集值优化问题的最优性条件
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摘要
新兴的向量集值优化理论在微分包含、逼近论、变分学与最优控制等领域均
有广泛的应用，集值优化问题在各种解意义下的最优性条件是其中的重要组成部
分，是建立现代优化算法的重要基础. 另一方面，凸性的概念在优化理论中扮演
着重要的角色，因而各种凸性的推广都倍受人们的关注. 本文旨在得到近似锥-
次类凸集值函数的性质，并在近似锥-次类凸假设下，分别得到集值优化问题关
于 Benson真有效元、赋范线性空间超有效元、强有效元、严有效元和局部凸空
间超有效元的非导数型最优性条件或导数型最优性条件. 具体结果可归纳如下：
1. 研究了凸锥的若干性质, 利用这些性质给出了近似锥-次类凸集值函数(也称
近似锥-次类凸集值映射)的几种等价刻画. 利用其中的一种等价刻画给出了
近似锥-次类凸函数的一个重要性质, 利用这个性质和择一性定理得到了集值
优化问题取得 Benson真有效元的一个 Lagrange 型必要条件.
2. 在赋范线性空间中讨论了集值优化问题的超有效性. 在近似锥-次类凸假设下,
得到了集值优化问题取得超有效元的一个标量化定理. 利用近似锥-次类凸函
数的一种等价刻画给出了近似锥-次类凸函数的另一个重要性质, 利用这个性
质和择一性定理得到了集值优化问题取得超有效元的一个Lagrange 型必要条
件. 利用超有效点集的性质给出了集值优化问题取得超有效元的一个
Lagrange 型充分条件. 给出了集值优化问题在超有效元意义下的一种
Lagrange 型无约束刻画.
3. 在局部凸空间讨论了集值优化问题的强有效性. 在近似锥-次类凸假设下, 利
用强有效点集的性质及择一性定理, 得到了集值优化问题取得强有效元的
Kuhn-Tucker 型必要条件. 利用基泛函的性质和双序线性空间的 Namioka 线
性泛函分解定理得到了集值优化问题取得强有效元的 Kuhn-Tucker 型充分条
件. 利用近似锥-次类凸函数的一种等价刻画给出了近似锥-次类凸函数的一
个重要性质, 利用这个性质和双序线性空间的 Namioka 线性泛函分解定理得
到了集值优化问题取得强有效元的 Lagrange 型必要条件.利用强有效点集的
性质给出了集值优化问题取得强有效元的 Lagrange 型充分条件. 给出了集值
优化问题在强有效元意义下的 Kuhn-Tucker 型和 Lagrange 型无约束刻画.
4. 在局部凸空间讨论了集值优化问题的严有效性. 在近似锥-次类凸假设下, 利
用凸集分离定理和标量化定理分别得到了集值优化问题取得严有效元的
Kuhn-Tucker 型必要和充分条件. 利用凸集分离定理和标量化定理得到了集
值优化问题取得严有效元的 Lagrange 型必要条件.利用严有效点集的性质和
定义给出了集值优化问题取得严有效元的 Lagrange 型充分条件. 给出了集值
优化问题在严有效元意义下的 Kuhn-Tucker 型和 Lagrange 型无约束刻画.
5. 在局部凸空间讨论了集值优化问题的超有效性. 在近似锥-次类凸假设下, 利
用择一性定理和利用标量化定理分别得到了集值优化问题取得超有效元的
Kuhn-Tucker 型必要和充分条件. 利用择一性定理和标量化定理得到了集值
优化问题取得超有效元的 Lagrange 型必要条件. 利用超有效点集的性质和定
义给出了集值优化问题取得超有效元的 Lagrange 型充分条件. 给出了集值优
化问题在超有效元意义下的 Kuhn-Tucker 型和 Lagrange 型无约束刻画.
6. 借助于修正的 Dubovitskij-Miljutin 切锥引进了一种新的集值函数切导数, M-
导数.给出了几种广义伪凸集值函数的概念.当目标函数和约束函数均 M-可导
时，在近似锥-次类凸假设下利用严有效点集的性质和凸集分离定理得到了集
值优化问题取得严有效元的 Fritz John和 Kuhn-Tucker 型必要条件. 在广义伪
凸假设下，得到了集值优化问题取得严有效元的 Kuhn-Tucker 型充分条件.
The theory of vector optimization with set-valued maps finds wide applications
in differential inclusions, approximation theory, variations, optimization control, and
so on, the optimality conditions for set-valued optimization problems in the sense of
various solutions are its important components and are the important base of
developing modern algorithms. On the other hand, the concept of convexity plays
important roles in the optimization theory, hence each of generalizations of convexity
receives researcher’s attentions. The thesis is to gain properties of nearly
cone-subconvexlike set-valued functions, and under the assumption of nearly
cone-subconvexlikeness, to obtain the optimality conditions with derivatives or
without derivatives for set-valued optimization problems in the sense of Benson
proper efficient element, superly efficient element of normed linear space, strongly
efficient element, strictly efficient element and superly efficient element of locally
convex space, respectively. For details, these results are givenin the following.
