Inmathematics, aquasiconvex function is areal-valuedfunction defined on aninterval or on aconvex subset of a realvector space such that theinverse image of any set of the form${\displaystyle (-\mathrm{\infty },a)}$ is aconvex 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 bequasiconcave.

Quasiconvexity is a more general property than convexity in that allconvex functions are also quasiconvex, but not all quasiconvex functions are convex.Univariateunimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiplearguments. For example, the 2-dimensionalRosenbrock function is unimodal but not quasiconvex and functions withstar-convex sublevel sets can be unimodal without being quasiconvex.

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

${\displaystyle f(\lambda x+(1-\lambda )y)\le max\{f(x),f(y)\}.}$

In words, if${\displaystyle f}$ 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${\displaystyle f}$ is quasiconvex. Note that the points${\displaystyle x}$ and${\displaystyle y}$, and the point directly between them, can be points on a line or more generally points inn-dimensional space.

If the inequality is strict, i.e.

for all${\displaystyle x\ne y}$ and${\displaystyle \lambda \in (0,1)}$, then${\displaystyle f}$ isstrictly 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 (see introduction) of defining a quasi-convex function${\displaystyle f(x)}$ is to require that each sublevel set${\displaystyle S_{\alpha }(f)=\{x\mid f(x)\le \alpha \}}$is a convex set.

Aquasiconcave function is a function whose negative is quasiconvex, and astrictly quasiconcave function is a function whose negative is strictly quasiconvex. Equivalently a function${\displaystyle f}$ is strictly quasiconcave if and only if

${\displaystyle f(\lambda x+(1-\lambda )y)\ge min\{f(x),f(y)\}.}$

A (strictly) quasiconvex function has (strictly) convexlower contour sets, while a (strictly) quasiconcave function has (strictly) convexupper 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 isquasilinear, and satisfies

${\displaystyle min\{f(x),f(y)\}\le f(\lambda x+(1-\lambda )y)\le max\{f(x),f(y)\}}$

For a quasilinear function defined on a plane, the levelsets are always lines. More generally, the level sets of a quasilinear function over${\displaystyle {\mathbb{R}}^n}$ are${\displaystyle n-1}$-dimensional planes.

Quasiconvex functions have applications inmathematical analysis, inmathematical optimization, and ingame theory andeconomics.

Innonlinear optimization, quasiconvex programming studiesiterative methods that converge to a minimum (if one exists) for quasiconvex functions. Quasiconvex programming is a generalization ofconvex 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 Lagrangiandual problems.^[2] Intheory, 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 classicalsubgradient 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 nonsmoothfilter methods.

Inmicroeconomics, quasiconcaveutility functions imply that consumers haveconvex preferences. Quasiconvex functions are importantalso ingame theory,industrial organization, andgeneral equilibrium theory, particularly for applications ofSion's minimax theorem. Generalizing aminimax theorem ofJohn von Neumann, Sion's theorem is also used in the theory ofpartial differential equations.

• maximum of quasiconvex functions (i.e.${\displaystyle f=max\left\{f_1,\dots ,f_n\right\}}$ ) is quasiconvex. Similarly, maximum of strict quasiconvex functions is strict quasiconvex.^[4] Similarly, theminimum ofquasiconcave functions is quasiconcave, and the minimum of strictly-quasiconcave functions is strictly-quasiconcave.
• composition with a non-decreasing function :${\displaystyle g:{\mathbb{R}}^n\to \mathbb{R}}$ quasiconvex,${\displaystyle h:\mathbb{R}\to \mathbb{R}}$ non-decreasing, then${\displaystyle f=h\circ g}$ is quasiconvex. Similarly, if${\displaystyle g:{\mathbb{R}}^n\to \mathbb{R}}$ quasiconcave,${\displaystyle h:\mathbb{R}\to \mathbb{R}}$ non-decreasing, then${\displaystyle f=h\circ g}$ is quasiconcave.
• minimization (i.e.${\displaystyle f(x,y)}$ quasiconvex,${\displaystyle C}$ convex set, then${\displaystyle h(x)=\underset{y\in C}{inf}f(x,y)}$ is quasiconvex)

• The sum of quasiconvex functions defined onthe same domain need not be quasiconvex: In other words, if${\displaystyle f(x),g(x)}$ are quasiconvex, then${\displaystyle (f+g)(x)=f(x)+g(x)}$ need not be quasiconvex. For example,${\displaystyle \pm \mathrm{sign}(x\pm 1)}$ are quasiconvex (in fact, quasilinear) functions whose sum is not quasiconvex.
• The sum of quasiconvex functions defined ondifferent domains (i.e. if${\displaystyle f(x),g(y)}$ are quasiconvex,${\displaystyle h(x,y)=f(x)+g(y)}$) need not be quasiconvex. Such functions are called "additively decomposed" in economics and "separable" inmathematical optimization. For example,${\displaystyle f(x)=x^2}$ and${\displaystyle g(y)=(y-1{)}^2}$ are quasiconvex (in fact, convex), but${\displaystyle h(x,y)=f(x)+g(y)}$ is not quasiconvex.

• Every convex function is quasiconvex.
• A concave function can be quasiconvex. For example,${\displaystyle x\mapsto \log (x)}$ is both concave and quasiconvex.
• Anymonotonic function is both quasiconvex and quasiconcave. More generally, a function which decreases up to a point and increases from that point on is quasiconvex (compareunimodality).
• Thefloor function${\displaystyle x\mapsto \lfloor x\rfloor }$ is an example of a quasiconvex function that is neither convex nor continuous.

• Convex function
• Concave function
• Logarithmically concave function
• Pseudoconvexity in the sense of several complex variables (not generalized convexity)
• Pseudoconvex function
• Invex function
• Concavification

• Avriel, M., Diewert, W.E., Schaible, S. and Zang, I.,Generalized Concavity, Plenum Press, 1988.
• Crouzeix, J.-P. (2008). "Quasi-concavity". In Durlauf, Steven N.; Blume, Lawrence E (eds.).The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. pp. 815–816.doi:10.1057/9780230226203.1375.ISBN 978-0-333-78676-5.
• Singer, IvanAbstract convex analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1997. xxii+491 pp. ISBN 0-471-16015-6

• SION, M., "On general minimax theorems", Pacific J. Math. 8 (1958), 171-176.
• Mathematical programming glossaryArchived 2006-07-15 at theWayback Machine
• Quasiconcavity and quasiconvexity - by Martin J. Osborne,University of Toronto Department of Economics

• Convex analysis
• Convex optimization
• Generalized convexity
• Real analysis
• Types of functions

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