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数学代写|凸优化作业代写Convex Optimization代考|Description of the Problem
The problem of visualization of a sequence flow was posed as a problem of combinatorial multi-objective optimization in [93], where the objectives correspond to the aesthetic criteria. The results of a psychological experiment, described in [93], substantiate the selection of aesthetic criteria that are most important for the potential users. The stated problem was attacked in [93] by a metaheuristic ant colony optimization algorithm. The solutions, found by means of that algorithm in reasonable time, were assessed as acceptable for applications. Nevertheless, the following reason motivated a further investigation: usually a few non-dominated solutions were found. Therefore, a hypothesis seems likely that there exist other Pareto optimal solutions, but they were not found by the metaheuristic algorithm used. To test that hypothesis all the global optima of the criteria in question should be found with a guarantee. To this end, the corresponding single-objective optimization problems were stated in the form of binary-linear optimization, and the CPLEX algorithm [38] was applied to solve them. A combination of CPLEX with the scalarization technique is also used to solve the multi-objective optimization solve small size problems, fails in the case of larger problems because of long computing time. A heuristic algorithm was proposed applicable to the problems of size sufficient for applications.
An example of elementary BPD is presented in Figure 10.1a. In geometric terms, it is requested to draw paths that consist of horizontal and vertical line segments, and connect the given geometric shapes (circles, rhombuses, and rectangles) in the plane. The shapes are located in “swim lanes,” at the centers of cells of a rectangular grid, and the paths are requested to consist of horizontal and vertical segments with the ends at circle markers, located on the boarders between “swim lanes,” as shown in Figure 10.1b. In terms of the graph theory we are interested in the paths between the given vertices of a graph, defined by a rectangular grid [92] of the type presented in Figure $10.1 \mathrm{~b}$. We search here for the paths with the minimum total length, minimum total number of bends, and minimum neighborhood; we refer to [93] for a detailed discussion on the criteria of path desirability. The argumentation presented there substantiates the consideration of the problem of aesthetic drawing of BPDs by means of the methods for multi-objective graph optimization.
Although some similarity is obvious between the considered problem and the classical bi-objective path problem $[64,77]$, the known methods for the latter do not seem applicable to the former one. This remark also holds for the similarity of the considered problem with routing problems in electronic design [32].
数学代写|凸优化作业代写Convex Optimization代考|Binary-Linear Model
To state a multi-objective optimization problem mathematically, we have to introduce variables that define the considered (reduced) graph. Let $p$ denote the number of rows, and $n$ denote the number of columns. The pivot vertices are marked by a double index $i j$ that indicates the crossing of the $i$-th row and $j$-thcolumn. The intermediate vertices are indicated by two neighboring pivot vertices. A path is defined by assigning value 1 to the indexed variable $x$, related to the edge which belongs to the path; the values of the variables related to edges not belonging to the path in question are equal to zero. The variable $x$ is indexed as follows: $\dot{x}{i j}$ and $\hat{x}{i j}$ are related to the top and bottom adjacent edges of the vertex $i j$, respectively; see Figure 10.3. Similarly $\overleftarrow{x}{i j}$ and $\vec{x}{i j}$ are related to the right and left adjacent edges. The values of $z_{i j}$ mark the path as follows: $z_{i j}=1$, if the vertex $i j$ is on the path, and $z_{i j}=0$, if it is not on the path. The values of the introduced variables should satisfy the following equalities:
$$
\begin{array}{r}
x_{i j}-\dot{x}{i+1, j}=0, \overleftarrow{x}{i, j+1}-\vec{x}{i j}=0 \ \dot{x}{i j}+\dot{x}{i j}+\overleftarrow{x}{i j}+\vec{x}{i j}-2 z{i j}=0 \
\overleftarrow{x}{i 1}=0, \vec{x}{i n}=0, \dot{x}{1 j}=0, \dot{x}{m j}=0 \
i-1, \ldots, p, j-1, \ldots, n
\end{array}
$$
Note that the zero length edges in (10.3) are redundant; but they are included into the model to unify the adjacency of all the pivot vertices.
A path is specified by the start and sink vertices, which are of the intermediate type; such vertices are presented in the mathematical model by the balance equalities as follows:
$$
\dot{x}{i j}+\dot{x}{i+1, j}=1,
$$
if the vertex is located on the $j$-th column between $i$ and $i+1$ rows, and
$$
\overleftarrow{x}{i, j+1}+\vec{x}{i j}=1
$$
if the vertex is located on the $i$-th row between $j$ and $j+1$ columns.
