An instance-weighted variant of the support vector machine (SVM) has attracted considerable attention recently since they are useful in various machine learning tasks such as non-stationary data analysis, heteroscedastic data modeling, transfer learning, learning to rank, and transduction. An important challenge in these scenarios is to overcome the computational bottleneck-instance weights often change dynamically or adaptively, and thus the weighted SVM solutions must be repeatedly computed. In this paper, we develop an algorithm that can efficiently and exactly up-date the weighted SVM solutions for arbitrary change of instance weights. Technically, this contribution can be regarded as an extension of the conventional solution-path algorithm for a single regularization parameter to multiple instance-weight parameters. However, this extension gives rise to a significant problem that breakpoints (at which the solution path turns) have to be identified in high-dimensional space. To facilitate this,we introduce a parametric representation of instance weights which allows us to find the breakpoints in high-dimensional space easily. Despite its simplicity, our parametrization covers various important machine learning tasks and it widens the applicability of the solution-path algorithm. Through extensive experiments on various practical applications, we demonstrate the usefulness of the proposed algorithm