# create.solve_asdp: Relaxed optimization for fixed-X and Gaussian knockoffs In knockoff: The Knockoff Filter for Controlled Variable Selection

## Description

This function solves the optimization problem needed to create fixed-X and Gaussian SDP knockoffs on a block-diagonal approximation of the covariance matrix. This will be less powerful than create.solve_sdp, but more computationally efficient.

## Usage

 1 2 3 4 5 6 7 create.solve_asdp( Sigma, max.size = 500, gaptol = 1e-06, maxit = 1000, verbose = FALSE ) 

## Arguments

 Sigma positive-definite p-by-p covariance matrix. max.size size of the largest block in the block-diagonal approximation of Sigma (default: 500). See Details. gaptol tolerance for duality gap as a fraction of the value of the objective functions (default: 1e-6). maxit the maximum number of iterations for the solver (default: 1000). verbose whether to display progress (default: FALSE).

## Details

Solves the following two-step semidefinite program:

(step 1)

\mathrm{maximize} \; \mathrm{sum}(s) \quad \mathrm{subject} \; \mathrm{to:} \; 0 ≤q s ≤q 1, \; 2 Σ_{\mathrm{approx}} - \mathrm{diag}(s) ≥q 0

(step 2)

\mathrm{maximize} \; γ \quad \mathrm{subject} \; \mathrm{to:} \; \mathrm{diag}(γ s) ≤q 2 Σ

Each smaller SDP is solved using the interior-point method implemented in dsdp.

The parameter max.size controls the size of the largest semidefinite program that needs to be solved. A larger value of max.size will increase the computation cost, while yielding a solution closer to that of the original semidefinite program.

If the matrix Sigma supplied by the user is a non-scaled covariance matrix (i.e. its diagonal entries are not all equal to 1), then the appropriate scaling is applied before solving the SDP defined above. The result is then scaled back before being returned, as to match the original scaling of the covariance matrix supplied by the user.

## Value

The solution s to the semidefinite program defined above.

Other optimization: create.solve_equi(), create.solve_sdp()