Identifying the Pareto-Optimal Solutions for Multi-point Distance Minimization Problem in Manhattan Space Justin Xu∗1,KalyanmoyDeb†2, and Abhinav Gaur‡3 1Senior, Troy High School Student, 4777 Northfield Parkway, Troy, MI 48098, USA 2Koenig Endowed Chair Professor, Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA 3Doctoral Student, Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA COIN Report Number 2015018 Computational Optimization and Innovation (COIN) Laboratory http://www.egr.msu.edu/˜kdeb/COIN.shtml Abstract Multi-point distance minimization problems (M-DMP) pose a number of theoretical challenges and simultaneously represent a number of practical applications, particularly in navigational and layout design problems. When the Euclidean distance measure is minimized for each target point, the resulting problem has a trivial solution, however such a consideration limits its application in practice. Since two-dimensional landscapes, floor spaces, or printed circuit boards are laid out in gridded structures for convenience, M-DMP problems are to be solved with Manhattan distance metric for their practical significance. Identification of Pareto-optimal solutions leading to trade-off minimal paths from multiple target points become a challenging task. In this paper, we suggest a systematic procedure for identifying Pareto-optimal solutions and provide a theoretical proof for validating our construction process. Thus, instead of devising generic multi-objective optimization algorithms for solving M-DMP problem under Manhattan space, which has been demonstrated to be a challenging task, this paper advocates the use of a computationally fast construction procedure. Further practicalities of the M-DMP problem are outlined and it is argued that future applications must involve a basic theoretical construction step similar to the procedure suggested here along with efficient algorithmic methods for handling those practicalities. 1 Introduction Multi-objective optimization problems give rise to a set of non-dominated trade-off solutions called a Pareto-optimal set, instead of a single optimal solution [2,4,12]. As a result, these problems are better ∗fi[email protected] †[email protected] ‡[email protected] 1 solved using a population-based optimization algorithm, such as an evolutionary algorithm, than a classical point-by-point optimization methods. Multi-point distance minimization problems (M-DMP) [9] in which simultaneous minimization of distances of a point from a pre-specified set of fixed target points have a number of advanced applications in navigational and layout optimization problems, such as factory layout or VLSI layout design problems. In such problems, identification of all trade-off Pareto-optimal points becomes an important task. There are two main reasons for finding them. First, these Pareto-optimal solutions give rise to minimal distance of any point and thus constitute the set of final solutions from a set of target points. Second, a knowledge of these solutions should provide a plethora of insights to the users about different possibilities of solving the problem. When Euclidean distances are used in the M-DMP, the Pareto-optimal set is the convex hull of all specified points and the task becomes trivial [10]. However, Euclidean distance from one point to another is not practically viable from a navigational point of view, as there may not exist a direct path connecting any two points. In street layouts and other gridded areas, travel is restricted to only specific orthogonal directions. Thus, from a practical viewpoint, the Manhattan distance between two points is more useful than Euclidean distance. However, the resulting Pareto-optimal set for a multi-point distance minimization problem under Manhattan distance measure becomes not so trivial to identify. In this paper, we develop a procedure for identifying the complete Pareto-optimal set and provide a proof for their optimality. A recent study using evolutionary multi-objective optimization (EMO) [18] has treated the problem in a generic multi-objective optimization problem and has shown difficulties of EMO methods in finding the entire Pareto-optimal set. However, the construction procedure outlined in this paper provides a systematic method of identifying the Pareto-optimal set, instead of using an iterative optimization algorithm. The following then documents the flow of the paper. Section 2 provides a description of the multi- point distance minimization problem with a few simple illustrative examples. Section 3 highlights the challenges in solving M-DMP. Section 4 describes our proposed construction procedure for identifying Pareto-optimal set for a generic M-DMP. This procedure is followed by a detailed proof for the Pareto- optimality of our proposed set in Section 5. Section 6 presents the application potential of the M-DMP, followed by our conclusions in Section 7. 