If a `Portfolio`

object is destroyed when modifying, remember to pass an
existing object into the `Portfolio`

object if you want to
modify it, otherwise it creates a new object. See Creating the Portfolio Object for details.

If the optimization fails with a "bad pivot" message from `lcprog`

,
try a larger value for `tolpiv`

which is a tolerance
for pivot selection in the `lcprog`

algorithm (try `1.0e-7`

,
for example) or try the `interior-point-convex`

version
of `quadprog`

. For details, see Choosing and Controlling the Solver for Mean-Variance Portfolio Optimization,
the help header for `lcprog`

, and `quadprog`

.

Although it is difficult to characterize when one algorithm is faster than the other, the
default solver, `lcprog`

is faster for smaller problems and the
`quadprog`

solver is faster for larger problems. If one solver
seems to take too much time, try the other solver. To change solvers, use `setSolver`

.

If you get matrix incompatibility or "non-conformable" errors, the representation of data in the tools follows a specific set of basic rules described in Conventions for Representation of Data.

If asset return data has missing or `NaN`

values,
the `estimateAssetMoments`

function
with the `'missingdata'`

flag set to `true`

may
fail with either too many iterations or a singular covariance. To
correct this problem, consider this:

If you have asset return data with no missing or

`NaN`

values, you can compute a covariance matrix that may be singular without difficulties. If you have missing or`NaN`

values in your data, the supported missing data feature requires that your covariance matrix must be positive-definite, that is, nonsingular.`estimateAssetMoments`

uses default settings for the missing data estimation procedure that might not be appropriate for all problems.

In either case, you might want to estimate the moments
of asset returns separately with either the ECM estimation functions
such as `ecmnmle`

or with your
own functions.

`mv_optim_transform`

ErrorsIf you obtain optimization errors such as:

Error using mv_optim_transform (line 233) Portfolio set appears to be either empty or unbounded. Check constraints. Error in Portfolio/estimateFrontier (line 63) [A, b, f0, f, H, g, lb] = mv_optim_transform(obj);

Error using mv_optim_transform (line 238) Cannot obtain finite lower bounds for specified portfolio set. Error in Portfolio/estimateFrontier (line 63) [A, b, f0, f, H, g, lb] = mv_optim_transform(obj);

`estimateBounds`

to examine your
portfolio set, and use `checkFeasibility`

to ensure that
your initial portfolio is either feasible and, if infeasible, that you have
sufficient turnover to get from your initial portfolio to the portfolio set.
To correct this problem, try solving your problem with larger values for turnover or tracking-error and gradually reduce to the value that you want.

If you obtain efficient portfolios that do not seem to make
sense, this can happen if you forget to set specific constraints or
you set incorrect constraints. For example, if you allow portfolio
weights to fall between `0`

and `1`

and
do not set a budget constraint, you can get portfolios that are 100%
invested in every asset. Although it may be hard to detect, the best
thing to do is to review the constraints you have set with display
of the object. If you get portfolios with 100% invested in each asset,
you can review the display of your object and quickly see that no
budget constraint is set. Also, you can use `estimateBounds`

and `checkFeasibility`

to determine if the
bounds for your portfolio set make sense and to determine if the portfolios
you obtained are feasible relative to an independent formulation of
your portfolio set.

If you obtain efficient frontiers that do not seem to make sense, this can happen for some
cases of mean and covariance of asset returns. It is possible for some mean-variance
portfolio optimization problems to have difficulties at the endpoints of the
efficient frontier. It is rare for standard portfolio problems, but this can occur.
For example, this can occur when using unusual combinations of turnover constraints
and transaction costs. Usually, the workaround of setting the hidden property
`enforcePareto`

produces a single portfolio for the entire
efficient frontier, where any other solutions are not Pareto optimal (which is what
efficient portfolios must be).