1. Properties for convex cones are discussed, which are used to obtain several
equivalent characterizations for nearly cone-subconvexlike functions (maps). By
applying an equivalent characterization of the nearly cone-subconvexlike function, an
important property of the nearly cone-subconvexlike function is presented, which and
alternative theorem are used to obtain a Lagrange necessary condition for set-valued
optimization problem to attain its Benson proper efficient element.
2. The super efficiency of normed linear space for set-valued optimization
problem is investigated. Under the assumption of nearly cone-subconvexlikeness, a
scaralization theorem of set-valued optimization problem to attain its superly efficient
element. By applying an equivalent characterization of the nearly cone-subconvexlike
function, another important property of the nearly cone-subconvexlike function is
presented, which and alternative theorem are used to obtain a Lagrange necessary
condition for set-valued optimization problem to attain its superly efficient element.
With the properties of the set of superly efficient points, a Lagrange sufficient
condition is obtained for set-valued optimization problem to attain its superly
efficient element. A kind of unconstrained characterization equivalent to set-valued
optimization problem is presented in the sense of superly efficient elements.
3. The strong efficiency of set-valued optimization problem in locally convex
spaces is investigated. Under the assumption of nearly cone-subconvexlikeness, with
properties of the set of strongly efficient points and alternative theorem, a
Kuhn-Tucker necessary condition is obtained for set-valued optimization problem to
obtain its strongly efficient element. By applying properties of base functional and a
Namioka decomposition theorem of a functional in a biordered linear space, a
Kuhn-Tucker sufficient condition is obtained for set-valued optimization problem to
obtain its strongly efficient element. By applying an equivalent characterization of the
nearly cone-subconvexlike function, another important property of the nearly
cone-subconvexlike function is presented, which and a Namioka decomposition
theorem of a functional in a biordered linear space are used to obtain a Lagrange
necessary condition for set-valued optimization problem to attain its strongly
efficient element. With the properties of the set of strongly efficient points, a
Lagrange sufficient conditionis obtained for set-valued optimization problem to attain
its strongly efficient element. Several kinds of Kuhn-Tucker and Lagrange
unconstrained characterizations equivalent to set-valued optimization problem are
presented in the sense of strongly efficient elements.
4. The strict efficiency of set-valued optimization problem in locally convex
spaces is investigated. Under the assumption of nearly cone-subconvexlikeness, with
a separation theorem for convex se
有广泛的应用，集值优化问题在各种解意义下的最优性条件是其中的重要组成部
分，是建立现代优化算法的重要基础. 另一方面，凸性的概念在优化理论中扮演
着重要的角色，因而各种凸性的推广都倍受人们的关注. 本文旨在得到近似锥-
次类凸集值函数的性质，并在近似锥-次类凸假设下，分别得到集值优化问题关
于 Benson真有效元、赋范线性空间超有效元、强有效元、严有效元和局部凸空
间超有效元的非导数型最优性条件或导数型最优性条件. 具体结果可归纳如下：
1. 研究了凸锥的若干性质, 利用这些性质给出了近似锥-次类凸集值函数(也称
近似锥-次类凸集值映射)的几种等价刻画. 利用其中的一种等价刻画给出了
近似锥-次类凸函数的一个重要性质, 利用这个性质和择一性定理得到了集值
优化问题取得 Benson真有效元的一个 Lagrange 型必要条件.
2. 在赋范线性空间中讨论了集值优化问题的超有效性. 在近似锥-次类凸假设下,
得到了集值优化问题取得超有效元的一个标量化定理. 利用近似锥-次类凸函
数的一种等价刻画给出了近似锥-次类凸函数的另一个重要性质, 利用这个性
质和择一性定理得到了集值优化问题取得超有效元的一个Lagrange 型必要条
件. 利用超有效点集的性质给出了集值优化问题取得超有效元的一个
Lagrange 型充分条件. 给出了集值优化问题在超有效元意义下的一种
Lagrange 型无约束刻画.
3. 在局部凸空间讨论了集值优化问题的强有效性. 在近似锥-次类凸假设下, 利
用强有效点集的性质及择一性定理, 得到了集值优化问题取得强有效元的
Kuhn-Tucker 型必要条件. 利用基泛函的性质和双序线性空间的 Namioka 线
性泛函分解定理得到了集值优化问题取得强有效元的 Kuhn-Tucker 型充分条
件. 利用近似锥-次类凸函数的一种等价刻画给出了近似锥-次类凸函数的一
个重要性质, 利用这个性质和双序线性空间的 Namioka 线性泛函分解定理得
到了集值优化问题取得强有效元的 Lagrange 型必要条件.利用强有效点集的
性质给出了集值优化问题取得强有效元的 Lagrange 型充分条件. 给出了集值
优化问题在强有效元意义下的 Kuhn-Tucker 型和 Lagrange 型无约束刻画.
