Article Directory
1 Convex set
1.1 Linear space
V is a non-empty set and F is a number field. If addition and number multiplication operations are defined on it, and the eight rules are satisfied, then the set V is called a linear space on the number field F.
1.2 Euclidean space
The inner product operation is defined on the n-dimensional linear space on the number field R, and the four rules are satisfied, then this n-dimensional linear space is called an n-dimensional Euclidean space.
1.3 Convex set
In Euclidean space, if for each pair of points in a set, each point on the straight line connecting the pair of points is also in the set, then the set is called a convex set. Such as the sphere is a convex set. Any hollow or indented set is not a convex set. For example, a five-pointed star is not a convex set and is called a concave set. In particular, in the vector space on the real number field R or the complex number field C, if the points on the line connecting any two points in the set S are all within S, then the set S is called a convex set.
1.4 Affine
1.4.1 Affine function
The affine function is a polynomial function with the highest degree of 1, and the general form is f(x)=Ax+b, A is an m×k order matrix, x is a k-dimensional column vector, b is an m-dimensional column vector, and f is from A mapping from k-dimensional vector space to m-dimensional vector space.
1.4.2 Linear functions
An affine function whose constant term is zero is called a linear function. The general form is f(x)=Ax.
1.4.3 Affine transformation
The mapping x→Ax+b from R k to R m is called affine transformation (when k≠m) or affine mapping (when k=m). If f is an affine transformation and S is a convex set, then f(S)={f(x)|x∈S} is a convex set. Conversely, if f is an affine transformation and f(S) is a convex set, then S is a convex set.
2 Convex function
2.1 Description
If the upper area of the image of the function is a convex set, then the function is a convex function. Pay attention to the top , because the concavity and convexity are the opposite for different viewing angles. For example, a bowl placed on the square is convex, and a bowl that is buckled upside down is concave.
2.2 Definition
If the domain dom f of the function f is a convex set, and satisfies f(x+(1-θ)y)<=θf(x)+(1-θ)f(y), where x,y∈dom f and 0 =<θ<=1.
2.3 Examples
(1) Exponential function: f(x)=e ax ;
(2) power function: F (X) X = A , x ∈ R + , or A ≧ 1 a≤0;
(3) Negative logarithmic function: f(x)=-lnx;
(4) Negative entropy function: f(x)=xlnx;
(5) Maximum value function: f(x)=max{x 1 ,x 2 ,...,x N }.
3 Convex optimization
Convex optimization problems are generally minimization problems. By adding a minus sign to the objective function, the maximization problem can be turned into a minimization problem.
The general form of the constrained optimization problem is as follows:
Suppose the domain of f(x) is domain f, and the domain of m i (x) is domain m i , then the feasible domain is D=domain f ∩ domain m i , that is, both The intersection.
If the objective function f(x) is a convex function and the feasible region is a convex set (that is, mi (x) in the inequality constraint is a convex function, and n j (x) in the equality constraint is an affine function), then this is called This kind of constrained optimization problem is convex optimization problem. The local optimal solution of the convex optimization problem is called the global optimal solution.
4 Convex quadratic programming
r and a i are n-dimensional real vectors, b i is a real number, i=1,2...,L,L+1,...L+M. For the following constrained optimization problem:
if the objective function f(x) is a quadratic function, G is a symmetric matrix, and the inequality constraints are all affine functions, then the above constrained optimization problem is called a quadratic programming problem, namely QP problem.
If the matrix G in f(x) is a positive semi-definite matrix, then the above QP problem is called a convex quadratic programming problem; if G is a positive definite matrix, it is a strictly convex quadratic programming problem.
If G is a positive semi-definite matrix, the feasible region is not empty, and the objective function f(x) has a lower bound in the feasible region, then the convex quadratic programming problem has a global minimum. Furthermore, if G is a positive definite matrix, the feasible region is not empty, and the objective function f(x) has a lower bound in the feasible region, then the strictly convex quadratic programming problem has a unique global minimum.
5 References
1. Baidu Encyclopedia;
2. Convex set, convex function, convex optimization and convex quadratic programming .
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