论文标题
Haverford College凸出优化，牛顿的方法和内部方法的高级论文
Senior Thesis for Haverford College Convex Optimization, Newton's Method and Interior Point Method
论文作者
论文摘要
本文由四个一般部分组成：凸集；凸功能；凸优化；和内点算法。我将首先引入凸集的定义，并给出三个常见的凸集示例，这些示例将在本文稍后使用，然后证明具有显着的分离和支持超平面定理。除了提供定义外，我还将逐渐介绍凸功能，我还将证明函数凸的一阶和二阶条件，然后引入一些示例，这些示例将在沿凸优化问题中使用。接下来，我将提供有关凸优化问题的官方定义，并证明它们具有某些特征，包括存在（通过最佳标准）和解决方案的唯一性。我还将产生两个凸优化问题，其中一个不能简单地解决，需要其他技能。之后，为了构建内点方法，我将引入二元性。在最后一节中，我将首先介绍下降方法和牛顿的方法，这些方法是内部方法的基础。然后，我将展示如何使用对数屏障函数和构建内点方法的中心路径。
This paper consists of four general parts: convex sets; convex functions; convex optimization; and the interior-point algorithm. I will start by introducing the definition of convex sets and give three common convex set examples which will be used later in this paper, then prove the significant separating and supporting hyperplane theorems. Stepping into convex functions, in addition to offering definitions, I will also prove the first order and second-order conditions for convexity of a function, and then introduce a couple of examples that will be used in a convex optimization problem later. Next, I will provide the official definition of convex optimization problems and prove some characteristics they have, including the existence (through optimality criterion) and the uniqueness of a solution. I will also generate two convex optimization problems, one of which cannot be simply solved and requires additional skills. Afterward, I will introduce duality, for the sake of constructing the interior-point method. In the last section, I will first present the descent method and Newton's method, which serve as the foundation of the interior-point method. Then, I will show how to use the logarithmic barrier function and the central path to build up the interior-point method.
