# solve

Solve optimization problem or equation problem

## Syntax

``sol = solve(prob)``
``sol = solve(prob,x0)``
``sol = solve(___,Name,Value)``
``[sol,fval] = solve(___)``
``[sol,fval,exitflag,output,lambda] = solve(___)``

## Description

example

````sol = solve(prob)` solves the optimization problem or equation problem `prob`.```

example

````sol = solve(prob,x0)` solves `prob` starting from the point `x0`.```

example

````sol = solve(___,Name,Value)` modifies the solution process using one or more name-value pair arguments in addition to the input arguments in previous syntaxes.```
````[sol,fval] = solve(___)` also returns the objective function value at the solution using any of the input arguments in previous syntaxes.```

example

````[sol,fval,exitflag,output,lambda] = solve(___)` also returns an exit flag describing the exit condition, an `output` structure containing additional information about the solution process, and, for non-integer optimization problems, a Lagrange multiplier structure.```

## Examples

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Solve a linear programming problem defined by an optimization problem.

```x = optimvar('x'); y = optimvar('y'); prob = optimproblem; prob.Objective = -x - y/3; prob.Constraints.cons1 = x + y <= 2; prob.Constraints.cons2 = x + y/4 <= 1; prob.Constraints.cons3 = x - y <= 2; prob.Constraints.cons4 = x/4 + y >= -1; prob.Constraints.cons5 = x + y >= 1; prob.Constraints.cons6 = -x + y <= 2; sol = solve(prob)```
```Solving problem using linprog. Optimal solution found. ```
```sol = struct with fields: x: 0.6667 y: 1.3333 ```

Find a minimum of the `peaks` function, which is included in MATLAB®, in the region ${x}^{2}+{y}^{2}\le 4$. To do so, convert the `peaks` function to an optimization expression.

```prob = optimproblem; x = optimvar('x'); y = optimvar('y'); fun = fcn2optimexpr(@peaks,x,y); prob.Objective = fun;```

Include the constraint as an inequality in the optimization variables.

`prob.Constraints = x^2 + y^2 <= 4;`

Set the initial point for `x` to 1 and `y` to –1, and solve the problem.

```x0.x = 1; x0.y = -1; sol = solve(prob,x0)```
```Solving problem using fmincon. 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. ```
```sol = struct with fields: x: 0.2283 y: -1.6255 ```

Compare the number of steps to solve an integer programming problem both with and without an initial feasible point. The problem has eight integer variables and four linear equality constraints, and all variables are restricted to be positive.

```prob = optimproblem; x = optimvar('x',8,1,'LowerBound',0,'Type','integer');```

Create four linear equality constraints and include them in the problem.

```Aeq = [22 13 26 33 21 3 14 26 39 16 22 28 26 30 23 24 18 14 29 27 30 38 26 26 41 26 28 36 18 38 16 26]; beq = [ 7872 10466 11322 12058]; cons = Aeq*x == beq; prob.Constraints.cons = cons;```

Create an objective function and include it in the problem.

```f = [2 10 13 17 7 5 7 3]; prob.Objective = f*x;```

Solve the problem without using an initial point, and examine the display to see the number of branch-and-bound nodes.

`[x1,fval1,exitflag1,output1] = solve(prob);`
```Solving problem using intlinprog. LP: Optimal objective value is 1554.047531. Cut Generation: Applied 8 strong CG cuts. Lower bound is 1591.000000. Branch and Bound: nodes total num int integer relative explored time (s) solution fval gap (%) 10000 1.06 0 - - 18027 1.81 1 2.906000e+03 4.509804e+01 21859 2.26 2 2.073000e+03 2.270974e+01 23546 2.39 3 1.854000e+03 1.180593e+01 24121 2.44 3 1.854000e+03 1.563342e+00 24294 2.45 3 1.854000e+03 0.000000e+00 Optimal solution found. Intlinprog stopped because the objective value is within a gap tolerance of the optimal value, options.AbsoluteGapTolerance = 0 (the default value). The intcon variables are integer within tolerance, options.IntegerTolerance = 1e-05 (the default value). ```

