## Working with Group Constraints Using PortfolioCVaR Object

Group constraints are optional linear constraints that group assets together and enforce bounds on the group weights (see Group Constraints). Although the constraints are implemented as general constraints, the usual convention is to form a group matrix that identifies membership of each asset within a specific group with Boolean indicators (either `true` or `false` or with `1` or `0`) for each element in the group matrix. Group constraints have properties `GroupMatrix` for the group membership matrix,``` LowerGroup``` for the lower-bound constraint on groups, and `UpperGroup` for the upper-bound constraint on groups.

### Setting Group Constraints Using the `PortfolioCVaR` Function

The properties for group constraints are set through the `PortfolioCVaR` object. Suppose that you have a portfolio of five assets and want to ensure that the first three assets constitute no more than 30% of your portfolio, then you can set group constraints:

```G = [ 1 1 1 0 0 ]; p = PortfolioCVaR('GroupMatrix', G, 'UpperGroup', 0.3); disp(p.NumAssets) disp(p.GroupMatrix) disp(p.UpperGroup)```
```5 1 1 1 0 0 0.3000```

The group matrix `G` can also be a logical matrix so that the following code achieves the same result.

```G = [ true true true false false ]; p = PortfolioCVaR('GroupMatrix', G, 'UpperGroup', 0.3); disp(p.NumAssets) disp(p.GroupMatrix) disp(p.UpperGroup)```
```5 1 1 1 0 0 0.3000```

### Setting Group Constraints Using the `setGroups` and `addGroups` Functions

You can also set the properties for group constraints using `setGroups`. Suppose that you have a portfolio of five assets and want to ensure that the first three assets constitute no more than 30% of your portfolio. Given a `PortfolioCVaR` object `p`, use `setGroups` to set the group constraints:

```G = [ true true true false false ]; p = PortfolioCVaR; p = setGroups(p, G, [], 0.3); disp(p.NumAssets) disp(p.GroupMatrix) disp(p.UpperGroup)```
```5 1 1 1 0 0 0.3000```

In this example, you would set the `LowerGroup` property to be empty (`[]`).

Suppose that you want to add another group constraint to make odd-numbered assets constitute at least 20% of your portfolio. Set up an augmented group matrix and introduce infinite bounds for unconstrained group bounds or use the `addGroups` function to build up group constraints. For this example, create another group matrix for the second group constraint:

```p = PortfolioCVaR; G = [ true true true false false ]; % group matrix for first group constraint p = setGroups(p, G, [], 0.3); G = [ true false true false true ]; % group matrix for second group constraint p = addGroups(p, G, 0.2); disp(p.NumAssets) disp(p.GroupMatrix) disp(p.LowerGroup) disp(p.UpperGroup)```
``` 5 1 1 1 0 0 1 0 1 0 1 -Inf 0.2000 0.3000 Inf```
`addGroups` determines which bounds are unbounded so you only need to focus on the constraints that you want to set.

The `PortfolioCVaR` object, `setGroups`, and `addGroups` implement scalar expansion on either the `LowerGroup` or `UpperGroup` properties based on the dimension of the group matrix in the property `GroupMatrix`. Suppose that you have a universe of 30 assets with 6 asset classes such that assets 1–5, assets 6–12, assets 13–18, assets 19–22, assets 23–27, and assets 28–30 constitute each of your asset classes and you want each asset class to fall from 0% to 25% of your portfolio. Let the following group matrix define your groups and scalar expansion define the common bounds on each group:

```p = PortfolioCVaR; G = blkdiag(true(1,5), true(1,7), true(1,6), true(1,4), true(1,5), true(1,3)); p = setGroups(p, G, 0, 0.25); disp(p.NumAssets) disp(p.GroupMatrix) disp(p.LowerGroup) disp(p.UpperGroup)```
```30 Columns 1 through 16 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 17 through 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500```