Which functions are convex?

Which functions are convex?

A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.

How do you know if a function is convex?

A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval. If a function is differentiable and convex then it is also continuously differentiable.

How do you determine if a function is convex or concave?

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).

What is a convex loss function?

Convex Loss Functions. All of these are convex upper bounds on 0-1 loss. Hinge loss: L(y, y) = max{0, 1 − yy} Exponential loss: L(y, y) = exp(−yy) Logistic loss: L(y, y) = log2(1 + exp(−yy))

Is logistic loss convex?

The method most commonly used for logistic regression is gradient descent. Gradient descent requires convex cost functions. Mean Squared Error, commonly used for linear regression models, isn't convex for logistic regression.

What does it mean when a cost function is non convex?

The cost function of a neural network is in general neither convex nor concave. This means that the matrix of all second partial derivatives (the Hessian) is neither positive semidefinite, nor negative semidefinite. Since the second derivative is a matrix, it's possible that it's neither one or the other.

What is convex cost function?

A convex function: given any two points on the curve there will be no intersection with any other points, for non convex function there will be at least one intersection. In terms of cost function with a convex type you are always guaranteed to have a global minimum, whilst for a non convex only local minima.

What is convex set with example?

Equivalently, a convex set or a convex region is a subset that intersect every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex.

What is convex objective function?

A convex optimization problem is a problem where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing. ... With a convex objective and a convex feasible region, there can be only one optimal solution, which is globally optimal.

What convex means?

Definition of Convex A convex shape is the opposite of a concave shape. ... Just like concave, convex can be used as a noun for a surface or line that curves outward, and it also has a use in geometry, where it describes a polygon with interior angles less than or equal to 180°.

Why convex optimization is important?

6 Answers. Machine learning algorithms use optimization all the time. ... Nonetheless, as mentioned in other answers, convex optimization is faster, simpler and less computationally intensive, so it is often easier to "convexify" a problem (make it convex optimization friendly), then use non-convex optimization.

Is e x convex?

The function ex is differentiable, and its second derivative is ex > 0, so that it is (strictly) convex.

What is another word for Convex?

Convex Synonyms - WordHippo Thesaurus....What is another word for convex?

What is a strongly convex function?

Intuitively speaking, strong convexity means that there exists a quadratic lower bound on the growth of the function. This directly implies that a strong convex function is strictly convex since the quadratic lower bound growth is of course strictly grater than the linear growth.

Is log a convex function?

, the composition of the logarithm with f, is itself a convex function.

Is log a concave function?

A function f(x) is log concave if log( f(x) ) is concave. The basic properties of convex functions are obvious. It's easy to show that the sum of two convex functions is convex, the maximum of two convex functions is convex, etc.

Is entropy a convex function?


Is xy a convex function?

And since convexity has iff relation with H being positive semi-definite (i.e., all eigenvalues greater than or equal to zero) , we can say that the xy is neither convex nor concave. Consider the values of f at (1,3),(2,2),(3,1) and also at (1,1),(2,2),(3,3).

Is a triangle convex?

A convex polygon is defined as a polygon with all its interior angles less than 180°. ... Note that a triangle (3-gon) is always convex. A convex polygon is the opposite of a concave polygon. See Concave Polygon.

Is a 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.

What is convex in science?

1. curving or bulging outwards. 2. (General Physics) physics having one or two surfaces curved or ground in the shape of a section of the exterior of a sphere, paraboloid, ellipsoid, etc: a convex lens.

What are convex figures?

A planar polygon is convex if it contains all the line segments connecting any pair of its points. Thus, for example, a regular pentagon is convex (left figure), while an indented pentagon is not (right figure). A planar polygon that is not convex is said to be a concave polygon.

What is convex and concave mirror?

What are convex and concave mirrors? If the inner side of the spherical mirror is reflecting, it is called a concave mirror. If the outer side of the spherical mirror is reflecting, it is called a convex mirror. Image. Concave mirrors can form inverted and real images and also virtual and erect images.

What is difference between concave and convex lens?

A concave lens is thinner in the middle and thicker at the edges. A convex lens is thicker in the middle and thinner at the edges. Used in the camera, focus sunlight, overhead projector, projector microscope, simple telescope, magnifying glasses, etc. It is also used for the correction of the problem in long sight.

Which lens is concave?

A concave lens is a lens that possesses at least one surface that curves inwards. It is a diverging lens, meaning that it spreads out light rays that have been refracted through it. A concave lens is thinner at its centre than at its edges, and is used to correct short-sightedness (myopia).

Where is convex lens used?

Convex lenses are used in eyeglasses for correcting farsightedness, where the distance between the eye's lens and retina is too short, as a result of which the focal point lies behind the retina. Eyeglasses with convex lenses increase refraction, and accordingly reduce the focal length.

What are the examples of convex lens?

Some examples of objects with convex lenses in include:

  • Binoculars and telescopes.
  • Cameras.
  • Eye glasses.
  • Flashlights.
  • Lasers ( CD, DVD players)

What devices use convex lenses?

Uses Of Convex Lens

  • Magnifying glasses.
  • Eyeglasses.
  • Cameras.
  • Microscopes.

Which is convex lens?

A convex lens is also known as a converging lens. A converging lens is a lens that converges rays of light that are traveling parallel to its principal axis. They can be identified by their shape which is relatively thick across the middle and thin at the upper and lower edges.

What is convex lens formula?

1. What is the Lens Formula for Convex Lens? Ans. According to the convex lens equation, 1/f = 1/v + 1/u. It relates the focal length of a lens with the distance of an object placed in front of it and the image formed of that object.