What is Invexity?
Recently it was shown that many results in Mathematical Programming involving convex functions actually hold for a wider class of functions, called invex. Here a simple characterization of invexity is given for both constrained and unconstrained problems.
What is pseudo concave?
In convex analysis and the calculus of variations, both branches of mathematics, a pseudoconvex function is a function that behaves like a convex function with respect to finding its local minima, but need not actually be convex. The property must hold in all of the function domain, and not only for nearby points.
Is every convex function differentiable?
(an example is shown in the examples section). A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval. If a function is differentiable and convex then it is also continuously differentiable. for all x and y in the interval.
What is the difference between convex and Quasiconvex?
is a convex set. A (strictly) quasiconvex function has (strictly) convex lower contour sets, while a (strictly) quasiconcave function has (strictly) convex upper contour sets. A function that is both quasiconvex and quasiconcave is quasilinear.
What is non convex function?
A non-convex function is wavy – has some ‘valleys’ (local minima) that aren’t as deep as the overall deepest ‘valley’ (global minimum). Optimization algorithms can get stuck in the local minimum, and it can be hard to tell when this happens.
What is convex to the origin?
“Mathematically, a ‘convex to the origin’ curve is described by saying that if A and B are any two points on the curve, then a ray passing through the origin O and any point X on the line segment AB will meet the curve at most at one point D between O and X”.
Is quasi convex convex?
The negative of a quasiconvex function is said to be quasiconcave. All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity.
How do you test for Quasiconvexity?
Reminder: A function f is quasiconcave if and only if for every x and y and every λ with 0 ≤ λ ≤ 1, if f(x) ≥ f(y) then f((1 − λ)x + λy) ≥ f(y).
What is convex and non convex functions?
A convex function has one minimum – a nice property, as an optimization algorithm won’t get stuck in a local minimum that isn’t a global minimum. Take x2−1, for example: A non-convex function is wavy – has some ‘valleys’ (local minima) that aren’t as deep as the overall deepest ‘valley’ (global minimum).
What is the difference between convex and non convex optimization?
The basic difference between the two categories is that in a) convex optimization there can be only one optimal solution, which is globally optimal or you might prove that there is no feasible solution to the problem, while in b) nonconvex optimization may have multiple locally optimal points and it can take a lot of …
Is sin a convex?
Since f”(−1)>0 , we see that sinx is convex (“concave up”) at x=−1 .
Which is the best definition of effective convexity?
Effective convexity is a discrete approximation of the second derivative of the bond’s value as a function of the interest rate:
Which is the most general type of invex function?
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 are differentiable functions.
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How is convexity used in the bond market?
Application of convexity Convexity is a risk management figure, used similarly to the way ‘gamma’ is used in derivatives risks management; it is a number used to manage the market risk a bond portfolio is exposed to. The second-order approximation of bond price movements due to rate changes uses the convexity: