How to derive the KKT condition
How to derive the KKT condition, in view of this problem, this article introduces the corresponding analysis and solution in detail, hoping to help more partners who want to solve this problem to find a more simple and feasible method.
-derive the KKT condition-
It is when solving the inequality constraints of convex optimization that the KKT condition is derived, which is derived step by step by graphics and symbols.
With solving problem
The minimum value of f (x) is 0, as shown in the following figure. At the same time, two conditions for the consistency of constrained minimum and unconstrained minimum are given (the second condition is positive definite quadratic form).
In the above case, we call this constraint not active, as shown in the following figure:
In order for the above constraints to take effect, redefine the objective function:
That is, it is equivalent to the movement of the center of the circle:
It is easy to see that if there is no constraint, the minimum value of the objective function is obtained at the center of the circle, but the constraint cannot be satisfied here:
Therefore, it is intuitive that the minimum value of the objective function is obtained just outside the boundary of the constrained region, as shown in the following figure:
To describe it with a mathematical formula, that is, to satisfy:
It is based on this equation that the famous Lagrange multiplier method is defined:
Summarize the above two situations (whether the unconstrained minimum acquisition position is in the feasible region):
Combine the above two kinds, pursue simplicity, and summarize the constraints, that is: KKT condition.
Specifically:
1)
Merge into KKT conditions:
2)
It is easier to observe
3)
Merge into KKT condition 4:
4)
Merge into condition 3:
The above equation is the reason why only two points really play a role in classification in support vector machines.
5) positive semidefinite quadratic constraint, which is equivalent to convex optimization.
The answer to the question on how to derive the KKT condition is shared here. I hope the above content can be of some help to you. If you still have a lot of doubts to be solved, you can follow the industry information channel to learn more about it.
