Optimization Toolbox    

Problems Covered by Large-Scale Methods

Not all possible problem formulations are covered by the large-scale algorithms. The following table describes what functionality is covered by the large-scale methods. For example, for fmincon, the large-scale algorithm covers the case where there are only bound constraints or only linear equalities. For each problem formulation, the third column describes what additional information is needed for the large-scale algorithms. For fminunc and fmincon, the gradient must be computed along with the objective in the user-supplied function (the gradient is not required for the medium-scale algorithms).

Since these methods can also be used on small- to medium-scale problems that are not necessarily sparse, the last column of the table emphasizes what conditions are needed for large-scale problems to run efficiently without exceeding your computer system's memory capabilities, e.g., the linear constraint matrices should be sparse. For smaller problems the conditions in the last column are unnecessary.

Several examples, which follow this table, clarify the contents of the table.

Table 1-4: Large-Scale Problem Coverage and Requirements 
Function
 Problem
Formulations
 Additional Information Needed
 For Large Problems
fminunc
Must provide gradient for f(x) in fun.
  • Provide sparsity structure of the Hessian, or compute the Hessian in fun.
  • The Hessian should be sparse.
fmincon

    • such that where

    • such that , and is an m-by-n matrix where

Must provide gradient for f(x) in fun.
  • Provide sparsity structure of the Hessian, or compute the Hessian in fun.
  • The Hessian should be sparse.
  • should be sparse.
lsqnonlin

    • such that where

F(x) must be overdetermined (have at least as many equations as variables).
 Not applicable.
  • Provide sparsity structure of the Jacobian, or compute the Jacobian in fun.
  • The Jacobian should be sparse.
lsqcurvefit
such that where
must be overdetermined (have at least as many equations as variables).
 Not applicable.
  • Provide sparsity structure of the Jacobian, or compute the Jacobian in fun.
  • The Jacobian should be sparse.
fsolve
must have the same number of equations as variables.
 Not applicable.
  • Provide sparsity structure of the Jacobian or compute the Jacobian in fun.
  • The Jacobian should be sparse.
lsqlin
such that where
is an m-by-n matrix where i.e., the problem must be overdetermined.
 Not applicable.
 should be sparse.
linprog
such that and , where
Not applicable.
 and should be sparse.
quadprog

    • such that where

    • such that , and is an m-by-n matrix where

Not applicable.
  • should be sparse.
  • should be sparse.

In the examples below, many of the M-file functions are available in the Optimization Toolbox optim directory. Most of these do not have a fixed problem size, i.e., the size of your starting point xstart will determine the size problem that is computed. If your computer system cannot handle the size suggested in the examples below, use a smaller dimension start point to run the problems. If the problems have upper or lower bounds or equalities, you will have to adjust the size of those vectors or matrices as well.


 Large-Scale Examples Nonlinear Equations with Jacobian