Contents
How do I know if my problem is convex?
Geometrically, a function is convex if a line segment drawn from any point (x, f(x)) to another point (y, f(y)) — called the chord from x to y — lies on or above the graph of f, as in the picture below: Algebraically, f is convex if, for any x and y, and any t between 0 and 1, f( tx + (1-t)y ) <= t f(x) + (1-t) f(y).
How do you show a convex optimization problem?
For an optimization problem to be convex, its hessian matrix must be positive definite in the whole search space. Hessian matrix is formed by the elements of partial second derivatives of Lagrangian function with respect to its control variables.
How do you know if a function of two variables is convex?
For a twice-differentiable function f, if the second derivative, f ”(x), is positive (or, if the acceleration is positive), then the graph is convex (or concave upward); if the second derivative is negative, then the graph is concave (or concave downward).
How do you show a convex set?
so [x,y] ⊆ B(x,r). If C1 and C2 are convex sets, so is their intersection C1 ∩C2; in fact, if C is any collection of convex sets, then OC (the intersection of all of them) is convex. The proof is short: if x,y ∈ OC, then x,y ∈ C for each C ∈ C. Therefore [x,y] ⊆ C for each C ∈ C, which means [x,y] ⊆ OC.
Which is the best way to check convexity?
The book “Convex Optimization” by Boyd, available free online here, describes methods to check. The standard definition is if f (θx + (1 − θ)y) ≤ θf (x) + (1 − θ)f (y) for 0≤θ≤1 and the domain of x,y is also convex.
How to calculate the convexity of a curve?
In fact, effective convexity is a secondary curve convexity statistic that measures the secondary effect of a change in the benchmark yield curve. Effective convexity = P V − +P V + −2P V 0 (ΔCurve)2P V 0 Effective convexity = P V − + P V + − 2 P V 0 ( Δ C u r v e) 2 P V 0. Where:
How can you tell if a function is convex?
So if you could prove that for your function, you would know it’s convex. The Hessian being positive semi-definite, as mentioned in comments, would also show that the function is convex. See page 67 of the book for more.
What does the convexity of a bond mean?
The term “convexity” refers to the higher sensitivity of the bond price to the changes in the interest rate. In other words, the convexity captures the inverse relationship between the yield of a bond and its price wherein the change in bond price is higher than the change in the interest rate.