Quasiconvex function

In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form

For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints.

The negative of a quasiconvex function is said to be quasiconcave.

Univariate unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple arguments.

For example, the 2-dimensional Rosenbrock function is unimodal but not quasiconvex and functions with star-convex sublevel sets can be unimodal without being quasiconvex.

An alternative way (see introduction) of defining a quasi-convex function

A function that is both quasiconvex and quasiconcave is quasilinear.

Quasiconvex functions have applications in mathematical analysis, in mathematical optimization, and in game theory and economics.

In nonlinear optimization, quasiconvex programming studies iterative methods that converge to a minimum (if one exists) for quasiconvex functions.

[1] Quasiconvex programming is used in the solution of "surrogate" dual problems, whose biduals provide quasiconvex closures of the primal problem, which therefore provide tighter bounds than do the convex closures provided by Lagrangian dual problems.

[2] In theory, quasiconvex programming and convex programming problems can be solved in reasonable amount of time, where the number of iterations grows like a polynomial in the dimension of the problem (and in the reciprocal of the approximation error tolerated);[3] however, such theoretically "efficient" methods use "divergent-series" step size rules, which were first developed for classical subgradient methods.

In microeconomics, quasiconcave utility functions imply that consumers have convex preferences.

Quasiconvex functions are important also in game theory, industrial organization, and general equilibrium theory, particularly for applications of Sion's minimax theorem.

Generalizing a minimax theorem of John von Neumann, Sion's theorem is also used in the theory of partial differential equations.

A quasiconvex function that is not convex
A function that is not quasiconvex: the set of points in the domain of the function for which the function values are below the dashed red line is the union of the two red intervals, which is not a convex set.
The probability density function of the normal distribution is quasiconcave but not concave.
The bivariate normal joint density is quasiconcave.
A quasilinear function is both quasiconvex and quasiconcave.
The graph of a function that is both concave and quasiconvex on the nonnegative real numbers.