Before formulating a multi-objective optimization problem, let us note that the important criterion of the total number of bends cannot be reduced to the criteria considered in the classical bi-objective path problem (see, e.g., [64]). We start the analysis from the single path and single objective which is the path length.
凸优化代考
数学代写|凸优化作业代写凸优化代考|问题描述
.凸优化 .凸优化
序列流的可视化问题在[93]中被提出为一个组合多目标优化问题，其中目标对应于美学标准。心理学实验的结果，如[93]所述，证实了对潜在用户来说最重要的审美标准的选择。该问题在[93]中被元启发式蚁群优化算法攻击。通过该算法在合理时间内找到的解被评估为可接受的应用。然而，以下原因促使我们进一步研究:通常会发现少数非支配解。因此，一个假设似乎很可能存在其他帕累托最优解，但它们没有被使用的元启发式算法发现。为了验证这一假设，所有有问题的标准的全局最优都必须有一个保证。为此，将相应的单目标优化问题以二元线性优化的形式表述出来，并采用CPLEX算法[38]求解。将CPLEX与尺度化技术相结合，解决了求解小规模问题的多目标优化问题，由于计算时间较长，在求解较大问题时失败。提出了一种启发式算法，该算法适用于应用程序大小足够大的问题
图10.1a给出了基本BPD的一个例子。在几何术语中，它被要求绘制由水平和垂直线段组成的路径，并连接平面中给定的几何形状(圆、菱形和矩形)。这些形状位于“泳道”中，位于矩形网格单元格的中心，路径要求由水平和垂直部分组成，两端位于“泳道”之间的圆形标记处，如图10.1b所示。就图论而言，我们感兴趣的是图的给定顶点之间的路径，由图$10.1 \mathrm{~b}$所示类型的矩形网格[92]定义。我们在这里寻找具有最小总长度、最小总弯道数和最小邻域的路径;我们参考[93]对路径可取性标准的详细讨论。文中提出的论证论证了利用多目标图优化方法对bpd的美观绘制问题的考虑
虽然所考虑的问题与经典的双目标路径问题$[64,77]$之间有一些明显的相似之处，但后者的已知方法似乎并不适用于前者。这句话同样适用于所考虑的问题与电子设计中的路由问题的相似性[32]
数学代写|凸优化作业代写凸优化代考|二元线性模型
.
要用数学方法表述一个多目标优化问题，我们必须引入定义考虑(约简)图的变量。让$p$表示行数，$n$表示列数。主顶点由一个双索引$i j$标记，该索引表示$i$ -th行和$j$ -thcolumn的交叉。中间顶点由两个相邻的主顶点表示。路径的定义是将值1赋给索引变量$x$，该变量与属于该路径的边相关;与不属于该路径的边相关的变量的值等于零。变量$x$的索引如下:$\dot{x}{i j}$和$\hat{x}{i j}$分别与顶点$i j$的上、下邻边相关;见图10.3。类似地，$\overleftarrow{x}{i j}$和$\vec{x}{i j}$与左右相邻边相关。$z_{i j}$的值以如下方式标记路径:$z_{i j}=1$，如果顶点$i j$在路径上，$z_{i j}=0$，如果顶点不在路径上。引入的变量值应满足以下等式:
$$
\begin{array}{r}
x_{i j}-\dot{x}{i+1, j}=0, \overleftarrow{x}{i, j+1}-\vec{x}{i j}=0 \ \dot{x}{i j}+\dot{x}{i j}+\overleftarrow{x}{i j}+\vec{x}{i j}-2 z{i j}=0 \
\overleftarrow{x}{i 1}=0, \vec{x}{i n}=0, \dot{x}{1 j}=0, \dot{x}{m j}=0 \
i-1, \ldots, p, j-1, \ldots, n
\end{array}
$$
注意(10.3)中的零长度边是冗余的;但它们被包含在模型中，以统一所有枢轴顶点的邻接关系
路径由中间类型的起始顶点和汇聚顶点指定;这些顶点在数学模型中通过如下平衡等式表示:
$$
\dot{x}{i j}+\dot{x}{i+1, j}=1,
$$
如果顶点位于$i$和$i+1$行之间的$j$ -th列上，
$$
\overleftarrow{x}{i, j+1}+\vec{x}{i j}=1
$$
如果顶点位于$j$和$j+1$列之间的$i$ -th行上。在阐述一个多目标优化问题之前，让我们注意到弯的总数这一重要准则不能简化为经典双目标路径问题中考虑的准则(参见，例如，[64])。我们从单一路径和单一目标开始分析，即路径长度。
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