# prob2struct

Convert optimization problem or equation problem to solver form

## Syntax

``problem = prob2struct(prob)``
``problem = prob2struct(prob,x0)``
``problem = prob2struct(___,Name,Value)``

## Description

Use `prob2struct` to convert an optimization problem or equation problem to solver form.

example

````problem = prob2struct(prob)` returns an optimization problem structure suitable for a solver-based solution. For nonlinear problems, `prob2struct` creates files for the objective function, and, if necessary, for nonlinear constraint functions and supporting files.```

example

````problem = prob2struct(prob,x0)` also converts the initial point structure `x0` and includes it in `problem`.```

example

````problem = prob2struct(___,Name,Value)`, for any input arguments, specifies additional options using one or more name-value pair arguments. For example, for a nonlinear optimization problem, ```problem = prob2struct(prob,'ObjectiveFunctionName','objfun1')``` specifies that `prob2struct` creates an objective function file named `objfun1.m` in the current folder.```

## Examples

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Convert an optimization problem object to a problem structure.

Input the basic MILP problem from Mixed-Integer Linear Programming Basics: Problem-Based.

```ingots = optimvar('ingots',4,1,'Type','integer','LowerBound',0,'UpperBound',1); alloys = optimvar('alloys',4,1,'LowerBound',0); weightIngots = [5,3,4,6]; costIngots = weightIngots.*[350,330,310,280]; costAlloys = [500,450,400,100]; cost = costIngots*ingots + costAlloys*alloys; steelprob = optimproblem; steelprob.Objective = cost; totalweight = weightIngots*ingots + sum(alloys); carbonIngots = [5,4,5,3]/100; molybIngots = [3,3,4,4,]/100; carbonAlloys = [8,7,6,3]/100; molybAlloys = [6,7,8,9]/100; totalCarbon = (weightIngots.*carbonIngots)*ingots + carbonAlloys*alloys; totalMolyb = (weightIngots.*molybIngots)*ingots + molybAlloys*alloys; steelprob.Constraints.conswt = totalweight == 25; steelprob.Constraints.conscarb = totalCarbon == 1.25; steelprob.Constraints.consmolyb = totalMolyb == 1.25;```

Convert the problem to an `intlinprog` problem structure.

`problem = prob2struct(steelprob);`

Examine the resulting linear equality constraint matrix and vector.

`Aeq = problem.Aeq`
```Aeq = (1,1) 1.0000 (2,1) 0.0800 (3,1) 0.0600 (1,2) 1.0000 (2,2) 0.0700 (3,2) 0.0700 (1,3) 1.0000 (2,3) 0.0600 (3,3) 0.0800 (1,4) 1.0000 (2,4) 0.0300 (3,4) 0.0900 (1,5) 5.0000 (2,5) 0.2500 (3,5) 0.1500 (1,6) 3.0000 (2,6) 0.1200 (3,6) 0.0900 (1,7) 4.0000 (2,7) 0.2000 (3,7) 0.1600 (1,8) 6.0000 (2,8) 0.1800 (3,8) 0.2400 ```
`beq = problem.beq`
```beq = 3×1 25.0000 1.2500 1.2500 ```

Examine the bounds.

`problem.lb`
```ans = 8×1 0 0 0 0 0 0 0 0 ```
`problem.ub`
```ans = 8×1 Inf Inf Inf Inf 1 1 1 1 ```

Solve the problem by calling `intlinprog`.

