Invex function

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. Consider a mathematical program of the form

\begin{array}{rl} \min & f(x)\ \text{s.t.} & g(x)\leq0 \end{array}

where f:\mathbb{R}^n\to\mathbb{R} and g:\mathbb{R}^n\to\mathbb{R}^m are differentiable functions. Let F={x\in\mathbb{R}^n;|;g(x)\leq0} 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{f(x_0)}

and

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

for all x\in{F}. 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 Karush–Kuhn–Tucker conditions are satisfied at x^* , then x^* is a global minimizer of f over F .

E-invex function

Let E from \mathbb{R}^n to \mathbb{R}^{n} and f from \mathbb{M} to \mathbb{R} be an E-differentiable function on a nonempty open set \mathbb{M} \subset \mathbb{R}^n. Then f is said to be an E-invex function at u if there exists a vector valued function \eta such that

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

for all x and u in \mathbb{M}.

E-invex functions were introduced by Abdulaleem as a generalization of differentiable convex functions.

E-type I Functions

Let E: \mathbb{R}^n \to \mathbb{R}^n , and M \subset \mathbb{R}^n be an open E-invex set. A vector-valued pair (f, g) , where f and g represent objective and constraint functions respectively, is said to be E-type I with respect to a vector-valued function \eta: M \times M \to \mathbb{R}^n , at u \in M , if the following inequalities hold for all x \in F_E={x\in\mathbb{R}^n;|;g(E(x))\leq 0} :

f_i(E(x)) - f_i(E(u)) \geq \nabla f_i(E(u)) \cdot \eta(E(x), E(u)),

-g_j(E(u)) \geq \nabla g_j(E(u)) \cdot \eta(E(x), E(u)).

Remark 1.

If f and g are differentiable functions and E(x) = x (E is an identity map), then the definition of E-type I functions reduces to the definition of type I functions introduced by Rueda and Hanson.

References

References

1. Hanson, Morgan A.. (1981). "On sufficiency of the Kuhn-Tucker conditions". Journal of Mathematical Analysis and Applications.
2. (1986). "What is invexity?". The ANZIAM Journal.
3. (1985). "Invex functions and duality". Journal of the Australian Mathematical Society.
4. Hanson, Morgan A.. (1999). "Invexity and the Kuhn–Tucker Theorem". Journal of Mathematical Analysis and Applications.
5. (1987). "Necessary and sufficient conditions in constrained optimization". Mathematical Programming.
6. Abdulaleem, Najeeb. (2019). "''E''-invexity and generalized ''E''-invexity in ''E''-differentiable multiobjective programming". ITM Web of Conferences.
7. Abdulaleem, Najeeb. (2023). "Optimality and duality for $ E $-differentiable multiobjective programming problems involving $ E $-type Ⅰ functions". Journal of Industrial and Management Optimization.
8. Rueda, Norma G. (1988-03-01). "Optimality criteria in mathematical programming involving generalized invexity". Journal of Mathematical Analysis and Applications.