The Supplement of the Paper “ Multi-Line Distance Minimization : A Visualized Many-Objective Test Problem Suite ”

Theorem 2. For an ML-DMP (Ω = R) with a regular polygon of m vertexes (A1, A2, ..., Am), points inside the polygon (including the boundary points) are the Pareto optimal solutions. In other words, for any point in the polygon, there is no point ∈ R that dominates it. Proof: Let us first consider the boundary points. It is clear that the m vertex points of the polygon are Pareto optimal since they are the intersection point of two target lines and have minimum distance (i.e., 0) to these two lines. Non-vertex boundary points have the minimum distance (0) to only one target line. Consider a five-objective ML-DMP in Fig. 1, where boundary point P1 is located on the target line ←−−→ A1A2 and has the best value on this objective. Clearly, the point that is able to dominate such a boundary point should be located on the same target line, if existing. Without the loss of generality, assume that P2 is the point dominating P1. According to the definition of the Pareto dominance, d(P2, ←−−→ A2A3) 6 d(P1, ←−−→ A2A3). However, since d(P2, A2) > d(P1, A2), we have d(P2, ←−−→ A2A3) > d(P1, ←−−→ A2A3), therefore, a contradiction. Now consider the points inside the polygon. For an interior point P3 in Fig. 1, assume there is a point P4 such that P4 ≺ P3. Draw a semi-straight line starting from P4 and passing through P3 (i.e., −−−→ P4P3). Since P3 is inside the polygon, there must be an intersection point of −−−→ P4P3 and the polygon boundary. This means that there exists at least one target