`x = intlinprog(problem)`
```Running HiGHS 1.6.0: Copyright (c) 2023 HiGHS under MIT licence terms Presolving model 3 rows, 8 cols, 24 nonzeros 3 rows, 8 cols, 18 nonzeros Solving MIP model with: 3 rows 8 cols (4 binary, 0 integer, 0 implied int., 4 continuous) 18 nonzeros Nodes | B&B Tree | Objective Bounds | Dynamic Constraints | Work Proc. InQueue | Leaves Expl. | BestBound BestSol Gap | Cuts InLp Confl. | LpIters Time 0 0 0 0.00% 0 inf inf 0 0 0 0 0.0s 0 0 0 0.00% 8125.6 inf inf 0 0 0 4 0.0s R 0 0 0 0.00% 8495 8495 0.00% 5 0 0 5 0.0s Solving report Status Optimal Primal bound 8495 Dual bound 8495 Gap 0% (tolerance: 0.01%) Solution status feasible 8495 (objective) 0 (bound viol.) 0 (int. viol.) 0 (row viol.) Timing 0.02 (total) 0.01 (presolve) 0.00 (postsolve) Nodes 1 LP iterations 5 (total) 0 (strong br.) 1 (separation) 0 (heuristics) Optimal solution found. Intlinprog stopped at the root node because the objective value is within a gap tolerance of the optimal value, options.AbsoluteGapTolerance = 1e-06. The intcon variables are integer within tolerance, options.ConstraintTolerance = 1e-06. ```
```x = 8×1 7.2500 0 0.2500 3.5000 1.0000 1.0000 0 1.0000 ```

Create a nonlinear problem in the problem-based framework.

```x = optimvar('x',2); fun = log(1 + 100*(x(2) - x(1)^2)^2 + (1 - x(1))^2); prob = optimproblem('Objective',fun); mycon = dot(x,x) <= 4; prob.Constraints.mycon = mycon; x0.x = [-1;1.5];```

Convert `prob` to an optimization problem structure. Name the generated objective function file `'logrosenbrock'` and the constraint function file `'circle2'`.

```problem = prob2struct(prob,x0,'ObjectiveFunctionName','logrosenbrock',... 'ConstraintFunctionName','circle2');```

`prob2struct` creates nonlinear objective and constraint function files in the current folder. To create these files in a different folder, use the `'FileLocation'` name-value pair.

Solve the problem.

`[x,fval] = fmincon(problem)`
```Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. ```
```x = 2×1 1.0000 1.0000 ```
```fval = 4.7244e-11 ```

## Input Arguments

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Optimization problem or equation problem, specified as an `OptimizationProblem` object or an `EquationProblem` object. Create an optimization problem by using `optimproblem`; create an equation problem by using `eqnproblem`.

Warning

The problem-based approach does not support complex values in the following: an objective function, nonlinear equalities, and nonlinear inequalities. If a function calculation has a complex value, even as an intermediate value, the final result might be incorrect.

Example: ```prob = optimproblem; prob.Objective = obj; prob.Constraints.cons1 = cons1;```

Example: `prob = eqnproblem; prob.Equations = eqs;`

Initial point, specified as a structure with field names equal to the variable names in `prob`.

For some Global Optimization Toolbox solvers, `x0` can be a vector of `OptimizationValues` objects representing multiple initial points. Create the points using the `optimvalues` function. These solvers are:

For an example using `x0` with named index variables, see Create Initial Point for Optimization with Named Index Variables.

Example: If `prob` has variables named `x` and `y`: `x0.x = [3,2,17]; x0.y = [pi/3,2*pi/3]`.

Data Types: `struct`

### Name-Value Arguments

Specify optional pairs of arguments as `Name1=Value1,...,NameN=ValueN`, where `Name` is the argument name and `Value` is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose `Name` in quotes.

Example: ```problem = prob2struct(prob,'FileLocation','C:\Documents\myproblem')```

Indication to use automatic differentiation (AD) for nonlinear constraint functions, specified as the comma-separated pair consisting of `'ConstraintDerivative'` and `'auto'` (use AD if possible), `'auto-forward'` (use forward AD if possible), `'auto-reverse'` (use reverse AD if possible), or `'finite-differences'` (do not use AD). Choices including `auto` cause the resulting constraint function file to use gradient information when solving the problem provided that the constraint functions are supported, as described in Supported Operations for Optimization Variables and Expressions. For an example, see Supply Derivatives in Problem-Based Workflow

Note

To use automatic derivatives in a problem converted by `prob2struct`, pass options specifying these derivatives.

