Search results
Definition 1. A function f : S ⊂ Rn → R defined on a convex set S is concave if for any two points x1 x2 ∈ , S and for any λ ∈ [0, 1] we have: λx1 (1 − λ) x2 ≥ λf(x1) (1 − λ)f(x2) + +. is called strictly concave if for any two points x1 , x2 ∈ S and for any λ ∈ (0, 1) we have: λx1 (1 − λ) x2 > λf(x1) (1 − λ)f(x2) + +.
In this lecture, we shift our focus to the other important player in convex optimization, namely, convex functions. Here are some of the topics that we will touch upon: Convex, concave, strictly convex, and strongly convex functions. First and second order characterizations of convex functions.
- 1MB
- 14
The terms “concave” and convex” are from the perspective of the object: A concave mirror caves in on the object; whereas. A convex mirror flexes away from the object. Each mirror has a radius of curvature r (which is infinite for the plane mirror) and a focal length f = 1 2r.
- 96KB
- 14
us.Proof. Let f be concave. Consider rst the case N = 1. Theorem 3 implies that for any a; b; c 2 C, with a < b < c, the graph of f is sandwiched between the graphs of two lines through the point (b; f(b)), one line through the points (a; f(a)) and (b; f(b)) and the other through the po. nts (b; f(b)) and (.
- 227KB
- 12
1 Concave and convex functions. Definition 1 A function f defined on the convex set C ⊂ Rn is called con-cave if for every x1, x2 ∈ C and 0 ≤ t ≤ 1, we have. f(tx1 + (1 − t)x2) ≥ tf(x1) + (1 − t)f(x2). Definition 2 A function f defined on the convex set C ⊂ Rn is called strictly concave if for every x1 6= x2,and 0 < t < 1, we have.
Definition 12.1: We say that the curve is convex downward bending up on the interval . if all points of the curve lie above the tangent at any point on the interval. Or when the curve turns anti-clock wise we call it is convex downward (concave upward) (see Fig. 1). Fig.1. (Convex downward/Bending up)
People also ask
Why are concave functions important?
Is concavity a necessary condition for a function?
How do you analyze concavity if n 1 is separable?
How do you prove a concave function?
How do we generalize concave functions?
Is F a concave function?
For flat mirror, M = 1. Concave Mirror (converging) A spherical mirror has the shape of a segment of a sphere. The mirror is characterized by its radius of curvature R, and center of curvature C. The mirror axis (or principal axis) passes through C and the vertex V of the mirror.
