Convex Optimization Notes
Convex Optimization
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2.1.1 Line
Suppose
are two points in . Points of the formwhere θ ∈ R, form the line passing through
and . -
2.1.2 Affine sets
(It is hyperplane (may not passing the origin))
A set
is affine if the line through any two distinct points in lies inWe refer to a point of the form
, as an affine combination of the pointsIf
is an affine set and , then the setis a subspace
The set of all affine combinations of points in some set
n is called the affine hull of , and denoted :The affine hull is the smallest affine set that contains
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2.1.3 Affine dimension and relative interior
We define the affine dimension of a set
as the dimension of its affine hullWe define the relative interior of the set
, denoted as its interior relative to :where
, the ball of radius and center in the normWe can then define the relative boundary of a set
as , where is the closure of C. -
2.1.4 Convex sets
A set
is convex iffRoughly speaking, a set is convex if every point in the set can be seen by every other point, along an unobstructed straight path between them, where unobstructed means lying in the set. Every affine set is also convex, since it contains the entire line between any two distinct points in it, and therefore also the line segment between the points.
We call a point of the form
, where and , a convex combination of the points . As with affine sets, it can be shown that a set is convex if and only if it contain contains every convex combination of its points. A convex combination of points can be thought of as a mixture or weighted average of the points, with the fraction of in the mixture.The convex hull of a set
, denoted , is the set of all convex combinations of points in :As the name suggests, the convex hull
is always convex. It is the smallest convex set that contains -
2.1.5 Cones
A set
is called a cone, or nonnegative homogeneous, if for every and , we have . A set is a convex cone if it is convex and a cone, which means that for any and , we haveA point of the form
with is called a conic combination (or a nonnegative linear combination) of . If are in a convex cone , then every conic combination of is in . Conversely, a set is a convex cone if and only if it contains all conic combinations of its elements.The conic hull of a set
is the set of all conic combinations of points in , i.e.,which is also the smallest convex cone that contains
.
Convex functions
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3.1 Basic properties and examples
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3.1.1 Definition
A function
is convex if is a convex set andWe say
is concave if is convex, and strictly concave if is strictly convex.A function is convex if and only if it is convex when restricted to any line that intersects its domain. In other words
is convex if and only if for all and all , the function is convex (on its domain, ). This property is very useful, since it allows us to check whether a function is convex by restricting it to a line. -
3.1.2 Extended-value extensions
It is often convenient to extend a convex function to all of
by defining its value to be outside its domain.If f is convex we define its extended-value extension
by -
3.1.3 First-order conditions
Suppose
is differentiable (i.e., its gradient exists at each point in , which is open). Then is convex if and only if is convex and holds for allStrict convexity can also be characterized by a first-order condition:
is strictly convex if and only if is convex and for , we have -
3.1.4 Second-order conditions
We now assume that
is twice differentiable, that is, its Hessian or second derivative exists at each point in , which is open. Then is convex if and only if is convex and its Hessian is positive semidefinite: for allStrict convexity can be partially characterized by second-order conditions. If
for all , then is strictly convex. The converse, however, is not true: for example, the function given by is strictly convex but has zero second derivative at .Remark 3.1 The separate requirement that
be convex cannot be dropped from the first- or second-order characterizations of convexity and concavity. For example, the function , with , satisfies for all , but is not a convex function. -
3.1.6 Sublevel sets
The
-sublevel set of a function is defined as . -
3.1.7 Epigraph
The epigraph of a function
is defined as -
3.1.8 Jensen’s inequality and extensions
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3.1.9 Inequalities
