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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). The most general portfolio set handled by
the portfolio optimization tools can have any of these constraints:

Linear inequality constraints

Linear equality constraints

`'Simple'`

Bound constraints`'Conditional'`

Bond constraintsBudget constraints

Group constraints

Group ratio constraints

Average turnover constraints

One-way turnover constraints

Cardinality constraints

*Linear inequality constraints* are general
linear constraints that model relationships among portfolio weights
that satisfy a system of inequalities. Linear inequality constraints
take the form

$${A}_{I}x\le {b}_{I}$$

where:

*x* is the portfolio (*n* vector).

*A _{I}* is the linear inequality
constraint matrix (

*b _{I}* is the linear inequality
constraint vector (

*n* is the number of assets in the universe
and *n _{I}* is the number of
constraints.

`PortfolioMAD`

object properties to specify linear inequality constraints are:

`AInequality`

for*A*_{I}`bInequality`

for*b*_{I}`NumAssets`

for*n*

The default is to ignore these constraints.

*Linear equality constraints* are general
linear constraints that model relationships among portfolio weights
that satisfy a system of equalities. Linear equality constraints
take the form

$${A}_{E}x={b}_{E}$$

where:

*x* is the portfolio (*n* vector).

*A _{E}* is the linear equality
constraint matrix (

*b _{E}* is the linear equality
constraint vector (

*n* is the number of assets in the universe
and *n _{E}* is the number of
constraints.

`PortfolioMAD`

object properties to specify linear equality constraints are:

`AEquality`

for*A*_{E}`bEquality`

for*b*_{E}`NumAssets`

for*n*

The default is to ignore these constraints.

`'Simple'`

*Bound constraints* are specialized linear constraints that
confine portfolio weights to fall either above or below specific bounds. Since every
portfolio set must be bounded, it is often a good practice, albeit not necessary, to
set explicit bounds for the portfolio problem. To obtain explicit bounds for a given
portfolio set, use the `estimateBounds`

function. Bound
constraints take the form

$${l}_{B}\le x\le {u}_{B}$$

where:

*x* is the portfolio (*n* vector).

*l _{B}* is the lower-bound
constraint (

*u _{B}* is the upper-bound
constraint (

*n* is the number of assets in the universe.

`PortfolioMAD`

object properties to specify bound constraints are:

`LowerBound`

for*l*_{B}`UpperBound`

for*u*_{B}`NumAssets`

for*n*

The default is to ignore these constraints.

The default portfolio optimization problem (see Default Portfolio Problem) has *l _{B}* =

`0`

with `'Conditional'`

Bound Constraints`'Conditional'`

*Bound constraints*, also called semicontinuous constraints,
are mixed-integer linear constraints that confine portfolio weights to fall either
above or below specific bounds *if* the asset is selected;
otherwise, the value of the asset is zero. Use `setBounds`

to
specify bound constraints with a `'Conditional'`

`BoundType`

. To mathematically formulate this type of constraints,
a binary variable *v*_{i} is
needed. *v*_{i} = 0
indicates that asset *i* is not selected and
*v*_{i} indicates
that the asset was selected. Thus

$${l}_{i}{v}_{i}\le {x}_{i}\le {u}_{i}{v}_{i}$$

where

*x* is the portfolio (*n* vector).

*l* is the conditional lower-bound constraint
(*n* vector).

*u* is the conditional upper-bound constraint
(*n* vector).

*n* is the number of assets in the universe.

`PortfolioMAD`

object properties to specify the bound constraint are:

`LowerBound`

for*l*_{B}`UpperBound`

for*u*_{B}`NumAssets`

for*n*

The default is to ignore this constraint.

*Budget constraints* are specialized linear
constraints that confine the sum of portfolio weights to fall either
above or below specific bounds. The constraints take the form

$${l}_{S}\le {1}^{T}x\le {u}_{S}$$

where:

*x* is the portfolio (*n* vector).

`1`

is the vector of ones (*n* vector).

*l _{S}* is the lower-bound
budget constraint (scalar).

*u _{S}* is the upper-bound
budget constraint (scalar).

*n* is the number of assets in the universe.

`PortfolioMAD`

object properties to specify budget constraints are:

`LowerBudget`

for*l*_{S}`UpperBudget`

for*u*_{S}`NumAssets`

for*n*

The default is to ignore this constraint.

The default portfolio optimization problem (see Default Portfolio Problem) has *l _{S}* =

`1`

,
which means that the portfolio weights sum to `1`

.
If the portfolio optimization problem includes possible movements
in and out of cash, the budget constraint specifies how far portfolios
can go into cash. For example, if `0`

and `1`

,
then the portfolio can have 0–100% invested in cash. If cash
is to be a portfolio choice, set `RiskFreeRate`

(*Group constraints* are specialized linear
constraints that enforce “membership” among groups of
assets. The constraints take the form

$${l}_{G}\le Gx\le {u}_{G}$$

where:

*x* is the portfolio (*n* vector).

