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 (-\infty,a) is a convex set. Informally, after cutting off the top of the function and looking down at the remaining part of the domain, an observer always sees something that is convex.

All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity. Quasiconvexity extends the notion of unimodality for functions with a single real argument.

Contents

Definition and properties

A function f:S \to \mathbb{R} defined on a convex subset S of a real vector space is quasiconvex if for all x, y \in S and \lambda \in [0,1] we have

f(\lambda x %2B (1 - \lambda)y)\leq\max\big(f(x),f(y)\big).

In words, if f is such that it is always true that a point directly between two other points does not give a higher a value of the function than do both of the other points, then f is quasiconvex. Note that the points x and y, and the point directly between them, can be points on a line or more generally points in n-dimensional space.

An alternative way (see introduction) of defining a quasi-convex function f(x) is to require that each sub-levelset S_\alpha(f) = \{x|f(x) \leq \alpha\} is a convex set.

If furthermore

f(\lambda x %2B (1 - \lambda)y)<\max\big(f(x),f(y)\big)

for all x \neq y and \lambda \in (0,1), then f 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.

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 f is quasiconcave if

f(\lambda x %2B (1 - \lambda)y)\geq\min\big(f(x),f(y)\big).

and strictly quasiconcave if

f(\lambda x %2B (1 - \lambda)y)>\min\big(f(x),f(y)\big)

A (strictly) quasiconvex function has (strictly) convex lower contour sets, while a (strictly) quasiconcave function has (strictly) convex upper contour sets.

A function that is both quasiconvex and quasiconcave is quasilinear.

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 (if one exists) for quasiconvex functions. Quasiconvex programming is a generalization of convex programming.[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" 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.

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

Operations not preserving quasiconvexity

In fact, if the sum of a finite set of (nonconstant) quasiconvex functions is quasiconvex, then all but either zero or one of the functions must be convex; this result holds for separable functions, in particular.[4][5][6][7]

Examples

See also

References

  1. ^ Di Guglielmo (1977, pp. 287–288): Di Guglielmo, F. (1977). "Nonconvex duality in multiobjective optimization". Mathematics of Operations Research 2 (3): 285–291. doi:10.1287/moor.2.3.285. JSTOR 3689518. MR484418. 
  2. ^ Di Guglielmo, F. (1981). "Estimates of the duality gap for discrete and quasiconvex optimization problems". In Schaible, Siegfried; Ziemba, William T.. Generalized concavity in optimization and economics: Proceedings of the NATO Advanced Study Institute held at the University of British Columbia, Vancouver, B.C., August 4–15, 1980. New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. pp. 281–298. ISBN 0-12-621120-5. MR652702. 
  3. ^ Kiwiel, Krzysztof C. (2001). "Convergence and efficiency of subgradient methods for quasiconvex minimization". Mathematical Programming (Series A) (Berlin, Heidelberg: Springer) 90 (1): pp. 1-25. doi:10.1007/PL00011414. ISSN 0025-5610. MR1819784.  Kiwiel acknowledges that Yuri Nesterov first established that quasiconvex minimization problems can be solved efficiently.
  4. ^ Gérard Debreu and Tjalling C. Koopmans. "Additively Decomposed Quasiconvex Functions", 1982, Mathematical Programming 24(l), 1–38.
  5. ^ Crouzeix, J. P. and Lindberg, P. O. 1986. Additively decomposed quasi-convex functions. Mathematical Programming 35(l), 42–57.
  6. ^ This theorem was proved for the special case of twice continuously differentiable functions by Arrow and Enthoven: Kenneth J. Arrow and A. C. Enthoven, Quasiconcave programming,Econometrica 29 (1961) 779-800.
  7. ^ Crouzeix, J.-P. (2008). "Quasi-concavity". In Durlauf, Steven N.; Blume, Lawrence E. The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. doi:10.1057/9780230226203.1375. http://www.dictionaryofeconomics.com/article?id=pde2008_Q000008. 

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