Is the Supremum a convex function?

Is the Supremum a convex function?

The point-wise maximum or supremum of convex functions is convex (this is a consequence of the fact that the intersection of convex epi-graphs is a convex epi-graph). If f is convex in (x, y) and C is a convex set then infy∈C f(x, y) is convex in x.

Are affine functions convex?

Affine functions: f(x) = aT x + b (for any a ∈ Rn,b ∈ R). They are convex, but not strictly convex; they are also concave: ∀λ ∈ [0,1], f(λx + (1 − λ)y) = aT (λx + (1 − λ)y) + b = λaT x + (1 − λ)aT y + λb + (1 − λ)b = λf(x) + (1 − λ)f(y). In fact, affine functions are the only functions that are both convex and concave.

What is the relationship between convex functions and convex sets?

A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets.

What is the difference between maximum and Supremum?

In terms of sets, the maximum is the largest member of the set, while the supremum is the smallest upper bound of the set.

What is Pointwise Supremum?

Pointwise Supremum: If f(x,y) is convex in x for each y ∈ A, then. g(x) = sup. y∈A. f(x,y) is convex.

Is sum of convex functions convex?

If f(x) is convex, then g(x) = f(ax+b) is also convex for any constants a, b ∈ R. If f(x) and g(x) are convex, then their sum h(x) = f(x) + g(x) is convex.

Is a function convex or concave?

A function that has an increasing first derivative bends upwards and is known as a convex function. On the other hand, a function, that has a decreasing first derivative is known as a concave function and bends downwards.

What is the Supremum of a function?

The supremum (abbreviated sup; plural suprema) of a subset of a partially ordered set is the least element in that is greater than or equal to all elements of if such an element exists. Consequently, the supremum is also referred to as the least upper bound (or LUB).

How do you prove that a function is convex?

A function is convex iff its epigraph is convex. See herefor a definition of the epigraph. It is clear that the epigraph of $\\sup g_i$ is the intersection of the epigraphs of all the $g_i$. Now the intersection of convex sets is convex, which yields a more geometric proof of the statement above.

Is a function convex if its epigraph is convex?

A function is convex iff its epigraph is convex. See herefor a definition of the epigraph. It is clear that the epigraph of $\\sup g_i$ is the intersection of the epigraphs of all the $g_i$.

Is the intersection of convex sets always convex?

A function is convex iff its epigraph is convex. See here for a definition of the epigraph. It is clear that the epigraph of sup gi is the intersection of the epigraphs of all the gi . Now the intersection of convex sets is convex, which yields a more geometric proof of the statement above.