Prior knowledge over arbitrary general sets is incorporated into nonlinear support vector machine approximation and classification problems as linear constraints of a linear program. The key tool in this incorporation is a theorem of the alternative for convex functions that converts nonlinear prior knowledge implications into linear inequalities without the need to kernelize these implications. Effectiveness of the proposed formulation is demonstrated on synthetic examples and on important breast cancer prognosis problems. All these problems exhibit marked improvements upon the introduction of prior knowledge over nonlinear kernel approaches that do not utilize such knowledge.