The concepts of $\varepsilon$-weakly efficient solution, $\varepsilon$-Henig efficient solution and $\varepsilon$-global efficient solution of set-valued equilibrium problem are introduced in locally convex Hausdorff topological vector spaces, the relationship between $\varepsilon$-Henig efficient solution set and Henig efficient solution set is discussed. By applying a nonlinear scalarization functional, necessary and sufficient optimality conditions are established for various approximation solutions of set-valued equilibrium problem with or without constraints, respectively. Related results in the literature are generalized.