The equation entitled "lasso dual" at the top of slide 8 in https://www.cs.cmu.edu/~ggordon/10725-F12/slides/17-dual-corresp.pdf DOES have the same value as the original lasso problem, but strong duality.

But then they transform to an "equivalent" problem which will attain its optimum at the same point, but will have a different optimal value/ In the equivalent version, they have dropped both the 1/2 scale factor and the norm of y term, as well as flipping the sign of everything to transform from a maximization to a minimization problem... so of course it will have a different optimal value.

This whole slide is to illustrate the "Dual Subtleties" they describe on slide 6.

The equation entitled "lasso dual" at the top of slide 8 in https://www.cs.cmu.edu/~ggordon/10725-F12/slides/17-dual-corresp.pdf DOES have the same value as the original lasso problem, by strong duality.

But then they transform to an "equivalent" problem, which will attain its optimum at the same point, but will have a different optimal value. In the equivalent version, they have dropped both the 1/2 scale factor and the norm of y term, and they have also flipped the sign of everything to transform from a maximization to a minimization problem... so of course it will have a different optimal value.

This whole slide is to illustrate the "Dual Subtleties" they describe on slide 6.