An example of a portfolio optimization problem that has difficulties at the endpoints of the efficient frontier is this standard mean-variance portfolio problem (long-only with a budget constraint) with the following mean and covariance of asset returns:

m = [ 1; 2; 3 ]; C = [ 1 1 0; 1 1 0; 0 0 1 ]; p = Portfolio; p = Portfolio(p, 'assetmean', m, 'assetcovar', C); p = Portfolio(p, 'lowerbudget', 1, 'upperbudget', 1); p = Portfolio(p, 'lowerbound', 0); plotFrontier(p);

To work around this problem, set the hidden Portfolio object
property for `enforcePareto`

. This property instructs
the optimizer to perform extra steps to ensure a Pareto-optimal solution.
This slows down the solver, but guarantees a Pareto-optimal solution.

p.enforcePareto = true; plotFrontier(p);

`'Conditional'`

`BoundType`

, `MinNumAssets`

, and
`MaxNumAssets`

ConstraintsWhen configuring a `Portfolio`

, `PortfolioCVaR`

, or `PortfolioMAD`

object to include
`'Conditional'`

`BoundType`

(semicontinuous) constraints using `setBounds`

or
`MinNumAssets`

and `MaxNumAssets`

(cardinality) constraints using `setMinMaxNumAssets`

, the values of the inputs that you supply can result
in warning messages.

