Projected Gradient Descent: Minimize (1/2)(x_1)^2 + (1/2)(x_2)^2, sub. to 2-x_1-x_2=0

[HD5450]

Part (a) and part (b) is pretty easy but for part (c) I don't know how you would start the problem, I tried to look online for examples but all I found are general equations.

Any help would be appreciated.

Edit: The projected gradient descent is defined as \(\displaystyle x_{k+1} = \prod_X (x_k - \tau_k \nabla f(x_k)) \) where \(\displaystyle \prod_X(x)\) is orthogonal projection of \(\displaystyle x\)on \(\displaystyle X\) and \(\displaystyle \tau_k\) is the step size. I have also attempted to run the first iteration but I am stuck to trying to do the projection. I don't know how to do \(\displaystyle \prod_X((1 -1)^T)\)

 

[Deleted member 4993]

View attachment 10658

Part (a) and part (b) is pretty easy but for part (c) I don't know how you would start the problem, I tried to look online for examples but all I found are general equations.

Any help would be appreciated.

Can you please define "gradient projection" method for us?

 

[HD5450]

Sorry I forgot to include the formula

Can you please define "gradient projection" method for us?

The formula that I have for this is \(\displaystyle x_{k+1} = \prod_X(x_k - \tau_k \nabla f(x_k)\) where \(\displaystyle \prod_X(X)\) is the orthangonal projection and \(\displaystyle \tau_k\) is the step size.

 

[HD5450]

Can you please define "gradient projection" method for us?

Gradient projection is defined as \(\displaystyle x_{k+1} = \prod_X (x_k - \tau_k \nabla f(x_k)) \)