How do you determine if a multivariable function is convex or concave?
Let f be a function of many variables, defined on a convex set S. We say that f is concave if the line segment joining any two points on the graph of f is never above the graph; f is convex if the line segment joining any two points on the graph is never below the graph.
How do you optimize a convex function?
Convex optimization problems can also be solved by the following contemporary methods:
- Bundle methods (Wolfe, Lemaréchal, Kiwiel), and.
- Subgradient projection methods (Polyak),
- Interior-point methods, which make use of self-concordant barrier functions and self-regular barrier functions.
- Cutting-plane methods.
How do you prove a multivariable function is concave?
Let f and g be functions of many variables. If f and g are concave and a ≥ 0 and b ≥ 0 then the function h defined by h(x) = af(x) + bg(x) for all x is concave.
How do you determine if a function is convex or nonconvex?
If in the whole range it is positive then it is convex if it is negative then it is concave, if it can be both positive and negative (for some sub-range) then it is neither convex nor concave. Linear functions (with second order derivative zero) are both convex and concave.
How do you determine if a set is convex?
Definition 3.1 A set C is convex if the line segment between any two points in C lies in C, i.e. ∀x1,x2 ∈ C, ∀θ ∈ [0, 1] θx1 + (1 − θ)x2 ∈ C.
How many ray diagrams does a convex mirror have?
To summarise
| Position of the object | Position of the image | Nature of the image |
|---|---|---|
| At infinity | At the focus F, behind the mirror | Virtual and erect |
| Between infinity and the pole P of the mirror | Between P and F, behind the mirror | Virtual and erect |
What happens to an image in a convex mirror?
The image produced by a convex mirror is always virtual, and located behind the mirror. When the object is far away from the mirror the image is upright and located at the focal point. As the object approaches the mirror the image also approaches the mirror and grows until its height equals that of the object.
Where is the image formed in a convex mirror?
Convex mirrors always form virtual images. This is because the focal point and the centre of curvature of the convex mirror are imaginary points and that cannot be reached. So the image is formed inside the mirror and cannot be projected on a screen.