`LowerBound`

Defined as Empty or ZeroWhen using `setBounds`

with
the `BoundType`

set to `'Conditional'`

and the
`LowerBound`

input argument is empty (```
[
]
```

) or `0`

, the `Conditional`

bound is not effective and is equivalent to a `Simple`

bound.

AssetMean = [ 0.0101110; 0.0043532; 0.0137058 ]; AssetCovar = [ 0.00324625 0.00022983 0.00420395; 0.00022983 0.00049937 0.00019247; 0.00420395 0.00019247 0.00764097 ]; p = Portfolio('AssetMean', AssetMean, 'AssetCovar', AssetCovar, 'Budget', 1); p = setBounds(p, 0, 0.5, 'BoundType', 'Conditional'); p = setMinMaxNumAssets(p, 3, 3); estimateFrontier(p, 10)

Warning: Conditional bounds with 'LowerBound' as zero are equivalent to simple bounds. Consider either using strictly positive 'LowerBound' or 'simple' as the 'BoundType' instead. > In internal.finance.PortfolioMixedInteger/checkBoundType (line 46) In Portfolio/checkarguments (line 204) In Portfolio/setBounds (line 80) Warning: The solution may have less than 'MinNumAssets' assets with nonzero weight. To enforce 'MinNumAssets' requirement, set strictly positive lower conditional bounds. > In internal.finance.PortfolioMixedInteger/hasIntegerConstraints (line 44) In Portfolio/estimateFrontier (line 51) ans = Columns 1 through 8 0.5000 0.3555 0.3011 0.3299 0.3585 0.3873 0.4160 0.4448 0.5000 0.5000 0.4653 0.3987 0.3322 0.2655 0.1989 0.1323 0.0000 0.1445 0.2335 0.2714 0.3093 0.3472 0.3850 0.4229 Columns 9 through 10 0.4735 0.5000 0.0657 0 0.4608 0.5000

In all the 10 optimal allocations, there are allocations (the first and last
ones) that only have two assets, which is in conflict with the
`MinNumAssets`

constraint that three assets should be
allocated. Also there are two warnings, which actually explain what happens. In
this case, the `'Conditional'`

bound constraints are defined as
`xi`

= `0`

or `0`

<=
`xi`

<= `0.5`

, which are internally
modeled as
`0`

**vi*<=xi<=`0.5`

**vi*,
where *vi* is `0`

or `1`

,
where `0`

indicates not allocated, and `1`

indicates allocated. Here, *vi*=`1`

, which
still allows the asset to have a weight of `0`

. In other words,
setting `LowerBound`

as `0`

or empty, doesn’t
clearly define the minimum allocation for an allocated asset. Therefore, a
`0`

weighted asset is also considered as an allocated
asset. To fix this warning, follow the instructions in the warning message, and
set a `LowerBound`

value that is strictly
positive.

AssetMean = [ 0.0101110; 0.0043532; 0.0137058 ]; AssetCovar = [ 0.00324625 0.00022983 0.00420395; 0.00022983 0.00049937 0.00019247; 0.00420395 0.00019247 0.00764097 ]; p = Portfolio('AssetMean', AssetMean, 'AssetCovar', AssetCovar, 'Budget', 1); p = setBounds(p, 0.3, 0.5, 'BoundType', 'Conditional'); p = setMinMaxNumAssets(p, 3, 3); estimateFrontier(p, 10)

ans = Columns 1 through 8 0.3000 0.3180 0.3353 0.3489 0.3580 0.3638 0.3694 0.3576 0.4000 0.3820 0.3642 0.3479 0.3333 0.3199 0.3067 0.3001 0.3000 0.3000 0.3005 0.3032 0.3088 0.3163 0.3240 0.3423 Columns 9 through 10 0.3289 0.3000 0.3000 0.3000 0.3711 0.4000

`'BoundType'`

Must Be Conformable with `NumAssets`

The `setBounds`

optional name-value argument for `'BoundType'`

must be defined
for all assets in a `Portfolio`

, `PortfolioCVaR`

, or `PortfolioMAD`

object. By default,
the `'BoundType'`

is `'Simple'`

and applies to
all assets. Using `setBounds`

, you
can choose to define a `'BoundType'`

for each asset. In this
case, the number of `'BoundType'`

specifications must match the
number of assets (`NumAssets`

) in the `Portfolio`

, `PortfolioCVaR`

, or `PortfolioMAD`

object. The
following example demonstrates the error when the number of
`'BoundType'`

specifications do not match the number of
assets in the `Portfolio`

object.

AssetMean = [ 0.0101110; 0.0043532; 0.0137058 ]; AssetCovar = [ 0.00324625 0.00022983 0.00420395; 0.00022983 0.00049937 0.00019247; 0.00420395 0.00019247 0.00764097 ]; p = Portfolio('AssetMean', AssetMean, 'AssetCovar', AssetCovar, 'Budget', 1); p = setBounds(p, 0.1, 0.5, 'BoundType',["simple"; "conditional"])

Cannot create bound constraints. Caused by: Error using internal.finance.PortfolioMixedInteger/checkBoundType (line 28) Length of 'BoundType' must be conformable with 'NumAssets'=3.

To correct this, modify the `BoundType`

to include three
specifications because the `Portfolio`

object has three
assets.

AssetMean = [ 0.0101110; 0.0043532; 0.0137058 ]; AssetCovar = [ 0.00324625 0.00022983 0.00420395; 0.00022983 0.00049937 0.00019247; 0.00420395 0.00019247 0.00764097 ]; p = Portfolio('AssetMean', AssetMean, 'AssetCovar', AssetCovar, 'Budget', 1); p = setBounds(p, 0.1, 0.5, 'BoundType',["simple"; "conditional";"conditional"]) p.BoundType

p = Portfolio with properties: BuyCost: [] SellCost: [] RiskFreeRate: [] AssetMean: [3×1 double] AssetCovar: [3×3 double] TrackingError: [] TrackingPort: [] Turnover: [] BuyTurnover: [] SellTurnover: [] Name: [] NumAssets: 3 AssetList: [] InitPort: [] AInequality: [] bInequality: [] AEquality: [] bEquality: [] LowerBound: [3×1 double] UpperBound: [3×1 double] LowerBudget: 1 UpperBudget: 1 GroupMatrix: [] LowerGroup: [] UpperGroup: [] GroupA: [] GroupB: [] LowerRatio: [] UpperRatio: [] BoundType: [3×1 categorical] MinNumAssets: [] MaxNumAssets: [] ans = 3×1 categorical array simple conditional conditional

`'BoundType'`

, `'MinNumAssets'`

, `'MaxNumAssets'`

ConstraintsWhen none of the constraints from `'BoundType'`

,
`'MinNumAssets'`

, or ` 'MaxNumAssets'`

are
active, the redundant constraints from `'BoundType'`

,
`'MinNumAssets'`

, `'MaxNumAssets'`

warning
occurs. This happens when you explicitly use `setBounds`

and
`setMinMaxNumAssets`

but with values that are inactive. That is,
the `'Conditional'`

`BoundType`

has a `LowerBound`

= ```
[
]
```

