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 is a convex set. 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.
Quasiconvexity is a more general property than convexity in that all convex functions are also quasiconvex, but not all quasiconvex functions are convex. 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.

Definition and properties

A function defined on a convex subset of a real vector space is quasiconvex if for all and we have
In words, if is such that it is always true that a point directly between two other points does not give a higher value of the function than both of the other points do, then is quasiconvex. Note that the points and, and the point directly between them, can be points on a line or more generally points in n-dimensional space.
If the inequality is strict, i.e.
for all and, then is strictly quasiconvex. That is, strict quasiconvexity requires that a point directly between two other points must give a lower value of the function than one of the other points does.
An alternative way of defining a quasi-convex function is to require that each sublevel set
is a convex set.
A quasiconcave function is a function whose negative is quasiconvex, and a strictly quasiconcave function is a function whose negative is strictly quasiconvex. Equivalently a function is strictly quasiconcave if and only if
A quasiconvex function has convex lower contour sets, while a quasiconcave function has convex upper contour sets.
Unimodal probability distributions like the Gaussian distribution are common examples of quasi-concave functions that are not concave.
A function that is both quasiconvex and quasiconcave is quasilinear, and satisfies
For a quasilinear function defined on a plane, the level
sets are always lines. More generally, the level sets of a quasilinear function over are -dimensional planes.

Applications

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

Mathematical optimization

In nonlinear optimization, quasiconvex programming studies iterative methods that converge to a minimum for quasiconvex functions. Quasiconvex programming is a generalization of convex programming. 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. 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 ; however, such theoretically "efficient" methods use "divergent-series" step size rules, which were first developed for classical subgradient methods. Classical subgradient methods using divergent-series rules are much slower than modern methods of convex minimization, such as subgradient projection methods, bundle methods of descent, and nonsmooth filter methods.

Economics and partial differential equations: Minimax theorems


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.

Preservation of quasiconvexity

Operations preserving quasiconvexity

  • maximum of quasiconvex functions is quasiconvex. Similarly, maximum of strict quasiconvex functions is strict quasiconvex. Similarly, the minimum of quasiconcave functions is quasiconcave, and the minimum of strictly-quasiconcave functions is strictly-quasiconcave.
  • composition with a non-decreasing function : quasiconvex, non-decreasing, then is quasiconvex. Similarly, if quasiconcave, non-decreasing, then is quasiconcave.
  • minimization

Operations not preserving quasiconvexity

  • The sum of quasiconvex functions defined on the same domain need not be quasiconvex: In other words, if are quasiconvex, then need not be quasiconvex. For example, are quasiconvex functions whose sum is not quasiconvex.
  • The sum of quasiconvex functions defined on different domains need not be quasiconvex. Such functions are called "additively decomposed" in economics and "separable" in mathematical optimization. For example, and are quasiconvex, but is not quasiconvex.

Examples

  • Every convex function is quasiconvex.
  • A concave function can be quasiconvex. For example, is both concave and quasiconvex.
  • Any monotonic function is both quasiconvex and quasiconcave. More generally, a function which decreases up to a point and increases from that point on is quasiconvex.
  • The floor function is an example of a quasiconvex function that is neither convex nor continuous.