## 3 Proposed Algorithm  
### Motivation  
Simulation results on several symmetric TSP benchmarks in [28] illustrate that OBBO seems to be relatively good at optimal solutions when compared with BBO. However, these experiments expose two serious problems, which may indicate that the initial conclusion is questionable. Although the terminal condition is set as a constant generation maximum for different comparison algorithms, such as 500 in that paper, the number of candidate solutions explored in a search space by different algorithms may be well-distinguished between each other. It is intuitively obvious that some opposite paths are explored and then considered in each generation for OBBO algorithm, but not for BBO algorithm. Obviously, to compare the performance of BBO and OBBO algorithm, it is unfair and inadvisable scheme, which instead by our termination criterion in Section 4.1. The other problem, hidden behind OBBO for TSP problems, is that the definition of opposite path is too simple to embody some important characteristics of the candidate path. According to our observations and understanding, the city sequences and the distances between adjacent cities are both the two most core features of a TSP path. It is to be noted clearly that the city sequences here means the relative order in TSP path alone. For example, we only concerned that city 25 is former than city 49, and latter than city 12 in a given TSP path (..., 12, 25, 49,...). However, we do not care about the coordinate of the cities 25, 49 and 12, not to mention the Euclidean distances between them. Furthermore, the Euclidean distance is directly computed based on the geometric coordinates of the nodes of the graph, and yet these TSP paths are differentiated by the sequences when only a graph is given. However, the authors used the former feature (city sequences) and ignored the latter one (distances between adjacent cities) unconsciously or consciously, when they defined the opposite path in [28]. Usually, using the original definition of opposite path presented in [28] will lead inevitably to the low utilization rate of opposite paths as shown in Section 4.2. It is obvious that the definition of opposite path can be considered as a key to promote the opposition-based soft computing for solving TSP problem. Therefore, an initial motivation of this paper is to further amend the definition of opposite path with the help of the Euclidean distances and city sequences in the graph. Therefore, the next question is that how to use together with Euclidean distances and city sequences of a candidate path. We think Opposition-Based Learning using the Current Optimum, a significant important variation of OBL in continuous domain, might be a good choice by careful observation of these definitions. As mentioned previously, it was first proposed for function optimization as follows [7]. **Definition 3. ** Let \(P=(x_{1},\,x_{2},\,...,\,x_{D})\) be a point in \(D\)-dimensional space, where \(x_{1},\,x_{2},\,...,\,x_{D}\,\in\,\mathbf{R}\) and \(x_{i}\in[a_{i},\,b_{i}],\,\forall\,\,i\in\{1,\,2,\,...,\,D\}\). The opposite point using the current optimum \(\bar{P}_{co}=(\,\bar{x}_{co1},\,\bar{x}_{co2},\,...,\,\,\bar{x}_{coD})\) is completely defined by its coordinates  
\[\bar{x}_{coi}=2x_{co}-x_{i} \tag{3}\]  
where \(x_{co}\) is the optimum solution in the current population. This definition has similarity, in style, with definition 1 proposed by [PERSON], but you will find that the opposite point using the current optimum may be outside the range of valid numbers definedby [_a_, _b_] if you analyze it carefully. Therefore, the possible solutions include recomputing based on Equation (3) until the new one falls in the range of valid numbers, reproducing a random point, and even using the left or right boundary of valid numbers as an alternative. Figure 2 illustrates the opposite point using the current optimum \(\bar{P}_{co}\) in one dimensional case.