or `0`

, `'MinNumAssets'`

is
`0`

, or `'MaxNumAssets'`

is the same value
as `NumAssets`

. In other words, if any of these three are
active, the warning will not show up when using the `estimate`

functions or `plotFrontier`

. The following two
examples show the rationale.

The first example is when the `BoundType`

is explicitly set
as `'Conditional'`

but the `LowerBound`

is
`0`

, and no `'MinNumAssets'`

and
`'MaxNumAssets'`

constraints are defined using `setMinMaxNumAssets`

.

AssetMean = [ 0.0101110; 0.0043532; 0.0137058 ]; AssetCovar = [ 0.00324625 0.00022983 0.00420395; 0.00022983 0.00049937 0.00019247; 0.00420395 0.00019247 0.00764097 ]; p = Portfolio('AssetMean', AssetMean, 'AssetCovar', AssetCovar, 'Budget', 1); p = setBounds(p, 0, 0.5, 'BoundType', 'Conditional'); estimateFrontier(p, 10)

Warning: Redundant constraints from 'BoundType', 'MinNumAssets', 'MaxNumAssets'. > In internal.finance.PortfolioMixedInteger/hasIntegerConstraints (line 24) In Portfolio/estimateFrontier (line 51) ans = Columns 1 through 8 0.5000 0.3555 0.3011 0.3299 0.3586 0.3873 0.4160 0.4448 0.5000 0.5000 0.4653 0.3987 0.3321 0.2655 0.1989 0.1323 0 0.1445 0.2335 0.2714 0.3093 0.3471 0.3850 0.4229 Columns 9 through 10 0.4735 0.5000 0.0657 0 0.4608 0.5000

The second example is when you explicitly set the three constraints, but all
with inactive values. In this example, the `BoundType`

is
`'Conditional'`

and the `LowerBound`

is
`0`

, thus specifying ineffective
`'Conditional'`

`BoundType`

constraints, and the
`'MinNumAssets'`

and `'MaxNumAssets'`

values are `0`

and `3`

, respectively. The
`setMinMaxNumAssets`

function specifies ineffective
`'MinNumAssets'`

and `'MaxNumAssets'`

constraints.

AssetMean = [ 0.0101110; 0.0043532; 0.0137058 ]; AssetCovar = [ 0.00324625 0.00022983 0.00420395; 0.00022983 0.00049937 0.00019247; 0.00420395 0.00019247 0.00764097 ]; p = Portfolio('AssetMean', AssetMean, 'AssetCovar', AssetCovar, 'Budget', 1); p = setBounds(p, 0, 0.5, 'BoundType', 'Conditional'); p = setMinMaxNumAssets(p, 0, 3); estimateFrontier(p, 10)

Warning: Redundant constraints from 'BoundType', 'MinNumAssets', 'MaxNumAssets'. > In internal.finance.PortfolioMixedInteger/hasIntegerConstraints (line 24) In Portfolio/estimateFrontier (line 51) ans = Columns 1 through 8 0.5000 0.3555 0.3011 0.3299 0.3586 0.3873 0.4160 0.4448 0.5000 0.5000 0.4653 0.3987 0.3321 0.2655 0.1989 0.1323 0 0.1445 0.2335 0.2714 0.3093 0.3471 0.3850 0.4229 Columns 9 through 10 0.4735 0.5000 0.0657 0 0.4608 0.5000

`'BoundType'`

, `'MinNumAssets'`

, `'MaxNumAssets'`

The `Portfolio`

, `PortfolioCVaR`

, or `PortfolioMAD`

object performs
validations of all the constraints that you set before solving any specific
optimization problems. The `Portfolio`

