Using our new Taylor formula, we obtain some high order subdifferential sufficient optimal conditions for Ck.1 functions in Banach spaces.
4. In Banach spaces, we prove that if the graph Gr(F) of the objective multifunction F is the union of finitely many generalized polyhedra and F(■)+C, the sum of the range F(■) and the order cone C, is convex or C is a polyhedron with nonempty interior, then its weak Pareto solution set and weak Pareto optimal value set are the unions of finitely many generalized polyhedra, respectively, and that if the graph of the objective multifunction F is the union of finitely many convex polyhedra, F(T)+Cis convex and the projection C2 of C, in finite dimensional subspace Y2 of Y with respect to F, is closed, then its Pareto solution set and Pareto optimal value set are the unions of finitely many convex polyhedra, respectively. Finally. dropping the assumption of the ordering cone C having a weakly base but requiring the objective multifunction being convex with respect to C, we prove that the Pareto solution set and Pareto optimal solution set of the objective multifunction are pathwise connected, respectively.
引文
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