vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...

, an invex function is a

differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...

f

from

\mathbb^n

\mathbb

for which there exists a vector valued function

\eta

such that :

f(x) - f(u) \geq \eta(x, u) \cdot \nabla f(u), \,

for all ''x'' and ''u''. Invex functions were introduced by Hanson as a generalization of

convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...

s. Ben-Israel and Mond provided a simple proof that a function is invex if and only if every

stationary point In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" inc ...

is a

global minimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...

, a theorem first stated by Craven and Glover. Hanson also showed that if the objective and the constraints of an

optimization problem In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables ...

are invex with respect to the same function

\eta(x,u)

, then the

Karush–Kuhn–Tucker conditions In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be o ...

are sufficient for a global minimum.

Type I invex functions

A slight generalization of invex functions called Type I invex functions are the most general class of functions for which the

are necessary and sufficient for a global minimum. Consider a mathematical program of the form

\begin
\min & f(x)\\
\text & g(x)\leq0
\end

where

f:\mathbb^n\to\mathbb

and

g:\mathbb^n\to\mathbb^m

are differentiable functions. Let

F=\

denote the feasible region of this program. The function

f

is a Type I objective function and the function

g

is a Type I constraint function at

x_0

with respect to

\eta

if there exists a vector-valued function

\eta

defined on

F

such that

f(x)-f(x_0)\geq\eta(x)\cdot\nabla

and

-g(x_0)\geq\eta(x)\cdot\nabla

for all

x\in

. Note that, unlike invexity, Type I invexity is defined relative to a point

x_0

. Theorem (Theorem 2.1 in): If

f

and

g

are Type I invex at a point

x^*

with respect to

\eta

, and the

are satisfied at

x^*

, then

x^*

is a global minimizer of

f

over

F

Type I invex functions

See also

References

Further reading