, `PortfolioCVaR`

, or `PortfolioMAD`

object first
considers all constraints other than `'Conditional'`

`BoundType`

, `'MinNumAssets'`

, and
`'MaxNumAssets'`

and issues an error message if they are
not compatible. Then the `Portfolio`

, `PortfolioCVaR`

, or `PortfolioMAD`

object adds the
three constraints to check if they are compatible with the already checked
constraints. This separation is natural because `'Conditional'`

`BoundType`

, `'MinNumAssets'`

, and
`'MaxNumAssets'`

require additional binary variables in the
mathematical formulation that leads to a MINLP, while other constraints only
need continuous variables. You can follow the error messages to check when the
infeasible problem occurs and take actions to fix the constraints.

One possible scenario is when the `BoundType`

is
`'Conditional'`

and Groups are defined for the
`Portfolio`

object. In this case, the Group definitions are
themselves in conflict. Consequently, the `'Conditional'`

bound
constraint cannot be applied when running `estimateFrontierLimits`

.

AssetMean = [ 0.0101110; 0.0043532; 0.0137058 ]; AssetCovar = [ 0.00324625 0.00022983 0.00420395; 0.00022983 0.00049937 0.00019247; 0.00420395 0.00019247 0.00764097 ]; p = Portfolio('AssetMean', AssetMean, 'AssetCovar', AssetCovar, 'Budget', 1); p = setBounds(p, 0.1, 0.5, 'BoundType','Conditional'); p = setGroups(p, [1,1,0], 0.3, 0.5); p = addGroups(p, [0,1,0], 0.6, 0.7); pwgt = estimateFrontierLimits(p)

Error using Portfolio/buildMixedIntegerProblem (line 31) Infeasible portfolio problem prior to considering 'BoundType', 'MinNumAssets', 'MaxNumAssets'. Verify if constraints from groups, bounds, group ratios, inequality, equality, etc. are compatible. Error in Portfolio/estimateFrontierLimits>int_frontierLimits (line 93) ProbStruct = buildMixedIntegerProblem(obj); Error in Portfolio/estimateFrontierLimits (line 73) pwgt = int_frontierLimits(obj, minsolution, maxsolution);

To correct this error, change the `LowerGroup`

in the
`addGroups`

function to also be
`0.3`

to match the `GroupMatrix`

input
from `setGroups`

.

AssetMean = [ 0.0101110; 0.0043532; 0.0137058 ]; AssetCovar = [ 0.00324625 0.00022983 0.00420395; 0.00022983 0.00049937 0.00019247; 0.00420395 0.00019247 0.00764097 ]; p = Portfolio('AssetMean', AssetMean, 'AssetCovar', AssetCovar, 'Budget', 1); p = setBounds(p, 0.1, 0.5, 'BoundType','Conditional'); p = setGroups(p, [1,1,0], 0.3, 0.5); p = addGroups(p, [0,1,0], 0.3, 0.7); pwgt = estimateFrontierLimits(p)

pwgt = 0 0.2000 0.5000 0.3000 0.5000 0.5000

A second possible scenario is when the `BoundType`

is
`'Conditional'`

and the `setEquality`

function is used
with the `bEquality`

parameter set to `0.04`

.
This sets an equality constraint to have *x*1 +
*x*3 = `0.04`

. At the same time, `setBounds`

also
set the semicontinuous constraints to have *xi* =
`0`

or `0.1`

<= *xi*
<= `2.5`

, which lead to *x*1 +
`x`

3 = `0`

or `0.1`

<=
*x*1 + *x*3 <= `5`

.
The semicontinuous constraints are not compatible with the equality constraint
because there is no way to get `x`

1 + `x`

3 to
equal `0.04`

. Therefore, the error message is
displayed.

