9. Obstacle penalty function method --- interior point, exterior point penalty function

9. Obstacle Penalty Function Method - Penalty Function of Inner Point and Outer Point

The basic idea of ​​the penalty function method is to transform the constrained problem into an unconstrained optimization problem with the help of a penalty function, and then obtain the solution of the original constrained problem by solving a series of unconstrained optimization problems. In the iterative process,
the penalty function method forces the iteration points to approach the feasible region by imposing penalties on the infeasible points. Once the iteration point becomes a feasible point, this feasible point is the optimal solution to the original problem

Penalty functions can be divided into exterior point method and interior point method:

• The outer point method is more general and can solve the situation where the constraint is a mixture of equality and inequality. The outer point method has no requirements for the initial point, and can arbitrarily select any point in the domain. The outer point method truly transforms the problem into an unconstrained problem;
• The initial point of the interior point method must be a point in the feasible region. When the constraints are more complex, it is difficult to choose the initial point of the interior point method, and the interior point method can only solve the situation where the constraints are inequalities.

Equality constraint penalty function:

According to the characteristics of the constraint conditions, the penalty penalty item is constructed, and then added to the objective function to convert it into an unconstrained problem

For the following constraint problems:

Construct an auxiliary function and find the minimum value of the auxiliary function:

σ is a penalty factor, which takes a large positive number, F(x,σ) is a penalty function, and σP(x) is a penalty item

Conditions to be met for penalty items

1. P(x) = 0 if and only if x is a feasible point of the problem—the feasible region/feasible point is the point or point set that satisfies the g(x) constraint
2. P(x) is a continuous function—convenient for derivation and extremum
3. For all x ∈ Rn, P(x) ≥ 0

Principle: When x is in the feasible region, there is no penalty, and the general optimization method seeks the minimum value; when x is not in the feasible region, a penalty is added to make F(x,σ) far from the optimal solution; that is, σ→∞, if the original problem has a solution, it must be in the feasible region, and minf(x)==minF(x,σ)

1. Outer point penalty function method

外点法是从可行区外慢慢接近边界，在接近的过程中计算每个阶段极值点，一旦到达可行区范围内，该极值点即为原带约束非线性规划的极值点
1）等式：
目标函数： min f(x)
约束集合： s.t. hi(x)=0 i=1,2,3,…m
增广为外点罚函数：

2）不等式：
目标函数： min f(x)
约束集合： s.t. gi(x)>=0 i=1,2,3,…m
增广为外点罚函数：

3)混合

!](https://img-blog.csdnimg.cn/034aec5bb6cc4c87a83bbe29aae22f23.png?x-oss-process=image/watermark,type_d3F5LXplbmhlaQ,shadow_50,text_Q1NETiBAanViYXJ5,size_20,color_FFFFFF,t_70,g_se,x_16)

2. Interior point penalty function

The interior point penalty function method is a kind of method that maintains strict feasibility. It always starts from the feasible point and keeps searching inside the feasible region. Therefore, this type of method is only suitable for nonlinear optimization problems with only inequality constraints

• The basic idea of ​​the interior point penalty function method is to introduce an obstacle term about constraints on the objective function. When the iteration point approaches the boundary of the feasible region from the interior of the feasible region, the obstacle term will tend to infinity to force the iteration point to return to the interior of the feasible region, thereby maintaining the strict feasibility of the iteration point. Then, the solution of the constrained problem is transformed into solving a series of easy sub-problems, so as to obtain the optimal approximation solution of the original problem. The method is also called the interior point barrier function method.

Objective function: min f(x)
Constraint set: st gi(x)>=0 i=1,2,3,…m
Augmented as outlier penalty function:

µ is a very small number. Then when x tends to the boundary, F(x, µ) → +∞, otherwise, when µ is small, the value of F(x, µ) is close to f(x).

The barrier function B(x) generally needs to satisfy:
(1) Continuous in the feasible region
(2) When x tends to the boundary of g(x)=0, B(x) → ∞.

Two commonly used barrier functions:
reciprocal barrier function,

logarithmic barrier function