Prior knowledge over arbitrary general sets is incorporated into nonlinear kernel approximation problems in the form of linear constraints in 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 two synthetic examples and an important lymph node metastasis prediction problem. All these problems exhibit marked improvements upon the introduction of prior knowledge over nonlinear kernel approximation approaches that do not utilize such knowledge. (Joint work with my PhD student Edward (Ted) W. Wild)