我們研究在希爾伯特空間 H 中的凸子集 C 之邊界點的支撐超平面存在性的等價條件，即是有限個定義在 H 上的凸函數之水平集，也就是說，

C ={ x∈H: φi(x) ≤ 0 ∀ i = 1, ...., m }

其中任意的φi 都是在 H 上的可微凸函數。為了研究集合 C 的性質，我們應該提出在閉凸集合上的次微分的概念。我們使用支撐超平面調查變分不等式的解的存在性，並且獲得替代定理。我們也研究了限制式為 a ≤ g(x) ≤ b 在 f(x) 上的非凸最小化問題，其中 f 和 g 是定義在 H 上的凸函數。

We study the explicit necessary and sufficient condition for the existence of supporting hyperplanes at boundary points of a convex subset C of a Hilbert space H that is determined by finitely many level sets of convex functions on H， that is

C ={ x∈H: φi(x) ≤ 0 ∀ i = 1, ...., m }

where each φi is a differentiable convex function on H. We shall propose the concept of subdifferential of a closed convex set to study properties of the set C. We use supporting hyperplanes to investigate the existence of solutions of variational inequalities and obtain an alternative theorem. We also study a nonconvex minimization problem min f(x) subject to a ≤ g(x) ≤ b, where f and g are convex functions on H.

論文審定書 ................................................................................................................i
Acknowledgements....................................................................................................ii
摘要.........................................................................................................................iii
Abstract...................................................................................................................iv
Table of Contents......................................................................................................v
List of Symbols........................................................................................................vi
1. Background...........................................................................................................1
1.1 Hilbert Space and Its Dual Space .........................................................................1
1.2 ConvexSetandConvexFunction.............................................................................3
1.3 Separation and Supporting Hyperplane...................................................................7
1.4 Variational Inequalities and Optimization Problems................................................11
2. The supporting hyperplane and an alternative to solutions of variational inequalities...14
2.1 Introduction........................................................................................................14
2.2 Preliminaries......................................................................................................16
2.3 Main results........................................................................................................19
2.4 Applications...................................................................................................... 23
2.5 Sub-differential of a closed convex set at boundary points......................................26
3. A non-convex problem..........................................................................................33
3.1 Introduction........................................................................................................33
3.2 Preliminaries.......................................................................................................34
3.3 Main results.......................................................................................................36
Bibliography.............................................................................................................39

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