4. 在局部凸空间讨论了集值优化问题的严有效性. 在近似锥-次类凸假设下, 利
用凸集分离定理和标量化定理分别得到了集值优化问题取得严有效元的
Kuhn-Tucker 型必要和充分条件. 利用凸集分离定理和标量化定理得到了集
值优化问题取得严有效元的 Lagrange 型必要条件.利用严有效点集的性质和
定义给出了集值优化问题取得严有效元的 Lagrange 型充分条件. 给出了集值
优化问题在严有效元意义下的 Kuhn-Tucker 型和 Lagrange 型无约束刻画.
5. 在局部凸空间讨论了集值优化问题的超有效性. 在近似锥-次类凸假设下, 利
用择一性定理和利用标量化定理分别得到了集值优化问题取得超有效元的
Kuhn-Tucker 型必要和充分条件. 利用择一性定理和标量化定理得到了集值
优化问题取得超有效元的 Lagrange 型必要条件. 利用超有效点集的性质和定
义给出了集值优化问题取得超有效元的 Lagrange 型充分条件. 给出了集值优
化问题在超有效元意义下的 Kuhn-Tucker 型和 Lagrange 型无约束刻画.
6. 借助于修正的 Dubovitskij-Miljutin 切锥引进了一种新的集值函数切导数, M-
导数.给出了几种广义伪凸集值函数的概念.当目标函数和约束函数均 M-可导
时，在近似锥-次类凸假设下利用严有效点集的性质和凸集分离定理得到了集
值优化问题取得严有效元的 Fritz John和 Kuhn-Tucker 型必要条件. 在广义伪
凸假设下，得到了集值优化问题取得严有效元的 Kuhn-Tucker 型充分条件.
The theory of vector optimization with set-valued maps finds wide applications
in differential inclusions, approximation theory, variations, optimization control, and
so on, the optimality conditions for set-valued optimization problems in the sense of
various solutions are its important components and are the important base of
developing modern algorithms. On the other hand, the concept of convexity plays
important roles in the optimization theory, hence each of generalizations of convexity
receives researcher’s attentions. The thesis is to gain properties of nearly
cone-subconvexlike set-valued functions, and under the assumption of nearly
cone-subconvexlikeness, to obtain the optimality conditions with derivatives or
without derivatives for set-valued optimization problems in the sense of Benson
proper efficient element, superly efficient element of normed linear space, strongly
efficient element, strictly efficient element and superly efficient element of locally
convex space, respectively. For details, these results are givenin the following.
1. Properties for convex cones are discussed, which are used to obtain several
equivalent characterizations for nearly cone-subconvexlike functions (maps). By
applying an equivalent characterization of the nearly cone-subconvexlike function, an
important property of the nearly cone-subconvexlike function is presented, which and
alternative theorem are used to obtain a Lagrange necessary condition for set-valued
optimization problem to attain its Benson proper efficient element.
2. The super efficiency of normed linear space for set-valued optimization
problem is investigated. Under the assumption of nearly cone-subconvexlikeness, a
scaralization theorem of set-valued optimization problem to attain its superly efficient
element. By applying an equivalent characterization of the nearly cone-subconvexlike
function, another important property of the nearly cone-subconvexlike function is
presented, which and alternative theorem are used to obtain a Lagrange necessary
condition for set-valued optimization problem to attain its superly efficient element.
With the properties of the set of superly efficient points, a Lagrange sufficient
condition is obtained for set-valued optimization problem to attain its superly
efficient element. A kind of unconstrained characterization equivalent to set-valued
optimization problem is presented in the sense of superly efficient elements.
3. The strong efficiency of set-valued optimization problem in locally convex
spaces is investigated. Under the assumption of nearly cone-subconvexlikeness, with
properties of the set of strongly efficient points and alternative theorem, a
Kuhn-Tucker necessary condition is obtained for set-valued optimization problem to
obtain its strongly efficient element. By applying properties of base functional and a
Namioka decomposition theorem of a functional in a biordered linear space, a
Kuhn-Tucker sufficient condition is obtained for set-valued optimization problem to
obtain its strongly efficient element. By applying an equivalent characterization of the
nearly cone-subconvexlike function, another important property of the nearly
cone-subconvexlike function is presented, which and a Namioka decomposition
theorem of a functional in a biordered linear space are used to obtain a Lagrange
necessary condition for set-valued optimization problem to attain its strongly
efficient element. With the properties of the set of strongly efficient points, a
Lagrange sufficient conditionis obtained for set-valued optimization problem to attain
its strongly efficient element. Several kinds of Kuhn-Tucker and Lagrange
unconstrained characterizations equivalent to set-valued optimization problem are
presented in the sense of strongly efficient elements.
4. The strict efficiency of set-valued optimization problem in locally convex
spaces is investigated. Under the assumption of nearly cone-subconvexlikeness, with
a separation theorem for convex se
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