Quasiconvexité

Quasiconvexité is a mathematical property of functions. A function f defined on a convex set S is quasiconvex if, for any two points x and y in S and for any scalar t between 0 and 1, the value of f at the convex combination of x and y is less than or equal to the maximum of f(x) and f(y). More formally, f(tx + (1-t)y) <= max(f(x), f(y)) for all x, y in S and t in [0, 1].

An alternative and often useful characterization of quasiconvexity is through its level sets. A function f is quasiconvex if and only if for every real number c, the set {x in S | f(x) <= c} is a convex set. This means that the "below c" regions of a quasiconvex function are convex.

While convex functions are always quasiconvex, the converse is not true. Quasiconvex functions are a broader class. A key difference is that for a convex function, the inequality is f(tx + (1-t)y) <= tf(x) + (1-t)f(y), which is a stricter condition.

Quasiconvexity plays a significant role in optimization theory, particularly in the analysis of minimization problems. If the objective function of a minimization problem is quasiconvex and the feasible region is convex, then any local minimum is also a global minimum. This property simplifies the search for optimal solutions. It is also encountered in economics, for instance, when dealing with utility functions that exhibit diminishing marginal rates of substitution.