在一般的数学模型中，由于要忽略一些次要因素，所建的模型往往是近似的，且对数学模型利用数值算法所求得的解大多是近似解。另一方面，在可行集非紧的情况下，精确解的解集往往是空集，而在较弱的条件下近似解集可以是非空的。在Hausdorff局部凸拓扑线性空间中分别研究了无约束和带约束集值均衡问题近似Benson真有效解。在没有任何凸性假设下，利用非线性泛函分别建立了最优性条件。

In the general mathematical model, because some secondary factors should be ignored, the established model is often approximate, and the solutions of mathematical model obtained by numerical algorithm are mostly approximate solutions. On the other hand, in the case of non-compact feasible set, the set of accurate solutions is often empty, while in the weaker case, the set of approximate solutions can be nonempty. In Hausdorff locally convex topological linear spaces, the approximate Benson proper efficient solutions of unconstrained and constrained set-valued equilibrium problems are studied respectively. Without any convexity assumption, necessary and sufficient optimality conditions for approximate Benson proper efficient solutions are established by using nonlinear functional.