Foreword
This paper records the convex optimization in several of the more basic concepts: convex set, affine set, convex hull, Cone, Cone.
Affine (affine sets)
Recall the definition of a straight line segment.
for
\[x_1 \not = x_2 \in R^n, \theta \in R\]
The straight line can be expressed as:
\ [Y = \ Theta x_1 + (l- \ Theta) x_2 \]
Similarly, for \ (\ Theta \) plus some limits can be derived segment defined:
\ [Y = \ Theta x_1 + (1- \ theta) x_2 , \ theta \ in R, \ theta \ in [0,1] \]
With the concept of the line, the set of affine defined below:
- Affine sets: a set of \ (C \) is an affine set, if \ (\ FORALL x_1, x_2 \ in C \) , the connection \ (x_1 \) and \ (x_2 \) of the straight line are set.
Mathematical language to describe the word is:
- Affine: Let \ (x_1, ..., x_k \ in C, \ theta_1, ..., \ theta_k \ in R & lt, \ theta_1 + ... + \ = theta_k. 1 \) , if the set \ (C \) is an affine set, if and only if \ (\ theta_1 x_1 + ... + \ theta_k x_k \ in C \)
We \ (\ theta_1 x_1 + ... + \ theta_k x_k \) is called an affine combination.
Associated with the subspace C
Here to talk about some of the properties of affine sets. According to the above definition, if \ (x_1, x_2 \ in C \) , \ (C \) is an affine set, \ (\ Theta x_1 + (l- \ Theta) x_2 \ in C \) , where \ ( \ Theta \ in R \) . Then ask, \ (\ Alpha x_1 + \ Beta x_2 \) belongs to \ (C \) it?
As shown below, when a straight line through the origin is not apparent in FIG \ (\ Alpha x_1 + \ Beta x_2 \ Not \ in C \) , when a straight line through the origin, there \ (\ alpha x_1 + \ beta x_2 \ in C \) .
We define: \ (V = C - V_0 = \ {the X--x_0 | the X-\ in C \}, \ FORALL x_0 \ in C \)
called \ (V \) with the \ (C \) related to the sub-space. We can be understood as \ (V \) is \ (C \) translation \ (v_0 \) a word space get.
Syndrome:
\ [\ FORALL V_1, V_2 \ in C, \ FORALL \ Alpha, \ Beta \ in R & lt \]
Since \ (V_1 + V_0 \ in C, V_2 + x_0 \ in C, x_0 \ in C \) , so : \
[\ Alpha (V_1 + x_0) + \ Beta (+ V_2 x_0) + (l- \ Alpha - \ Beta) x_0 \ in C \]
i.e.
\ [\ alpha v_1 + \ beta v_2 + x_0 \ in C \ ]
At this time, there is:
\ [\ + Alpha V_1 \ Beta V_2 \ in V \]
one example
- Solution set of linear equations are affine set
\ [C = \ {x | AX = b \}, A \ in R ^ {m \ times n}, b \ in R ^ m, x \ in R ^ n \ ]
And \ (C \) associated subspace \ (V = \ {x - x_0 | x \ in C \}, \ forall x_0 \ in C \) also happens matrix \ (A \) null space.
Affine package
For any set \ (C \) , whether constructed affine set minimum. If you can, this is called a minimum set of affine affine package .
Affine hull: \ (AFF \ C = \ {\ theta_1 x_1 + ... + \ theta_k x_k | \ FORALL x_1, ..., x_k \ in C, \ theta_1 + \ theta_2 + ... + \ = theta_k. 1 \ } \)
Affine Affine package is that it itself.
Convex sets Convex Set
- A convex set is set, i.e., when a line segment between any two points of any course (C \) \ within
- Mathematical expression is then: \ (\ FORALL x_1, x_2 \ in C, \ FORALL \ Theta, \ Theta \ in [0,1], \ Theta x_1 + (l- \ Theta) x_2 \ in C \)
\ (x_1, ..., x_k \ ) convex combination of
\ (x_1, ..., x_k \ ) convex combination is expressed as:
\[\theta_1 x_1 + \theta_2 x_2 + ... + \theta_k x_k \in C\]
\[\theta_1, ... ,\theta_k \in R, \theta_1+...+\theta_k = 1\]
\[ \theta_1,...,\theta_k \in [0,1]\]
- \ (C \) is a convex set is equivalent to any \ (C \) is a convex combination are \ (C \) inside.
Convex hull
For any set \ (C \ ^ in n-R & lt \) , which is referred to as a convex hull:
\ [Cov C = \ {\ theta_1 x_1 + ..., + \ theta_k x_k | \ FORALL X1, ..., x_k \ in C, \ forall \ theta_1 , ... \ theta_k \ in [0,1], \ theta_1 + ... + \ theta_k = 1 \} \]
The Convex Cone Cone Cone and convex cone
- \ (C \) is equivalent to the taper \ (\ FORALL X \ C in, \ Theta> 0 \) , there are \ (\ theta x \ in C \)
- \ (C \) is equivalent to a convex cone \ (\ FORALL x_1, x_2 \ in C, \ theta_1, \ theta_2 \ GEQ 0 \) , there are \ (\ theta_1 x_1 + \ theta_2 x_2 \ in C \)
A combination of convex cone
\[\theta_1 x_1 + ... + \theta_k x_k, \theta_1,...,\theta_k \geq 0\]
Convex Cone
\[x_1,...,x_k \in C, \{\theta_1 x_1 + ... + \theta_k x_k | x_1,...,x_k \in C, \theta_1,...,\theta_k \geq 0\}\]
to sum up
Affine combination
\ [\ forall \ theta_1, ... , \ theta_k, \ theta_1 + ... + \ theta_k = 1 \]Convex combination
\ [\ theta_1, ..., \ theta_k, \ theta_1 + ... + \ theta_k = 1, \ theta_1, ..., \ theta_k \ in [0,1] \]Convex cone assembly
\ [\ forall \ theta_1, ... \ theta_k, \ theta_1, ..., \ theta_k \ geq 0 \]- Any of a set of affine, it must be convex.
- Convex cone must also be convex.
- If the set only one element: \ (C = X} {\) , the collection is a set of affine. If this point is the origin, i.e. \ (X = 0 \) , then it is a convex cone.
Affine is the empty set, is also convex sets and cones.