Choosing and Controlling the Solver for Mean-Variance Portfolio Optimization

The default solver for mean-variance portfolio optimization is quadprog from Optimization Toolbox™ software, which, by default, implements an interior-point-convex algorithm. Although quadprog works for most problems, you can adjust arguments to control the algorithm. For details about quadprog and quadratic programming algorithms and options, see Quadratic Programming Algorithms.

Alternatively, the mean-variance portfolio optimization tools let you use setSolver to specify a solverType of lcprog which implements a linear complementarity programming (LCP) algorithm.

Using 'quadprog' and 'lcprog'

To modify either the default solver quadprog or to specify lcprog as your solver, use the setSolver function to set the hidden properties solverType and solverOptions that specify and control the solver. Because the solver properties are hidden, you cannot set these options using the Portfolio object. The default solver is quadprog so you do not need to use setSolver to specify this solver. When using quadprog, the default interior-point-convex algorithm of quadprog is used. You can verify that you are using the interior-point-convex algorithm with this code:

p = Portfolio;
display(p.solverType)
display(p.solverOptions)

  quadprog options:

   Options used by current Algorithm ('interior-point-convex'):
   (Other available algorithms: 'active-set', 'trust-region-reflective')

   Set properties:
                Display: 'off'
    OptimalityTolerance: 1.0000e-12

   Default properties:
              Algorithm: 'interior-point-convex'
    ConstraintTolerance: 1.0000e-08
           LinearSolver: 'auto'
          MaxIterations: 200
           ScaleProblem: 0
          StepTolerance: 1.0000e-12

   Show options not used by current Algorithm ('interior-point-convex')

You can switch back to lcprog with this code:

p = setSolver(p, 'lcprog');
display(p.solverType);
display(p.solverOptions)

lcprog
  struct with fields:

     MaxIter: []
    TieBreak: []
      TolPiv: 5.0000e-08

In both cases, setSolver sets up default options associated with either solver. If you want to specify additional options associated with a given solver, setSolver accepts these options with name-value arguments in the function call. For example, if you intend to use quadprog and want to use the 'trust-region-reflective' algorithm, call setSolver with this code:

p = Portfolio;
p = setSolver(p, 'quadprog', 'Algorithm', 'trust-region-reflective');
display(p.solverOptions)

  quadprog options:

   Options used by current Algorithm ('trust-region-reflective'):
   (Other available algorithms: 'active-set', 'interior-point-convex')

   Set properties:
              Algorithm: 'trust-region-reflective'

   Default properties:
                Display: 'final'
      FunctionTolerance: 'default dependent on problem'
     HessianMultiplyFcn: []
          MaxIterations: 'default dependent on problem'
    OptimalityTolerance: 'default dependent on problem'
          StepTolerance: 2.2204e-14
    SubproblemAlgorithm: 'cg'
               TypicalX: 'ones(numberOfVariables,1)'

   Show options not used by current Algorithm ('trust-region-reflective')

In addition, if you want to specify any of the options for quadprog that you typically set through optimoptions from Optimization Toolbox, setSolver accepts an optimoptions object as the second argument. For example, you can start with the default options for quadprog and then use setSolver to change the algorithm to 'trust-region-reflective' with no displayed output:

p = Portfolio;
options = optimoptions('quadprog', 'Algorithm', 'trust-region-reflective', 'Display', 'off');
p = setSolver(p, 'quadprog', options);
display(p.solverOptions.Algorithm)
display(p.solverOptions.Display)

trust-region-reflective
off

Using the Mixed Integer Nonlinear Programming (MINLP) Solver

The mixed integer nonlinear programming (MINLP) solver, configured using setSolverMINLP, enables you to specify associated solver options for portfolio optimization for a Portfolio object. The MINLP solver is used when any one, or any combination of 'Conditional' BoundType, MinNumAssets, or MaxNumAssets constraints are active. In this case, you formulate the portfolio problem by adding NumAssets binary variables, where 0 indicates not invested, and 1 is invested. For more information on using 'Conditional' BoundType, see setBounds. For more information on specifying MinNumAssets and MaxNumAssets, see setMinMaxNumAssets.

When using the estimate functions with a Portfolio object where 'Conditional' BoundType, MinNumAssets, or MaxNumAssets constraints are active, the mixed integer nonlinear programming (MINLP) solver is automatically used.

Solver Guidelines for Portfolio Objects

The following table provides guidelines for using setSolver and setSolverMINLP.

