Английская Википедия:Invex function
In vector calculus, an invex function is a differentiable function <math>f</math> from <math>\mathbb{R}^n</math> to <math>\mathbb{R}</math> for which there exists a vector valued function <math>\eta</math> such that
- <math>f(x) - f(u) \geq \eta(x, u) \cdot \nabla f(u), \, </math>
for all x and u.
Invex functions were introduced by Hanson as a generalization of convex functions.[1] Ben-Israel and Mond provided a simple proof that a function is invex if and only if every stationary point is a global minimum, a theorem first stated by Craven and Glover.[2][3]
Hanson also showed that if the objective and the constraints of an optimization problem are invex with respect to the same function <math>\eta(x,u) </math>, then the Karush–Kuhn–Tucker conditions 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 Karush–Kuhn–Tucker conditions are necessary and sufficient for a global minimum.[4] Consider a mathematical program of the form
<math>\begin{array}{rl} \min & f(x)\\ \text{s.t.} & g(x)\leq0 \end{array}</math>
where <math>f:\mathbb{R}^n\to\mathbb{R}</math> and <math>g:\mathbb{R}^n\to\mathbb{R}^m</math>are differentiable functions. Let <math>F=\{x\in\mathbb{R}^n\;|\;g(x)\leq0\}</math>denote the feasible region of this program. The function <math>f</math> is a Type I objective function and the function <math>g</math> is a Type I constraint function at <math>x_0</math>with respect to <math>\eta</math> if there exists a vector-valued function <math>\eta</math> defined on <math>F</math> such that
<math>f(x)-f(x_0)\geq\eta(x)\cdot\nabla{f(x_0)}</math>
and
<math>-g(x_0)\geq\eta(x)\cdot\nabla{g(x_0)}</math>
for all <math>x\in{F}</math>.[5] Note that, unlike invexity, Type I invexity is defined relative to a point <math>x_0</math>.
Theorem (Theorem 2.1 in[4]): If <math>f </math> and <math>g </math> are Type I invex at a point <math>x^* </math>with respect to <math>\eta </math>, and the Karush–Kuhn–Tucker conditions are satisfied at <math>x^* </math>, then <math>x^* </math>is a global minimizer of <math>f </math> over <math>F </math>.
E-invex function
Let <math>E</math> from <math>\mathbb{R}^n</math> to <math>\mathbb{R}^{n}</math> and <math>f</math> from <math>\mathbb{M}</math> to <math>\mathbb{R}</math> be an <math>E</math>-differentiable function on a nonempty open set <math>\mathbb{M} \subset \mathbb{R}^n</math>. Then <math>f</math> is said to be an E-invex function at <math>u</math> if there exists a vector valued function <math>\eta</math> such that
- <math>(f\circ E)(x) - (f\circ E)(u) \geq \nabla (f\circ E)(u) \cdot \eta(E(x), E(u)) , \, </math>
for all <math>x</math> and <math>u</math> in <math>\mathbb{M}</math>.
E-invex functions were introduced by Abdulaleem as a generalization of differentiable convex functions.[6]
See also
References
Further reading
- S. K. Mishra and G. Giorgi, Invexity and optimization, Nonconvex Optimization and Its Applications, Vol. 88, Springer-Verlag, Berlin, 2008.
- S. K. Mishra, S.-Y. Wang and K. K. Lai, Generalized Convexity and Vector Optimization, Springer, New York, 2009.
Шаблон:Convex analysis and variational analysis
