# SLOPE

SLOPE assigns larger weights to larger coefficient magnitudes. Here $|\beta|_{(j)}$ denotes the $j$th largest absolute value. Its corners can encourage equal magnitudes, including coefficients with opposite signs. Supply exactly one finite, nonnegative, nonincreasing weight per coefficient; the weights already include the penalty strength. 

$$

  R(\beta) = \sum_{j=1}^p w_j|\beta|_{(j)},\qquad w_1
           \geq \cdots \geq w_p \geq 0.

$$

<a id="constraint-region"></a>

## Constraint region

![Feasible region for slope; the shaded set includes its boundary.](<../assets/368ca855b22083587228648a1aa4995922f33a0557d15d788818f9940c386d67.svg>) 

$w = (1,0.5)$: $\max(|\beta_1|,|\beta_2|) + 0.5\min(|\beta_1|,|\beta_2|) \leq 1$. See the [plot conventions](<./#read-the-constraint-plots>) for the fixed budget and coefficient scale. 

<a id="use-the-penalty"></a>

## Use the penalty

```rust
use shrinkage::{Penalty, ProximalPenalty, Slope};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let penalty = Slope::new(vec![0.5, 0.25]);
    let coefficients = [2.0, -1.0];
    println!("Penalty: {}", penalty.value(&coefficients)?);
    let mut shrunk = [0.0; 2];
    penalty.prox(&coefficients, 1.0, &mut shrunk)?;
    println!("Proximal result: {shrunk:?}");
    Ok(())
}
```

Equal weights recover lasso. Repeated weights and trailing zeros are valid; all-zero weights give no penalty. Use [proximal gradient](<../reference/solvers/proximal-gradient>) or the [hybrid solver](<../reference/solvers/hybrid>) with Gaussian or binomial datafits. The [SLOPE fitting guide](<../models/slope>) explains magnitude clusters and advanced controls. 

API: [`shrinkage::Slope`](<https://shrinkage.cc/packages/rust/reference/b160a8286206ec7afad932df7849dd0eb34050fc3d7a58465f4671287121ba7c>).