AssetMean = [ 0.05; 0.1; 0.12; 0.18 ]; AssetCovar = [ 0.0064 0.00408 0.00192 0; 0.00408 0.0289 0.0204 0.0119; 0.00192 0.0204 0.0576 0.0336; 0 0.0119 0.0336 0.1225 ]; p = Portfolio('AssetMean', AssetMean, 'AssetCovar', AssetCovar, 'Budget', 1); A = [ 1 0 1 0 ]; b = 0.04; p = setEquality(p, A, b); p = setBounds(p, 0.1, 2.5, 'BoundType','Conditional'); p = setMinMaxNumAssets(p, 2, 2); pwgt = estimateFrontierLimits(p)

Error using Portfolio/buildMixedIntegerProblem (line 109) Infeasible portfolio problem when considering 'BoundType', 'MinNumAssets', 'MaxNumAssets'. Verify if these are compatible with constraints from groups, bounds, group ratios, inequality, equality, etc. Error in Portfolio/estimateFrontierLimits>int_frontierLimits (line 93) ProbStruct = buildMixedIntegerProblem(obj); Error in Portfolio/estimateFrontierLimits (line 73) pwgt = int_frontierLimits(obj, minsolution, maxsolution);

To correct this error, change the `bEquality`

parameter from
`0.04`

to `.4`

.

AssetMean = [ 0.05; 0.1; 0.12; 0.18 ]; AssetCovar = [ 0.0064 0.00408 0.00192 0; 0.00408 0.0289 0.0204 0.0119; 0.00192 0.0204 0.0576 0.0336; 0 0.0119 0.0336 0.1225 ]; p = Portfolio('AssetMean', AssetMean, 'AssetCovar', AssetCovar, 'Budget', 1); A = [ 1 0 1 0 ]; b = 0.4; p = setEquality(p, A, b); p = setBounds(p, 0.1, 2.5, 'BoundType','Conditional'); p = setMinMaxNumAssets(p, 2, 2); pwgt = estimateFrontierLimits(p)

pwgt = 0.4000 0 0.6000 0 0 0.4000 0 0.6000

This error occurs when you are using a `Portfolio`

, `PortfolioCVaR`

, or `PortfolioMAD`

object and there is
no `UpperBound`

defined in `setBounds`

and
you are using `setMinMaxNumAssets`

. In this case, this is formulated as a mixed
integer programming problem and an ` UpperBound`

is required to
enforce `MinNumAssets`

and `MaxNumAssets`

constraints.

The optimizer first attempts to estimate the upper bound of each asset, based
on all the specified constraints. If the `UpperBound`

cannot be
found, an error message occurs which instructs you to set an explicit
`UpperBound`

. In most cases, as long as you set some upper
bounds to the problem using any `set`

function, the optimizer
can successfully find a good
estimation.

AssetMean = [ 0.05; 0.1; 0.12; 0.18 ]; AssetCovar = [ 0.0064 0.00408 0.00192 0; 0.00408 0.0289 0.0204 0.0119; 0.00192 0.0204 0.0576 0.0336; 0 0.0119 0.0336 0.1225 ]; p = Portfolio('AssetMean', AssetMean, 'AssetCovar', AssetCovar); p = setBounds(p, 0.1, 'BoundType','Conditional'); p = setGroups(p, [1,1,1,0], 0.3, 0.5); p = setMinMaxNumAssets(p, 3, 3); pwgt = estimateFrontierLimits(p)

Error using Portfolio/buildMixedIntegerProblem (line 42) Unbounded portfolio problem. Upper bounds cannot be inferred from the existing constraints. Set finite upper bounds using 'setBounds'. Error in Portfolio/estimateFrontierLimits>int_frontierLimits (line 93) ProbStruct = buildMixedIntegerProblem(obj); Error in Portfolio/estimateFrontierLimits (line 73) pwgt = int_frontierLimits(obj, minsolution, maxsolution);

To correct this error, specify an `UpperBound`

value for
`setBounds`

.

