Lesson plan
Invex function
In vector calculus, an invex function is a differentiable function f from \mathbb{R}^n to \mathbb{R} 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 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 \eta(x,u) , then the Karush–Kuhn–Tucker conditions are sufficient for a global minimum.
Article summary
In vector calculus, an invex function is a differentiable function f from \mathbb{R}^n to \mathbb{R} 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 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 \eta(x,u) , then the Karush–Kuhn–Tucker conditions are sufficient for a global minimum.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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