2 Multi-Point Distance Minimization Problem (M-DMP) For our study in this paper, we have considered two-dimensional decision space. In a generic M-DMP, 2 every point in the decision space has a unique distance to a set of K predefined target points (Ti ∈ R , 2 i =1, 2,...,K). Any point Pi =(xi,yi) ∈ R has K distance values (d1,d2,...,dK ), which we call here as an objective vector. Thus, the goal in an M-DMP is to minimize all K distances simultaneously, or, minimize {d1,d2, ..., dK }. It is clear that, in general, there may not exist a single point P on the decision space that will minimize all K distances, no matter what distance metric is used to define the M-DMP. Thus, the M-DMP problem is an unconstrained multi-objective optimization problem having at most K conflicting objectives. As dictated by the multi-objective optimization literature [2, 12], solutions to such problems give rise to a set of Pareto-optimal solutions, instead of a single solution. For a generic K-objective optimization problem, a Pareto-optimal solution is defined as follows: Definition 1. AsolutionX is said to be Pareto-optimal, if there does not exist a solution Y (= X)in the feasible search space satisfying dj(Y ) ≤ dj(X) for all objectives j =1, 2,...,K. 2 A collection of all Pareto-optimal solutions is called the Pareto-optimal set. For a generic M-DMP problem, the Pareto-optimal set is bounded and has infinite points. The Pareto-optimal set depends on the chosen distance metric used to compute individual distances di. Figures 1a and 1b show a two- target and a three-target M-DMP problem. For the M-DMP problem with Euclidean distance metric, the distance of a point X =(x, y) with a target point a =(xa,ya) is calculated, as follows: 2 2 d(X,a)= (x − xa) +(y − ya) . (1) For this problem, the Pareto-optimal set is the convex hull constructed by the target points, as marked in the figure. This has been proven elsewhere [10]. O O 2 1 O3 O1 O 2 (a) Two target points. (b) Three target points. Figure 1: Pareto-optimal sets for M-DMP with Euclidean distance for two illustrative examples. However, this paper considers M-DMP problem with the Manhattan distance, given as follows: d(X,a)=|(x − xa)| + |(y − ya)|. (2) 2.1 Illustrative Examples of M-DMP with Manhattan Distance We take the same two examples as above, and mark the Pareto-optimal set in Figures 2a and 2b. [18] shows that the Pareto-optimal set for the two and three-objective cases (with two and three target points) can be easily mathematically calculated. For the left example having two target points (O1 and O2), all points within the rectangular box formed with two target points as diagonally opposite points are Pareto-optimal, as shown in Figure 2a. For three objectives, the Pareto-optimal set is the combination of the pairwise intersections of the Pareto-optimal sets formed by all two-objective Pareto-optimal sets. For each pair of two objectives, we get rectangular Pareto-optimal sets S12, S13, S23. A little thought will reveal that intersection sets are: A = {S12 ∩ S13}, B = {S12 ∩ S23}, C = {S13 ∩ S23}. The Pareto-optimal set for the overall problem is then P = A ∩ B ∩ C, as shown in Figure 2b. Thus, it is clear that Pareto-optimal set for a M-DMP with Manhattan distance is not straightfor- ward and is more complex to identify than a M-DMP with Euclidean distance measure. We outline the challenges in the next section. 3 O1 O 2 A C O3 B O1 O 2 (a) Two target points. (b) Three target points. Figure 2: Pareto-optimal sets for M-DMP with Manhattan distance for two illustrative examples. 3 Challenges in Solving M-DMP While the Pareto-optimal set is always convex for Euclidean distance metric, as shown in Figure 2b, the Pareto-optimal set can be non-convex for the Manhattan distance metric. Additionally, the Pareto- optimal set can be point-connected, meaning that two convex Pareto-optimal subsets can share a common point. The non-convexity and degeneracy to a single dimension are the challenges that an optimization algorithm must have to overcome before identifying the complete Pareto-optimal set for a M-DMP problem with Manhattan distance. A recent study [18] demonstrated the difficulties that a numerical optimization algorithm faces with to identify the M-DMP with Manhattan distance measure, even for two and three objectives. Even after multiple generations, there were still many spurious and dominated solutions is hard to eliminate due to domination of points possible only along specific directions. Thus, a different approach must be taken to identify the Pareto-optimal set. Here, we borrow results from an existing study on a related problem and suggest a construction procedure that will directly result in identifying the complete Pareto-optimal set. In Section 5, we also provide a detail proof of our construction procedure.