The SVDD (support vector data description) is one of the most well-known one-class support vector learning methods, in which one tries the strategy of utilizing balls defined on the feature space in order to distinguish a set of normal data from all other possible abnormal objects. Recently, with the objective of generalizing the SVDD which treats all training data with equal importance, the so-called D-SVDD (density-induced support vector data description) was proposed incorporating the idea that the data in a higher density region are more significant than those in a lower density region. In this paper, we consider the problem of further improving the D-SVDD toward the use of a partial reference set for testing, and propose an LMI (linear matrix inequality)-based optimization approach to solve the improved version of the D-SVDD problems. Our approach utilizes a new class of density-induced distance measures based on the RSDE (reduced set density estimator) along with the LMI-based mathematical formulation in the form of the SDP (semi-definite programming) problems, which can be efficiently solved by interior point methods. The validity of the proposed approach is illustrated via numerical experiments using real data sets.