# Penalties

A penalty assigns a cost to optimization-scale coefficients. Intercepts are excluded, and strengths are already included in every penalty's value. All built-in strengths must be finite and nonnegative. 

| Penalty | Encourages |
| --- | --- |
| [L1](<l1>) | Sparse individual coefficients |
| [Ridge](<ridge>) | Small coefficients through a squared L2 cost |
| [Elastic net](<elastic-net>) | Sparsity together with quadratic shrinkage |
| [SLOPE](<slope>) | Sparsity and equal coefficient magnitudes |
| [Group LASSO](<group-lasso>) | Sparse, explicitly defined groups |
| [Clustered lasso](<clustered-lasso>) | Sparsity and equal signed effects |
| [Chain fused lasso](<chain-fused-lasso>) | Sparsity and adjacent equal effects |

Use [Scaled](<scaled>) to multiply a complete penalty, [Masked](<masked>) to exclude supported coordinates, and [AxisPenalty](<axis>) to apply a vector penalty along coefficient-matrix axes. 

<a id="read-the-constraint-plots"></a>

## Read the constraint plots

Each page shows the feasible region for a two-coordinate constrained problem, 

$$

  \min_{\beta_0,\beta} F(\beta_0,\beta) \quad\text{subject to}\quad R(\beta)
  \leq t.

$$

The plots fix $t = 1$ and shade the feasible region, including its boundary. Axes have equal scales and represent optimization coefficients, not reported original-scale coefficients. Captions specify every strength, weight, and group. The intercept does not appear in these plots. 

For a penalized solution, choosing $t = R(\widehat\beta)$ makes that solution optimal for the corresponding constrained problem. The budget $t$ is not the penalty multiplier: the appropriate correspondence depends on the data and the solution. Our fixed budget illustrates geometry rather than fitting a dataset. Zero strengths and unpenalized directions can give unbounded regions; arrows indicate continuation beyond the plotted window. 

<a id="compose-supported-capabilities"></a>

## Compose supported capabilities

[`shrinkage::Penalty`](<https://shrinkage.cc/packages/rust/reference/a4f4243001e8d9121fca7a92bcb924b17aa5375aba23987fae95b2a05faf5245>) evaluates the complete cost. A compatible [`shrinkage::ProximalPenalty`](<https://shrinkage.cc/packages/rust/reference/45434a0394c5d718086e7e777f7db8271e2fb2b568e2519720168e5ea2877ed0>) also minimizes 

$$

  \operatorname{prox}_{sR}(v)
  = \mathop{\arg\min}_{z}\left\{ \frac{\|z - v\|_2^2}{2s} + R(z) \right\}, \qquad s
                              > 0.

$$

All seven base penalties provide this exact complete proximal map and work with [proximal gradient](<../reference/solvers/proximal-gradient>). A sum does not acquire an exact map simply because its terms have individual maps. Other solvers need additional capabilities; see the [compatibility table](<../reference/solvers/>).