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STAT 6020 Fall 2021 HW 2


1. (1 pt) Show that one solution to the following problem (from Section 3 page 17)

where  is the right singular vector corresponding to the largest singular value of matrix . [Hint: consider the connection between the  of and the eigen decomposition of .]


2. (1 pt) Show that any can be equivalently formulated as the standard form


3. (2 pts) Given the problem

Write out the dual function and the corresponding dual problem.


4. (2 pts) In the lecture, we have shown that the standard problem

has the dual function

Write out the corresponding dual problem for the LP and then derive the corresponding dual problem of the dual problem.


5. (4 pt) is a free and publicly available solver for convex programing. It is widely applicable and very easy to use. As we will see later, it is almost never the fastest software you can use for solving convex problem, and it is a great tool if you just want to try some ideas on your optimization formulation or if your problem size is small and the timing is not a concern. Check http://cvxr.com/cvx/ for its information. It is based on Matlab, but it also has available version in Python and R . You can use any version of them as you like. Simply Google it to find the link. They provide very detailed documentations and examples. So learn how to use it by yourself. In this problem, you will be asked to use as the basic tool to solve convex optimization problem.

Use the prostate data from the Elements of Statistical Learning by Hastie, Tibshirani and Friedman , downloadable from (https://web.stanford.edu/ hastie/ElemStatLearn/). Use lpsa as the response and lcavol, lweight, age, lbph, svi, lcp, gleason, pgg45 as the predictors. Standardize all the predictors before model fitting and do not include an intercept. Fit a linear regression model based on the Danzig selector (Section 3 Page 24). Use to solve the problem for a sequence of values. As you increase , you will see that fewer predictors are selected in the final model (with nonzero coeffi-cients). For your sequence of values, plot the optimal objective agains the number of nonzeros in . Also submit your code for the part (the optimization formulation). Make sure your range covers the situation with all predictors in the model and the situation when no variables are selected at all.