Exercise 26 ProblemSet4

I don't understand how did we continue to solve the question after defining \(y_{epsilon}\). And if I remember correctly during the exercise session this term was a bit different from what was written in the solution now. Could you please explain what actually is it and why we needed to introduce it and how we got the equation shown in the red circle?

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The \(\nabla f(x)\) is certainly a subgradient of f. Then this exercise is equivalent to show any subgradient g of f must be \(\nabla f(x)\) which means the difference between g and \(\nabla f(x)\) must be 0 in each coordinate. That is, we need to show that middle term in the red circle is 0 for each coordinate i. To prove this, we utilize the first inequality in the picture which holds true for all y and therefore holds true for our defined \(y_\epsilon\). Then we can show that the middle term in the red circle has to be 0 by taking epsilon to 0.

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