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Invex function

In vector calculus, an invex function is a differentiable function f {\displaystyle f} from R n {\displaystyle \mathbb {R} ^{n}} to R {\displaystyle \mathbb {R} } for which there exists a vector valued function η {\displaystyle \eta } such that

f ( x ) − f ( u ) ≥ η ( x , u ) ⋅ ∇ f ( u ) , {\displaystyle 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 functions. 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.

Hanson also showed that if the objective and the constraints of an optimization problem are invex with respect to the same function η ( x , u ) {\displaystyle \eta (x,u)} , then the Karush–Kuhn–Tucker conditions are sufficient for a global minimum.

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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

min f ( x ) s.t. g ( x ) ≤ 0 {\displaystyle {\begin{array}{rl}\min &f(x)\\{\text{s.t.}}&g(x)\leq 0\end{array}}}

where f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } and g : R n → R m {\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} ^{m}} are differentiable functions. Let F = { x ∈ R n | g ( x ) ≤ 0 } {\displaystyle F=\{x\in \mathbb {R} ^{n}\;|\;g(x)\leq 0\}} denote the feasible region of this program. The function f {\displaystyle f} is a Type I objective function and the function g {\displaystyle g} is a Type I constraint function at x 0 {\displaystyle x_{0}} with respect to η {\displaystyle \eta } if there exists a vector-valued function η {\displaystyle \eta } defined on F {\displaystyle F} such that

f ( x ) − f ( x 0 ) ≥ η ( x ) ⋅ ∇ f ( x 0 ) {\displaystyle f(x)-f(x_{0})\geq \eta (x)\cdot \nabla {f(x_{0})}}

and

− g ( x 0 ) ≥ η ( x ) ⋅ ∇ g ( x 0 ) {\displaystyle -g(x_{0})\geq \eta (x)\cdot \nabla {g(x_{0})}}

for all x ∈ F {\displaystyle x\in {F}} .5 Note that, unlike invexity, Type I invexity is defined relative to a point x 0 {\displaystyle x_{0}} .

Theorem (Theorem 2.1 in6): If f {\displaystyle f} and g {\displaystyle g} are Type I invex at a point x ∗ {\displaystyle x^{*}} with respect to η {\displaystyle \eta } , and the Karush–Kuhn–Tucker conditions are satisfied at x ∗ {\displaystyle x^{*}} , then x ∗ {\displaystyle x^{*}} is a global minimizer of f {\displaystyle f} over F {\displaystyle F} .

E-invex function

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

( f ∘ E ) ( x ) − ( f ∘ E ) ( u ) ≥ ∇ ( f ∘ E ) ( u ) ⋅ η ( E ( x ) , E ( u ) ) , {\displaystyle (f\circ E)(x)-(f\circ E)(u)\geq \nabla (f\circ E)(u)\cdot \eta (E(x),E(u)),\,}

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

E-invex functions were introduced by Abdulaleem as a generalization of differentiable convex functions.7

E-type I Functions

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

f i ( E ( x ) ) − f i ( E ( u ) ) ≥ ∇ f i ( E ( u ) ) ⋅ η ( E ( x ) , E ( u ) ) , {\displaystyle f_{i}(E(x))-f_{i}(E(u))\geq \nabla f_{i}(E(u))\cdot \eta (E(x),E(u)),}

− g j ( E ( u ) ) ≥ ∇ g j ( E ( u ) ) ⋅ η ( E ( x ) , E ( u ) ) . {\displaystyle -g_{j}(E(u))\geq \nabla g_{j}(E(u))\cdot \eta (E(x),E(u)).}

Remark 1.

If f {\displaystyle f} and g {\displaystyle g} are differentiable functions and E ( x ) = x {\displaystyle E(x)=x} ( E {\displaystyle E} is an identity map), then the definition of E-type I functions8 reduces to the definition of type I functions introduced by Rueda and Hanson.9

See also

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.

References

  1. Hanson, Morgan A. (1981). "On sufficiency of the Kuhn-Tucker conditions". Journal of Mathematical Analysis and Applications. 80 (2): 545–550. doi:10.1016/0022-247X(81)90123-2. hdl:10338.dmlcz/141569. ISSN 0022-247X. /wiki/Doi_(identifier) ↩

  2. Ben-Israel, A.; Mond, B. (1986). "What is invexity?". The ANZIAM Journal. 28 (1): 1–9. doi:10.1017/S0334270000005142. ISSN 1839-4078. https://doi.org/10.1017%2FS0334270000005142 ↩

  3. Craven, B. D.; Glover, B. M. (1985). "Invex functions and duality". Journal of the Australian Mathematical Society. 39 (1): 1–20. doi:10.1017/S1446788700022126. ISSN 0263-6115. https://doi.org/10.1017%2FS1446788700022126 ↩

  4. Hanson, Morgan A. (1999). "Invexity and the Kuhn–Tucker Theorem". Journal of Mathematical Analysis and Applications. 236 (2): 594–604. doi:10.1006/jmaa.1999.6484. ISSN 0022-247X. https://doi.org/10.1006%2Fjmaa.1999.6484 ↩

  5. Hanson, M. A.; Mond, B. (1987). "Necessary and sufficient conditions in constrained optimization". Mathematical Programming. 37 (1): 51–58. doi:10.1007/BF02591683. ISSN 1436-4646. S2CID 206818360. /wiki/Doi_(identifier) ↩

  6. Hanson, Morgan A. (1999). "Invexity and the Kuhn–Tucker Theorem". Journal of Mathematical Analysis and Applications. 236 (2): 594–604. doi:10.1006/jmaa.1999.6484. ISSN 0022-247X. https://doi.org/10.1006%2Fjmaa.1999.6484 ↩

  7. Abdulaleem, Najeeb (2019). "E-invexity and generalized E-invexity in E-differentiable multiobjective programming". ITM Web of Conferences. 24 (1) 01002. doi:10.1051/itmconf/20192401002. https://doi.org/10.1051%2Fitmconf%2F20192401002 ↩

  8. Abdulaleem, Najeeb (2023). "Optimality and duality for $ E $-differentiable multiobjective programming problems involving $ E $-type Ⅰ functions". Journal of Industrial and Management Optimization. 19 (2): 1513. doi:10.3934/jimo.2022004. ISSN 1547-5816. https://www.aimsciences.org/article/doi/10.3934/jimo.2022004 ↩

  9. Rueda, Norma G; Hanson, Morgan A (1988-03-01). "Optimality criteria in mathematical programming involving generalized invexity". Journal of Mathematical Analysis and Applications. 130 (2): 375–385. doi:10.1016/0022-247X(88)90313-7. ISSN 0022-247X. https://linkinghub.elsevier.com/retrieve/pii/0022247X88903137 ↩