```options = optimoptions('fmincon','SpecifyObjectiveGradient',true,... 'SpecifyConstraintGradient',true); problem.options = options;```

Example: `'finite-differences'`

Data Types: `char` | `string`

Name of the nonlinear constraint function file created by `prob2struct` for an optimization problem, specified as the comma-separated pair consisting of `'ConstraintFunctionName'` and a file name. This argument applies to `fmincon` or `fminunc` problems; see `problem`. Do not include the file extension `.m` in the file name. `prob2struct` appends the file extension when it creates the file.

If you do not specify `ConstraintFunctionName`, then `prob2struct` overwrites `'generatedConstraints.m'`. If you do not specify `FileLocation`, then `prob2struct` creates the file in the current folder.

The returned `problem` structure refers to this function file.

Example: `"mynlcons"`

Data Types: `char` | `string`

Name of the nonlinear equation function file created by `prob2struct` for an equation problem, specified as the comma-separated pair consisting of `'EquationFunctionName'` and a file name. This argument applies to `fsolve`, `fzero`, or `lsqnonlin` equations; see `problem`. Do not include the file extension `.m` in the file name. `prob2struct` appends the file extension when it creates the file.

If you do not specify `EquationFunctionName`, then `prob2struct` overwrites `'generatedEquation.m'`. If you do not specify `FileLocation`, then `prob2struct` creates the file in the current folder.

The returned `problem` structure refers to this function file.

Example: `"myequation"`

Data Types: `char` | `string`

Location for generated files (objective function, constraint function, and other subfunction files), specified as the comma-separated pair consisting of `'FileLocation'` and a path to a writable folder. All the generated files are stored in this folder; multiple folders are not supported.

Example: `'C:Documents\MATLAB\myproject'`

Data Types: `char` | `string`

Indication to use automatic differentiation (AD) for nonlinear objective function, specified as the comma-separated pair consisting of `'ObjectiveDerivative'` and `'auto'` (use AD if possible), `'auto-forward'` (use forward AD if possible), `'auto-reverse'` (use reverse AD if possible), or `'finite-differences'` (do not use AD). Choices including `auto` cause the resulting objective function file to include derivative information when solving the problem provided that the objective function is supported, as described in Supported Operations for Optimization Variables and Expressions. For an example, see Supply Derivatives in Problem-Based Workflow.

