Additive Constant Methods For Multidimensional Scaling

Multidimensional Scaling (MDS) is a set of methods for discovering "hidden" structures in multidimensional data to make scientific decisions by analyzing the input data, which are typically a matrix of similarities measured on objects. Applications of MDS are found in a wide range of areas, including taxonomy, management, economics, geodesy, geography, genetics, psychology, linguistics, political science, biochemistry, etc. Recently it becomes a research topic of common interest in both areas of management and optimization. This paper is based on the idea about using additive multi-constant to adjust the dissimilarity matrix to make its B-matrix to be positive semidefinit. In [1], Benasseni studied the partial additive constant problem in multidimensional scaling, this is a quite challenging to solve and Benasseni proposed a numerical procedure for the model, which is under rather restrictive assumptions, which prevents it from being widely used. This paper proposes a new model called partial additive multi-constant problem, which is more close to the real problem. By using the quadratic positive semidefinite programming, we can find the optimal solution under very weak conditions. We can also use the ready-to-use numerical package such as QSDP Solver in [22], allowing a great deal of flexibility in choosing parameters, which can get the solution more efficiency.