It is well-known that the sum of two quasiconvex functions is not quasiconvex in general, and the same occurs with the minimum. Although apparently these two statements (for the sum or minimum) have nothing in common, they are related, as we show in this paper. To develop our study, the notion of quasiconvex family is introduced, and we establish various characterizations of such a concept: one of them being the quasiconvexity of the pointwise infimum of arbitrary translations of quasiconvex functions in the family; another is the convexity of the union of any two of their sublevel sets; a third one is the quasiconvexity of the sum of the quasiconvex functions, composed with arbitrary nondecreasing functions. As a by-product, any of the aforementioned characterizations, besides providing quasiconvexity of the sum, also implies the semistrict quasiconvexity of the sum if every function in the family has the same property. Three concrete applications in quasiconvex optimization are presented: First, we establish the convexity of the (Benson) proper efficient solution set to a quasiconvex vector optimization problem; second, we derive conditions that allow us to reduce a constrained optimization problem to one with a single inequality constraint, and finally, we show a class of quasiconvex minimization problems having zero duality gap.

众所周知，两个拟凸函数之和通常不是拟凸的，而最小值也相同。尽管显然这两个语句（总和或最小值）没有共同点，但它们是相关的，正如我们在本文中所展示的那样。为了发展我们的研究，引入了拟凸族的概念，并且我们建立了这种概念的各种特征：其中之一是该族中拟凸函数的任意翻译的有向无穷的点凸的拟凸性；另一个是其子级集中任何两个子集的并集的凸性；第三个是拟凸函数之和的拟凸性，它由任意的非递减函数组成。作为副产品，上述任何特征除了提供总和的拟凸性外，如果族中的每个函数都具有相同的属性，则也意味着和的半严格拟凸性。提出了在拟凸优化中的三个具体应用：首先，建立拟凸矢量优化问题的（Benson）适当有效解集的凸性；其次，推导条件，使我们可以将约束优化问题简化为具有单个不等式约束的条件，最后，我们展示了一类具有零对偶间隙的拟凸最小化问题。