# Sorted L1 solver

### Description

Solves the sorted L1 penalized regression problem: given a matrix *A*,
a vector *b*, and a decreasing vector *λ*, find the vector
*x* minimizing

*\frac{1}{2}\Vert Ax - b \Vert_2^2 +
∑_{i=1}^p λ_i |x|_{(i)}.*

### Usage

1 2 3 | ```
SLOPE_solver(A, b, lambda, initial = NULL, prox = prox_sorted_L1,
max_iter = 10000, grad_iter = 20, opt_iter = 1, tol_infeas = 1e-06,
tol_rel_gap = 1e-06)
``` |

### Arguments

`A` |
an |

`b` |
vector of length |

`lambda` |
vector of length |

`initial` |
initial guess for |

`prox` |
function that computes the sorted L1 prox |

`max_iter` |
maximum number of iterations in the gradient descent |

`grad_iter` |
number of iterations between gradient updates |

`opt_iter` |
number of iterations between checks for optimality |

`tol_infeas` |
tolerance for infeasibility |

`tol_rel_gap` |
tolerance for relative gap between primal and dual problems |

### Details

This optimization problem is convex and is solved using an accelerated proximal gradient descent method.

### Value

An object of class `SLOPE_solver.result`

. This object is a list
containing at least the following components:

`x` |
solution vector |

`optimal` |
logical: whether the solution is optimal |

`iter` |
number of iterations |