## Working with MAD Portfolio Constraints Using Defaults

The final element for a complete specification of a portfolio optimization problem is the set of feasible portfolios, which is called a portfolio set. A portfolio set $X\subset {R}^{n}$ is specified by construction as the intersection of sets formed by a collection of constraints on portfolio weights. A portfolio set necessarily and sufficiently must be a nonempty, closed, and bounded set.

When setting up your portfolio set, ensure that the portfolio set satisfies these conditions. The most basic or “default” portfolio set requires portfolio weights to be nonnegative (using the lower-bound constraint) and to sum to 1 (using the budget constraint). For information on the workflow when using PortfolioMAD objects, see PortfolioMAD Object Workflow.

### Setting Default Constraints for Portfolio Weights Using PortfolioMAD Object

The “default” MAD portfolio problem has two constraints on portfolio weights:

• Portfolio weights must be nonnegative.

• Portfolio weights must sum to 1.

Implicitly, these constraints imply that portfolio weights are no greater than 1, although this is a superfluous constraint to impose on the problem.

#### Setting Default Constraints Using the PortfolioMAD Function

Given a portfolio optimization problem with NumAssets = 20 assets, use the PortfolioMAD object to set up a default problem and explicitly set bounds and budget constraints:

p = PortfolioMAD('NumAssets', 20, 'LowerBound', 0, 'Budget', 1);
disp(p)

SellCost: []
RiskFreeRate: []
Turnover: []
SellTurnover: []
NumScenarios: []
Name: []
NumAssets: 20
AssetList: []
InitPort: []
AInequality: []
bInequality: []
AEquality: []
bEquality: []
LowerBound: [20×1 double]
UpperBound: []
LowerBudget: 1
UpperBudget: 1
GroupMatrix: []
LowerGroup: []
UpperGroup: []
GroupA: []
GroupB: []
LowerRatio: []
UpperRatio: []
MinNumAssets: []
MaxNumAssets: []
BoundType: []

#### Setting Default Constraints Using the setDefaultConstraints Function

An alternative approach is to use the setDefaultConstraints function. If the number of assets is already known in a PortfolioMAD object, use setDefaultConstraints with no arguments to set up the necessary bound and budget constraints. Suppose that you have 20 assets to set up the portfolio set for a default problem:

p = setDefaultConstraints(p);
disp(p)

SellCost: []
RiskFreeRate: []
Turnover: []
SellTurnover: []
NumScenarios: []
Name: []
NumAssets: 20
AssetList: []
InitPort: []
AInequality: []
bInequality: []
AEquality: []
bEquality: []
LowerBound: [20×1 double]
UpperBound: []
LowerBudget: 1
UpperBudget: 1
GroupMatrix: []
LowerGroup: []
UpperGroup: []
GroupA: []
GroupB: []
LowerRatio: []
UpperRatio: []
MinNumAssets: []
MaxNumAssets: []
BoundType: [20×1 categorical]

If the number of assets is unknown, setDefaultConstraints accepts NumAssets as an optional argument to form a portfolio set for a default problem. Suppose that you have 20 assets:

p = setDefaultConstraints(p, 20);
disp(p)

SellCost: []
RiskFreeRate: []
Turnover: []
SellTurnover: []
NumScenarios: []
Name: []
NumAssets: 20
AssetList: []
InitPort: []
AInequality: []
bInequality: []
AEquality: []
bEquality: []
LowerBound: [20×1 double]
UpperBound: []
LowerBudget: 1
UpperBudget: 1
GroupMatrix: []
LowerGroup: []
UpperGroup: []
GroupA: []
GroupB: []
LowerRatio: []
UpperRatio: []
MinNumAssets: []
MaxNumAssets: []
BoundType: [20×1 categorical]