In this work we perform Reliability-Based Design Optimization (RBDO) by classifying the limit states by using soft non-linear Support Vector Machines (SVM). By adopting the kernel trick in the dual formulation, by using e.g. the Gaussian kernel,we classify non-linear states of fail or safe obtained from design of experiments. The Most Probable Point (MPP) of the SVM is established in the physical space where the distance is minimized in the metric of Hasofer-Lind. The solution to the corresponding optimality conditions is obtained by using Newton's method with an inexact Jacobian and a line-search of Armijo type. At the MPP, we perform Taylor expansions of the SVM using intermediate variables defined by the iso-probabilistic transformation. In such manner, we derive a Quadratic Programming (QP) problem which is solved in the standard normal space. This is done for several probability distributions such as e.g. lognormal, Gumbel, gamma and Weibull. The optimal solution to the QP problem is mapped back to the physical space and new Taylor expansions of the SVM are derived and a new QP problem is formulated and solved. This procedure continues in sequence until we obtain convergence of our RBDO problem. The steps presented above constitute our proposed FORM-based sequential QP approach for RBDO by using SVM. The target of reliability appearing in the FORM-based QP problem might also be adjusted using different SORM formulas such as e.g. Breitung, Hohenbichler or Tvedt, or by applying importance-based Halton or Hammersley sampling. A nice feature of the proposed SVM-based RBDO approach is that several limit state functions can be represented simultaneously by only one single SVM. Thus, the proposed SVM-based RBDO methodology might be considered to be a rational approach for the treatment of RBDO problems including system reliability. This is demonstrated by solving established RBDO benchmarks.