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# 计算机代写|计算机视觉代写Computer Vision代考|CS766 Discrete Optimization

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## 计算机代写|计算机视觉代写Computer Vision代考|Discrete Optimization

Discrete optimization deals with problems where the elements of the solution set $S$ take discrete values, e.g., $S \subseteq \mathbb{Z}^n=\left{i_1, i_2, \ldots, i_n\right} ; i_n \in \mathbb{Z}$.

Usually, discrete optimization problems are $N P$-hard to solve, which, informally speaking, in essence states that there is no known algorithm which finds the correct solution in polynomial time. Therefore, execution times soon become infeasible as the size of the problem (the number of unknowns) grows.

As a consequence, many discrete optimization methods aim at finding approximate solutions, which can often be proven to be located within some reasonable bounds to the “true” optimum. These methods are often compared in terms of the quality of the solution they provide, i.e., how close the approximate solution gets to the “true” optimal solution. This is in contrast to continuous optimization problems, which aim at optimizing their rate of convergence to local minima of the objective function.
In practice it turns out that the fact that the solution can only take discrete values, which acts as an additional constraint, often complicates matters when we efficiently want to find a solution. Therefore, a technique called relaxation can be applied, where the discrete problem is transformed into its continuous version: The objective function remains unchanged, but now the solution can take continuous values, e.g., by replacing $S_{\mathrm{d}} \subseteq \mathbb{Z}^n$ with $S_{\mathrm{c}} \subseteq \mathbb{R}^n$, i.e., the (additional) constraint that the solution has to take discrete values is dropped. The continuous representation can be solved with an appropriate continuous optimization technique. A simple way of deriving the discrete solution $x_{\mathrm{d}}^$ from the thus obtained continuous one $x_{\mathrm{c}}^$ is to choose that element of the discrete solution set $S_{\mathrm{d}}$ which is closest to $x_{\mathrm{c}}^$. Please note that there is no guarantee that $x_{\mathrm{d}}^$ is the optimal solution of the discrete problem, but under reasonable conditions it should be sufficiently close to it.

## 计算机代写|计算机视觉代写Computer Vision代考|Combinatorial Optimization

In combinatorial optimization, the solution set $S$ has a finite number of elements, too. Therefore, any combinatorial optimization problem is also a discrete problem. Additionally, however, for many problems it is impractical to build $S$ as an explicit enumeration of all possible solutions. Instead, a (combinatorial) solution can be expressed as a combination of some other representation of the data.

To make things clear, consider to the satnav example again. Here, $S$ is usually not represented by a simple enumeration of all possible routes from the start to a destination location. Instead, the data consists of a map of the roads, streets, motorways, etc., and each route can be obtained by combining these entities (or parts of them). Observe that this allows a much more compact representation of the solution set.

This representation leads to an obvious solution strategy for optimization problems: we “just” have to try all possible combinations and find out which one yields the minimum value of the objective function. Unfortunately, this is infeasible due to the exponential growth of the number of possible solutions when the number of elements to combine increases (a fact which is sometimes called combinatorial explosion).
An example of combinatorial optimization methods used in computer vision are the so-called graph cuts, which can, e.g., be utilized in segmentation problems: consider an image showing an object in front of some kind of background. Now we want to obtain a reasonable segmentation of the foreground object from the background. Here, the image can be represented by a graph $G=(V, E)$, where each pixel $i$ is represented by a vertex $v_i \in V$, which is connected to all of its neighbors via an edge $e_{i j} \in E$ (where pixels $i$ and $j$ are adjacent pixels; typically a 4-neighborhood is considered).

A solution $s$ of the segmentation problem which separates the object region from the background consists of a set of edges (where each of these edges connects a pixel located at the border of the object to a background pixel) and can be called a cut of the graph. In order to find the best solution, a cost $c_{i j}$ can be assigned to each edge $e_{i j}$, which can be derived from the intensity difference between pixel $i$ and $j$ : the higher the intensity difference, the higher $c_{i j}$. Hence, the solution of the problem is equal to find the cut which minimizes the overall cost along the cut. As each cut defines a combination of edges, graph cuts can be used to solve combinatorial optimization problems. This combinatorial strategy clearly is superior to enumerate all possible segmentations and seek the solution by examination of every element of the enumeration.

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。