Using invexity, D. V. Luu and N. X. Ha investigated in [4], [5], and [8] the problem
$$\begin{array}{l l} \text{minimize }& f(x)=\inf\limits_{\alpha\in A}f_{\alpha}(x),\\ \text{subject to }& g(x)=\inf\limits_{\beta\in B}g_{\beta}(x)\leq 0,\quad x\in C.\end{array}$$ We show that their results concerned with this problem are wrong or too weak or redundant, while a lot of too strong and unnecessary assumptions are used. For correction, a necessary condition and a sufficient condition for local minima are given in this paper. Some comments and sufficient conditions for local minima of the other problems considered in [5] and [8] are also presented.