Quasiconvex function

Mathematical function with convex lower level sets
the unrelated generalization of convexity used in the calculus of variations

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.

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. 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 (and in the reciprocal of the approximation error tolerated); however, such theoretically "efficient" methods use "divergent series" stepsize 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.

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.

maximum of quasiconvex functions (i. e. ) 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 (i. e. quasiconvex, convex set, then is quasiconvex)

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. The sum of quasiconvex functions defined on different domains (i. e. if are quasiconvex, ) need not be quasiconvex. Such functions are called "additively decomposed" in economics and "separable" in mathematical optimization.

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 (compare unimodality). The floor function is an example of a quasiconvex function that is neither convex nor continuous.