Constrained Least Square Optimization

In summary, the conversation is about finding the solution to a given equation with a constraint and using Matlab's inverse operator. The equation involves a column vector and a scalar, and the constraint is represented using double absolute value marks. The type of math being used is not specified, but the person asking for clarification is curious about the notation and the matrix norm being used in the constraint.
  • #1
ahmadnajeeb
1
0
Hi,
I want to know the solution of the following equation.
[tex]
a = argmin_{a}[\sum{||a^Tx_i - y_i||^2}+\alpha ||a||^2] \\
[/tex]
where [tex]x_i, y_i[/tex] are column vectors of dimensions m and n respectively where [tex]m>n[/tex]. [tex]\alpha[/tex] is a scalar and
[tex]Y = a^T X[/tex] where [tex]X=[x_1 x_2 ... x_k], Y = [y_1 y_2 ... y_k] [/tex]

I know that without this constraint [tex] \alpha ||a||^2 [/tex], its a simple least square optimization problem and I can solve it using Matlab's inverse operator. I want to use the same inverse operator but don't know how this constraint changes my original model.
 
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  • #2
Sorry, but I'm curious- What is this? I don't recognize the aT thing or the double absolute value marks. Could you tell me what type of math is this? Is it Calculus III?
 
  • #3
What matrix norm are you using for your constraint?
 

1. What is Constrained Least Square Optimization?

Constrained Least Square Optimization is a mathematical technique used to find the best possible solution for a problem while satisfying a set of constraints. It is often used in data analysis and machine learning to find the optimal values for a set of parameters.

2. How does Constrained Least Square Optimization differ from Unconstrained Least Square Optimization?

The main difference between Constrained and Unconstrained Least Square Optimization is the inclusion of constraints. In Unconstrained Least Square Optimization, there are no restrictions on the values that the parameters can take, while in Constrained Least Square Optimization, the parameters must satisfy certain constraints.

3. What are some examples of constraints in Constrained Least Square Optimization?

Constraints in Constrained Least Square Optimization can include conditions such as upper and lower bounds on the parameter values, linear or nonlinear inequalities, and equality constraints. These constraints can be used to incorporate prior knowledge or domain-specific information into the optimization process.

4. What are the advantages of using Constrained Least Square Optimization?

One of the main advantages of Constrained Least Square Optimization is that it allows for the incorporation of additional information or prior knowledge into the optimization process. This can lead to more accurate and interpretable results. It also ensures that the optimized solution satisfies the specified constraints, which may be necessary in certain applications.

5. What are some common algorithms used for Constrained Least Square Optimization?

There are several popular algorithms used for Constrained Least Square Optimization, including the interior-point method, active set method, and sequential quadratic programming. The choice of algorithm depends on the specific problem and constraints, and different algorithms may perform better for different types of constraints.

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