*l _{G}* is the lower-bound
group constraint (

*u _{G}* is the upper-bound
group constraint (

*G* is the matrix of group membership indexes
(*n _{G}*-by-

Each row of *G* identifies which assets belong
to a group associated with that row. Each row contains either `0`

s
or `1`

s with `1`

indicating that
an asset is part of the group or `0`

indicating that
the asset is not part of the group.

`PortfolioMAD`

object properties to specify group constraints are:

`GroupMatrix`

for*G*`LowerGroup`

for*l*_{G}`UpperGroup`

for*u*_{G}`NumAssets`

for*n*

The default is to ignore these constraints.

*Group ratio constraints* are specialized
linear constraints that enforce relationships among groups of assets.
The constraints take the form

$${l}_{Ri}{({G}_{B}x)}_{i}\le {({G}_{A}x)}_{i}\le {u}_{Ri}{({G}_{B}x)}_{i}$$

for *i* = 1,..., *n _{R}* where:

*x* is the portfolio (*n* vector).

*l _{R}* is the vector of
lower-bound group ratio constraints (

*u _{R}* is the vector matrix
of upper-bound group ratio constraints (

*G _{A}* is the matrix of
base group membership indexes (

*G _{B}* is the matrix of
comparison group membership indexes (

*n* is the number of assets in the universe
and *n _{R}* is the number of
constraints.

Each row of *G _{A}* and

Each row contains either `0`

s or `1`

s
with `1`

indicating that an asset is part of the
group or `0`

indicating that the asset is not part
of the group.

`PortfolioMAD`

object properties to specify group ratio constraints are:

`GroupA`

for*G*_{A}`GroupB`

for*G*_{B}`LowerRatio`

for*l*_{R}`UpperRatio`

for*u*_{R}`NumAssets`

for*n*

The default is to ignore these constraints.

*Turnover constraint* is a linear absolute value constraint that ensures
estimated optimal portfolios differ from an initial portfolio by no more than a
specified amount. Although portfolio turnover is defined in many ways, the turnover
constraints implemented in Financial Toolbox™ compute portfolio turnover as the average of purchases and sales.
Average turnover constraints take the form

$$\frac{1}{2}{1}^{T}|x-{x}_{0}|\le \tau $$

where:

*x* is the portfolio (*n* vector).

`1`

is the vector of ones (*n* vector).

*x _{0}* is the initial
portfolio (

*τ* is the upper bound for turnover (scalar).

*n* is the number of assets in the universe.

`PortfolioMAD`

object properties to specify the average turnover constraint are:

`Turnover`

for*τ*`InitPort`

for*x*_{0}`NumAssets`

for*n*

The default is to ignore this constraint.

*One-way turnover constraints* ensure that
estimated optimal portfolios differ from an initial portfolio by no
more than specified amounts according to whether the differences are
purchases or sales. The constraints take the forms

$${1}^{T}\times \mathrm{max}\left\{0,x-{x}_{0}\right\}\le {\tau}_{B}$$

$${1}^{T}\times \mathrm{max}\left\{0,{x}_{0}-x\right\}\le {\tau}_{S}$$

where:

*x* is the portfolio (*n* vector)

`1`

is the vector of ones (*n* vector).

*x _{0}* is the Initial
portfolio (

τ_{B} is the upper
bound for turnover constraint on purchases (scalar).

τ_{S} is the upper
bound for turnover constraint on sales (scalar).

To specify one-way turnover constraints, use the following properties in the
`PortfolioMAD`

object:

`BuyTurnover`

for τ_{B}`SellTurnover`

for τ_{S}`InitPort`

for*x*_{0}

The default is to ignore this constraint.

**Note**

The average turnover constraint (see Working with Average Turnover Constraints Using PortfolioMAD Object) with τ is not a
combination of the one-way turnover constraints with τ =
τ_{B} =
τ_{S}.

*Cardinality constraint* limits the number of assets in the
optimal allocation for an `PortfolioMAD`

object. Use `setMinMaxNumAssets`

to specify the `'MinNumAssets'`

and
`'MaxNumAssets'`

constraints. To mathematically formulate this
type of constraints, a binary variable
*v*_{i} is needed.
*v*_{i} = 0 indicates
that asset *i* is not selected and
*v*_{i} = 1
indicates that the asset was selected. Thus

$$MinNumAssets\le {\displaystyle \sum _{i=1}^{NumAssets}{v}_{i}\le MaxNumAssets}$$

The default is to ignore this constraint.

`Portfolio`

| `PortfolioCVaR`

| `PortfolioMAD`