AssetMean = [ 0.05; 0.1; 0.12; 0.18 ]; AssetCovar = [ 0.0064 0.00408 0.00192 0; 0.00408 0.0289 0.0204 0.0119; 0.00192 0.0204 0.0576 0.0336; 0 0.0119 0.0336 0.1225 ]; p = Portfolio('AssetMean', AssetMean, 'AssetCovar', AssetCovar); p = setBounds(p, 0.1, .9, 'BoundType','Conditional'); p = setGroups(p, [1,1,1,0], 0.3, 0.5); p = setMinMaxNumAssets(p, 3, 3); pwgt = estimateFrontierLimits(p)

pwgt = 0.1000 0 0.1000 0.1000 0.1000 0.4000 0 0.9000

`MaxNumAssets`

SpecifiedWhen using a `Portfolio`

, `PortfolioCVaR`

, or `PortfolioMAD`

object, the optimal
allocation `w`

may contain some very small values that leads to
`sum`

(`w`

>`0`

) larger
than `MaxNumAssets`

, even though the
`MaxNumAssets`

constraint is specified using `setMinMaxNumAssets`

. For example, in the following code when
`setMinMaxNumAssets`

is used to set
`MaxNumAssets`

to `15`

, the
`sum`

(`w`

>`0`

) indicates
that there are `19`

assets. A close examination of the weights
shows that the weights are extremely small and are actually
0.

T = readtable('dowPortfolio.xlsx'); symbol = T.Properties.VariableNames(3:end); assetReturn = tick2ret(T{:,3:end}); p = Portfolio('AssetList', symbol, 'budget', 1); p = setMinMaxNumAssets(p, 10, 15); p = estimateAssetMoments(p,assetReturn); p = setBounds(p,0.01,0.5,'BoundType','Conditional','NumAssets',30); p = setTrackingError(p,0.05,ones(1, p.NumAssets)/p.NumAssets); w = estimateFrontierLimits(p,'min'); % minimum risk portfolio sum(w>0) % Number of assets that are allocated in the optimal portfolio w(w<eps) % Check the weights of the very small weighted assets

ans = 19 ans = 1.0e-20 * -0.0000 0 0 0.0293 0 0.3626 0.2494 0 0.0926 -0.0000 0 0.0020 0 0 0 0

This situation only happens when the `OuterApproximation`

algorithm is used with `setSolverMINLP`

to
solve a MINLP portfolio optimization problem. The
`OuterApproximation`

internally fixes the latest solved
integer variables and runs an NLP with `quadprog`

or `fmincon`

, which introduces numerical
issues and leads to weights that are very close to 0.

If you do not want to deal with very small values, you can use `setSolverMINLP`

to
select a different algorithm. In this example, the
`'TrustRegionCP'`

algorithm is
specified.

T = readtable('dowPortfolio.xlsx'); symbol = T.Properties.VariableNames(3:end); assetReturn = tick2ret(T{:,3:end}); p = Portfolio('AssetList', symbol, 'budget', 1); p = setMinMaxNumAssets(p, 10, 15); p = estimateAssetMoments(p,assetReturn); p = setBounds(p,0.01,0.5,'BoundType','Conditional','NumAssets',30); p = setTrackingError(p,0.05,ones(1, p.NumAssets)/p.NumAssets); p = setSolverMINLP(p,'TrustRegionCP'); w = estimateFrontierLimits(p,'min'); % minimum risk portfolio sum(w>0) % Number of assets that are allocated in the optimal portfolio w(w<eps) % The weights of the very small weighted assets are strictly zeros

ans = 14 ans = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

`Portfolio`

| `checkFeasibility`

| `estimateAssetMoments`

| `setBounds`

| `setMinMaxNumAssets`

- Postprocessing Results to Set Up Tradable Portfolios
- Creating the Portfolio Object
- Working with Portfolio Constraints Using Defaults
- Estimate Efficient Portfolios for Entire Efficient Frontier for Portfolio Object
- Estimate Efficient Frontiers for Portfolio Object
- Asset Allocation Case Study
- Portfolio Optimization Examples
- Portfolio Optimization with Semicontinuous and Cardinality Constraints
- Black-Litterman Portfolio Optimization
- Portfolio Optimization Using Factor Models