Note

To use automatic derivatives in a problem converted by `prob2struct`, pass options specifying these derivatives.

```options = optimoptions('fmincon','SpecifyObjectiveGradient',true,... 'SpecifyConstraintGradient',true); problem.options = options;```

Example: `'finite-differences'`

Data Types: `char` | `string`

Name of the objective function file created by `prob2struct` for an optimization problem, specified as the comma-separated pair consisting of `'ObjectiveFunctionName'` and a file name. This argument applies to `fmincon` or `fminunc` problems; see `problem`. Do not include the file extension `.m` in the file name. `prob2struct` appends the file extension when it creates the file.

If you do not specify `ObjectiveFunctionName`, then `prob2struct` overwrites `'generatedObjective.m'`. If you do not specify `FileLocation`, then `prob2struct` creates the file in the current folder.

The returned `problem` structure refers to this function file.

Example: `"myobj"`

Data Types: `char` | `string`

Optimization solver, specified as the name of a listed solver. For optimization problems, this table contains the available solvers for each problem type, including solvers from Global Optimization Toolbox. Details for equation problems appear below the optimization solver details.

For converting nonlinear problems with integer constraints using `prob2struct`, the resulting problem structure can depend on the chosen solver. If you do not have a Global Optimization Toolbox license, you must specify the solver. See Integer Constraints in Nonlinear Problem-Based Optimization.

The default solver for each optimization problem type is listed here.

Problem TypeDefault Solver
Linear Programming (LP)`linprog`
Mixed-Integer Linear Programming (MILP)`intlinprog`
Quadratic Programming (QP)`quadprog`
Second-Order Cone Programming (SOCP)`coneprog`
Linear Least Squares`lsqlin`
Nonlinear Least Squares`lsqnonlin`
Nonlinear Programming (NLP)

`fminunc` for problems with no constraints, otherwise `fmincon`

Mixed-Integer Nonlinear Programming (MINLP)`ga` (Global Optimization Toolbox)
Multiobjective`gamultiobj` (Global Optimization Toolbox)

In this table, means the solver is available for the problem type, x means the solver is not available.

Problem Type

LPMILPQPSOCPLinear Least SquaresNonlinear Least SquaresNLPMINLP
Solver
`linprog`

xxxxxxx
`intlinprog`

xxxxxx
`quadprog`

x

xxx
`coneprog`

xx

xxxx
`lsqlin`xxxx

xxx
`lsqnonneg`xxxx

xxx
`lsqnonlin`xxxx

xx
`fminunc`

x

x

x
`fmincon`

x

x
`fminbnd`xxxx

x
`fminsearch`xxxx

x
`patternsearch` (Global Optimization Toolbox)

x

x
`ga` (Global Optimization Toolbox)

`particleswarm` (Global Optimization Toolbox)

x

x

x
`simulannealbnd` (Global Optimization Toolbox)

x

x

x
`surrogateopt` (Global Optimization Toolbox)

`gamultiobj` (Global Optimization Toolbox)

`paretosearch` (Global Optimization Toolbox)

x

x

Note

If you choose `lsqcurvefit` as the solver for a least-squares problem, `solve` uses `lsqnonlin`. The `lsqcurvefit` and `lsqnonlin` solvers are identical for `solve`.

Caution

For maximization problems (`prob.ObjectiveSense` is `"max"` or `"maximize"`), do not specify a least-squares solver (one with a name beginning `lsq`). If you do, `solve` throws an error, because these solvers cannot maximize.

For equation solving, this table contains the available solvers for each problem type. In the table,

• * indicates the default solver for the problem type.

• Y indicates an available solver.

• N indicates an unavailable solver.

Supported Solvers for Equations

Equation Type`lsqlin``lsqnonneg``fzero``fsolve``lsqnonlin`
Linear*NY (scalar only)YY
Linear plus bounds*YNNY
Scalar nonlinearNN*YY
Nonlinear systemNNN*Y
Nonlinear system plus boundsNNNN*

Example: `'intlinprog'`

Data Types: `char` | `string`

## Output Arguments

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Problem structure, returned as an `fmincon` `problem` structure, `fminunc` `problem` structure, `fsolve` `problem` structure, `intlinprog` `problem` structure, `linprog` `problem` structure, `lsqlin` `problem` structure, `lsqnonlin` `problem` structure, `quadprog` `problem` structure, or `ga` `problem` (Global Optimization Toolbox) structure.

The following table gives the resulting default problem type for optimization problems. You can also obtain nondefault problem types. For example, for nonlinear bound-constrained problems, you can select most Global Optimization Toolbox solvers by using the `solver` argument.

Optimization Objective and Constraint Types (Linear Constraints Include Bounds)

Resulting Problem Type

Linear objective and constraint functions.

At least one problem variable has the `'integer'` type.

`intlinprog`

Linear objective and constraint functions.

No problem variable has the `'integer'` type.

`linprog`

Linear constraint functions.

The objective function is a constant plus a sum of squares of linear expressions.

`lsqlin`

Bound constraints.

The objective function is a constant plus a sum of squares of general nonlinear expressions.

`lsqnonlin`

Linear constraint functions.

`quadprog`

General nonlinear objective function.

No constraints.

`fminunc`

General nonlinear objective function, and there is at least one constraint of any type.

Or, there is at least one general nonlinear constraint function.

`fmincon`

Nonlinear objective function or constraint function, and there is at least one integer variable.

`ga`

The following table gives the resulting problem type for equation solving problems.

Equation Types

Resulting Problem Type

Linear system with or without bounds

`lsqlin`

Scalar (single) nonlinear equation

