ABSTRACT
Most of the objective functions used in optimization problems are generally quasi-concave. In many problems, both quasi-concave and quasi-convex functions characterize a constraint set, which is a convex set. There are quasi-concave functions which are not concave functions, although the converse need not be true. Upper-level sets, also known as upper contour sets, are convex sets for quasi-concave functions, and they are used in problems involving consumer's utility maximization and a company's cost minimization. Quasi-concavity is a weaker assumption than concavity in the sense that, although every concave function is quasi-concave, the converse is not true. However, economists sometimes demand something more than quasi-concavity. Note that in general a quasi-concave function that is also concave has its graph approximately shaped like a bell, or part thereof, and a quasi-convex function has its graph shaped like an inverted bell, or a part of it.
