In statistical research, regression models based on data play a central role; one of these models is the linear regression model. However, this model may give misleading results when data contain outliers. The outliers in linear regression can be resolved in two stages: by using the Mean Shift Outlier Model (MSOM) and by providing a new solution for this model. First, we construct a Tikhonov regularization problem for the MSOM. Then, we treat this problem using convex optimization techniques, specifically conic quadratic programming, permitting the use of interior point methods. We present numerical examples, which reveal very good results, and we conclude with an outlook to future studies.