`fzero`

Nonlinear system without constraints

`fsolve`

Nonlinear system with bounds

`lsqnonlin`

Note

For nonlinear problems, `prob2struct` creates function files for the objective and nonlinear constraint functions. For objective and constraint functions that call supporting functions, `prob2struct` also creates supporting function files and stores them in the `FileLocation` folder. To access extra parameters in generated functions, see Obtain Generated Function Details.

For linear and quadratic optimization problems, the problem structure includes an additional field, `f0`, that represents an additive constant for the objective function. If you solve the problem structure using the specified solver, the returned objective function value does not include the `f0` value. If you solve `prob` using the `solve` function, the returned objective function value includes the `f0` value.

If the ObjectiveSense of `prob` is `'max'` or `'maximize'`, then `problem` uses the negative of the objective function in `prob` because solvers minimize. To maximize, they minimize the negative of the original objective function. In this case, the reported optimal function value from the solver is the negative of the value in the original problem. See Maximizing an Objective. You cannot use `lsqlin` for a maximization problem.

## Tips

• If you call `prob2struct` multiple times in the same MATLAB® session for nonlinear problems, use the `ObjectiveFunctionName` or `EquationFunctionName` argument and, if appropriate, the `ConstraintFunctionName` argument. Specifying unique names ensures that the resulting problem structures refer to the correct objective and constraint functions. Otherwise, subsequent calls to `prob2struct` can cause the generated nonlinear function files to overwrite existing files.

• To avoid causing an infinite recursion, do not call `prob2struct` inside an objective or constraint function.

• When calling `prob2struct` in parallel for nonlinear problems, ensure that the resulting objective and constraint function files have unique names. Doing so avoids each pass of the loop writing to the same file or files.

## Algorithms

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### Conversion to Solver Form

The basis for the problem structure is an implicit ordering of all problem variables into a single vector. The order of the problem variables is the same as the order of the `Variables` property in `prob`. See `OptimizationProblem`. You can also find the order by using `varindex`.

For example, suppose that the problem variables are in this order:

• `x` — a 3-by-2-by-4 array

• `y` — a 3-by-2 array

In this case, the implicit variable order is the same as if the problem variable is `vars = [x(:);y(:)]`.

The first 24 elements of `vars` are equivalent to `x(:)`, and the next six elements are equivalent to `y(:)`, for a total of 30 elements. The lower and upper bounds correspond to this variable ordering, and each linear constraint matrix has 30 columns.

For problems with general nonlinear objective or constraint functions, `prob2struct` creates function files in the current folder or in the folder specified by `FileLocation`. The returned `problem` structure refers to these function files.

### Automatic Differentiation

Automatic differentiation (AD) applies to the `solve` and `prob2struct` functions under the following conditions:

When AD AppliesAll Constraint Functions SupportedOne or More Constraints Not Supported
Objective Function SupportedAD used for objective and constraintsAD used for objective only

Note

For linear or quadratic objective or constraint functions, applicable solvers always use explicit function gradients. These gradients are not produced using AD. See Closed Form.

When these conditions are not satisfied, `solve` estimates gradients by finite differences, and `prob2struct` does not create gradients in its generated function files.

Solvers choose the following type of AD by default:

• For a general nonlinear objective function, `fmincon` defaults to reverse AD for the objective function. `fmincon` defaults to reverse AD for the nonlinear constraint function when the number of nonlinear constraints is less than the number of variables. Otherwise, `fmincon` defaults to forward AD for the nonlinear constraint function.

• For a general nonlinear objective function, `fminunc` defaults to reverse AD.

• For a least-squares objective function, `fmincon` and `fminunc` default to forward AD for the objective function. For the definition of a problem-based least-squares objective function, see Write Objective Function for Problem-Based Least Squares.

• `lsqnonlin` defaults to forward AD when the number of elements in the objective vector is greater than or equal to the number of variables. Otherwise, `lsqnonlin` defaults to reverse AD.

• `fsolve` defaults to forward AD when the number of equations is greater than or equal to the number of variables. Otherwise, `fsolve` defaults to reverse AD.

Note

To use automatic derivatives in a problem converted by `prob2struct`, pass options specifying these derivatives.

```options = optimoptions('fmincon','SpecifyObjectiveGradient',true,... 'SpecifyConstraintGradient',true); problem.options = options;```

Currently, AD works only for first derivatives; it does not apply to second or higher derivatives. So, for example, if you want to use an analytic Hessian to speed your optimization, you cannot use `solve` directly, and must instead use the approach described in Supply Derivatives in Problem-Based Workflow.

## Version History

Introduced in R2017b

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