Portfolio Problem	Portfolio Function	Type of Optimization Problem	Main Solver	Helper Solver
Portfolio without tracking error constraints	estimateFrontierByRisk	Optimizing a portfolio for a certain risk level introduces a nonlinear constraint. Therefore, this problem has a linear objective with linear and nonlinear constraints.	'fmincon' using setSolver	For 'min': quadratic objective, 'quadprog' or 'lcprog' using setSolver For 'max': linear objective, 'linprog' or 'lcprog' using setSolver
Portfolio without tracking error constraints	estimateFrontierByReturn	Quadratic objective with linear constraints	'quadprog' or 'lcprog' using setSolver	For ‘min’: quadratic objective, 'quadprog' or 'lcprog' using setSolver For ‘max’: linear objective, 'linprog' or 'lcprog' using setSolver
Portfolio without tracking error constraints	estimateFrontierLimits	Quadratic or linear objective with linear constraints	For ‘min’: quadratic objective, 'quadprog' or 'lcprog' using setSolver For ‘max’: linear objective, 'linprog' or 'lcprog' using setSolver	Not applicable
Portfolio without tracking error constraints	estimateMaxSharpeRatio	Quadratic objective with linear constraints	'quadprog' using setSolver	Because estimateMaxSharpeRatio internally calls estimateFrontierLimits, all solvers needed by estimateFrontierLimits are the helper solvers
Portfolio with tracking error constraints	estimateFrontierByRisk	Linear objective with linear and nonlinear constraints	'fmincon' using setSolver	Not applicable
Portfolio with tracking error constraints	estimateFrontierByReturn	Linear objective with linear and nonlinear constraints	'fmincon' using setSolver	Not applicable
Portfolio with tracking error constraints	estimateFrontierLimits	Quadratic (minimum risk problem) or linear (maximum return problem) objective with linear and nonlinear constraints	'fmincon' using setSolver	Not applicable
Portfolio with tracking error constraints	estimateMaxSharpeRatio	Quadratic objective with linear and nonlinear constraints	'fmincon' using setSolver	Not applicable
Portfolio with active 'Conditional' BoundType, MinNumAssets, MaxNumAssets, or conditional budget constraints	estimateFrontierByRisk	The problem is formulated by introducing NumAssets binary variables to indicate whether the corresponding asset is invested or not. Therefore, it requires a mixed integer nonlinear programming solver. Three types of MINLP solvers are offered, see setSolverMINLP.	Mixed integer nonlinear programming solver (MINLP) using setSolverMINLP	'quadprog' or 'fmincon' are used when the estimate functions reduce the problem into NLP. These two solvers can be configured through setSolver.
Portfolio with active 'Conditional' BoundType, MinNumAssets, MaxNumAssets, or conditional budget constraints	estimateFrontierByReturn	The problem is formulated by introducing NumAssets binary variables to indicate whether the corresponding asset is invested or not. Therefore, it requires a mixed integer nonlinear programming solver. Three types of MINLP solvers are offered, see setSolverMINLP.	Mixed integer nonlinear programming solver (MINLP) using setSolverMINLP	'quadprog' or 'fmincon' are used when the estimate functions reduce the problem into NLP. These two solvers can be configured through setSolver.
Portfolio with active 'Conditional' BoundType, MinNumAssets, MaxNumAssets, or conditional budget constraints	estimateFrontierLimits	The problem is formulated by introducing NumAssets binary variables to indicate whether the corresponding asset is invested or not. Therefore, it requires a mixed integer nonlinear programming solver. Three types of MINLP solvers are offered, see setSolverMINLP.	Mixed integer nonlinear programming solver (MINLP) using setSolverMINLP	'quadprog' or 'fmincon' are used when the estimate functions reduce the problem into NLP. These two solvers can be configured through setSolver.
Portfolio with active 'Conditional' BoundType, MinNumAssets, MaxNumAssets, or conditional budget constraints	estimateMaxSharpeRatio	The problem is formulated by introducing NumAssets binary variables to indicate whether the corresponding asset is invested or not. Therefore, it requires a mixed integer nonlinear programming solver. Three types of MINLP solvers are offered, see setSolverMINLP.	Mixed integer nonlinear programming solver (MINLP) using setSolverMINLP	'quadprog' or 'fmincon' are used when the estimate functions reduce the problem into NLP. These two solvers can be configured through setSolver.

Solver Guidelines for Custom Objective Problems Using Portfolio Objects

The following table provides guidelines for using setSolver and setSolverMINLP when using estimateCustomObjectivePortfolio.

Portfolio Problem	Portfolio Function	Type of Optimization Problem	Main Solver	Helper Solver
Portfolio with custom linear objective without tracking error constraints	estimateCustomObjectivePortfolio	Linear objective with linear constraints	'linprog' using setSolver	Not applicable
Portfolio with custom quadratic objective without tracking error constraints	estimateCustomObjectivePortfolio	Quadratic objective with linear constraints	'quadprog' using setSolver	Not applicable
Portfolio with custom nonlinear objective without tracking error constraints	estimateCustomObjectivePortfolio	Nonlinear objective with linear constraints	'fmincon' using setSolver	Not applicable
Portfolio with custom linear objective with tracking error constraints	estimateCustomObjectivePortfolio	Linear objective with linear and nonlinear constraints	'fmincon' using setSolver	Not applicable
Portfolio with custom quadratic objective with tracking error constraints	estimateCustomObjectivePortfolio	Quadratic objective with linear and nonlinear constraints	'fmincon' using setSolver	Not applicable
Portfolio with custom nonlinear objective with tracking error constraints	estimateCustomObjectivePortfolio	Nonlinear objective with linear and nonlinear constraints	'fmincon' using setSolver	Not applicable
Portfolio with active 'Conditional' BoundType, MinNumAssets, MaxNumAssets, or conditional budget constraints	estimateCustomObjectivePortfolio	The problem is formulated by introducing NumAssets binary variables to indicate whether the corresponding asset is invested or not. Therefore, it requires a mixed integer nonlinear programming solver. Three types of MINLP solvers are offered, see setSolverMINLP.	Mixed integer nonlinear programming solver (MINLP) using setSolverMINLP	'intlinprog' is used when the problem reduces to a mixed-integer linear problem. This solver can be configured using setSolverMINLP and its name-value argument IntMainSolverOptions.