What is jointly convex function?
A real valued function f(A, B) defined on B(H) × B(H) is said to be. jointly convex in (A, B) if. f(λA1 + (1 − λ)A2, λB1 + (1 − λ)B2) ⩽ λf(A1,B1) + (1 − λ)f(A2,B2) for all Ai,Bi ∈ B(H), 1 ⩽ i ⩽ 2 and λ ∈ [0,1]. f(A, B) is said to be jointly concave if −f(A, B) is jointly convex in (A, B).
How do you prove a problem is convex?
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). A function is concave if -f is convex — i.e. if the chord from x to y lies on or below the graph of f.
What does it mean if a function is strictly convex?
A strictly convex function is a function that the straight line between any pair of points on the curve is above the curve. except for the intersection points between the straight line and the curve.
Is the composition of two convex functions convex?
real analysis – The composition of two convex functions is convex – Mathematics Stack Exchange. Stack Overflow for Teams – Start collaborating and sharing organizational knowledge.
Is the sum of convex functions convex?
According to the definition of a convex set, the set S = ∩iSi is also a convex set. Exercise 2 Show that if f(x) and g(x) are convex functions on a convex set S, then their sum h(x) = f(x) + g(x) (4) is also a convex function on S.
How do you know if a function is convex?
To find out if it is concave or convex, look at the second derivative. If the result is positive, it is convex. If it is negative, then it is concave.
What is convex set in linear programming problem?
The convex set is a set of points in a plane that is said to be convex, the line segment joining any two points in the set, completely lies in the set. A bounded feasible region will have both a maximum value and minimum value for the objective function. It is bounded if it can be enclosed in any shape.
Is the minimum of two convex functions convex?
True or False: the minimum of two convex functions is convex. False: Consider (x + 1)2 and (x − 1)2. f(x) convex, g(y) increasing.
How do we transform a convex problem into an equivalent one?
We can transform a convex problem into an equivalent one via a number of transformations. Sometimes the transformation is useful to obtain an explicit solution, or is done for algorithmic purposes. The transformation does not necessarily preserve the convexity properties of the problem. Here is a list of transformations that do preserve convexity.
What is the problem of minimizing convex functions?
Specifically, if is a convex function of the variable , and is partitioned as , with , , , then the function is convex (look at the epigraph of this function). Hence the problem of minimizing can be reduced to one involving only: The reduction may not be easy to carry out explicitly. is positive semi-definite, is positive-definite, and , .
Specifically, if is a convex function of the variable , and is partitioned as , with , , , then the function is convex (look at the epigraph of this function). Hence the problem of minimizing can be reduced to one involving only:
How do you preserve convexity of a problem?
We may ‘‘eliminate’’ some variables of the problem and reduce it to one with fewer variables. This operation preserves convexity. Specifically, if is a convex function of the variable , and is partitioned as , with , , , then the function