# 数学代写|凸优化作业代写Convex Optimization代考|MATH620

## 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

.凸优化 .凸优化

## 数学代写|凸优化作业代写凸优化代考|二元线性模型

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$$\begin{array}{r} x_{i j}-\dot{x}{i+1, j}=0, \overleftarrow{x}{i, j+1}-\vec{x}{i j}=0 \ \dot{x}{i j}+\dot{x}{i j}+\overleftarrow{x}{i j}+\vec{x}{i j}-2 z{i j}=0 \ \overleftarrow{x}{i 1}=0, \vec{x}{i n}=0, \dot{x}{1 j}=0, \dot{x}{m j}=0 \ i-1, \ldots, p, j-1, \ldots, n \end{array}$$

$$\dot{x}{i j}+\dot{x}{i+1, j}=1,$$

$$\overleftarrow{x}{i, j+1}+\vec{x}{i j}=1$$

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