For comparison, find the solution using an initial feasible point.

```x0.x = [8 62 23 103 53 84 46 34]'; [x2,fval2,exitflag2,output2] = solve(prob,x0);```
```Solving problem using intlinprog. LP: Optimal objective value is 1554.047531. Cut Generation: Applied 8 strong CG cuts. Lower bound is 1591.000000. Relative gap is 59.20%. Branch and Bound: nodes total num int integer relative explored time (s) solution fval gap (%) 3627 0.56 2 2.154000e+03 2.593968e+01 5844 0.72 3 1.854000e+03 1.180593e+01 6204 0.75 3 1.854000e+03 1.455526e+00 6400 0.77 3 1.854000e+03 0.000000e+00 Optimal solution found. Intlinprog stopped because the objective value is within a gap tolerance of the optimal value, options.AbsoluteGapTolerance = 0 (the default value). The intcon variables are integer within tolerance, options.IntegerTolerance = 1e-05 (the default value). ```
`fprintf('Without an initial point, solve took %d steps.\nWith an initial point, solve took %d steps.',output1.numnodes,output2.numnodes)`
```Without an initial point, solve took 24294 steps. With an initial point, solve took 6400 steps. ```

Giving an initial point does not always improve the problem. For this problem, using an initial point saves time and computational steps. However, for some problems, an initial point can cause `solve` to take more steps.

Solve the problem

`$\underset{x}{\mathrm{min}}\left(-3{x}_{1}-2{x}_{2}-{x}_{3}\right)\phantom{\rule{0.2777777777777778em}{0ex}}subject\phantom{\rule{0.2777777777777778em}{0ex}}to\left\{\begin{array}{l}{x}_{3}\phantom{\rule{0.2777777777777778em}{0ex}}binary\\ {x}_{1},{x}_{2}\ge 0\\ {x}_{1}+{x}_{2}+{x}_{3}\le 7\\ 4{x}_{1}+2{x}_{2}+{x}_{3}=12\end{array}$`

without showing iterative display.

```x = optimvar('x',2,1,'LowerBound',0); x3 = optimvar('x3','Type','integer','LowerBound',0,'UpperBound',1); prob = optimproblem; prob.Objective = -3*x(1) - 2*x(2) - x3; prob.Constraints.cons1 = x(1) + x(2) + x3 <= 7; prob.Constraints.cons2 = 4*x(1) + 2*x(2) + x3 == 12; options = optimoptions('intlinprog','Display','off'); sol = solve(prob,'Options',options)```
```sol = struct with fields: x: [2x1 double] x3: 1 ```

Examine the solution.

`sol.x`
```ans = 2×1 0 5.5000 ```
`sol.x3`
```ans = 1 ```

Force `solve` to use `intlinprog` as the solver for a linear programming problem.

```x = optimvar('x'); y = optimvar('y'); prob = optimproblem; prob.Objective = -x - y/3; prob.Constraints.cons1 = x + y <= 2; prob.Constraints.cons2 = x + y/4 <= 1; prob.Constraints.cons3 = x - y <= 2; prob.Constraints.cons4 = x/4 + y >= -1; prob.Constraints.cons5 = x + y >= 1; prob.Constraints.cons6 = -x + y <= 2; sol = solve(prob,'Solver', 'intlinprog')```
```Solving problem using intlinprog. LP: Optimal objective value is -1.111111. Optimal solution found. No integer variables specified. Intlinprog solved the linear problem. ```
```sol = struct with fields: x: 0.6667 y: 1.3333 ```

Solve the mixed-integer linear programming problem described in Solve Integer Programming Problem with Nondefault Options and examine all of the output data.

```x = optimvar('x',2,1,'LowerBound',0); x3 = optimvar('x3','Type','integer','LowerBound',0,'UpperBound',1); prob = optimproblem; prob.Objective = -3*x(1) - 2*x(2) - x3; prob.Constraints.cons1 = x(1) + x(2) + x3 <= 7; prob.Constraints.cons2 = 4*x(1) + 2*x(2) + x3 == 12; [sol,fval,exitflag,output] = solve(prob)```
```Solving problem using intlinprog. LP: Optimal objective value is -12.000000. Optimal solution found. Intlinprog stopped at the root node because the objective value is within a gap tolerance of the optimal value, options.AbsoluteGapTolerance = 0 (the default value). The intcon variables are integer within tolerance, options.IntegerTolerance = 1e-05 (the default value). ```
```sol = struct with fields: x: [2x1 double] x3: 1 ```
```fval = -12 ```
```exitflag = OptimalSolution ```
```output = struct with fields: relativegap: 0 absolutegap: 0 numfeaspoints: 1 numnodes: 0 constrviolation: 0 message: 'Optimal solution found....' solver: 'intlinprog' ```

For a problem without any integer constraints, you can also obtain a nonempty Lagrange multiplier structure as the fifth output.

Create and solve an optimization problem using named index variables. The problem is to maximize the profit-weighted flow of fruit to various airports, subject to constraints on the weighted flows.

```rng(0) % For reproducibility p = optimproblem('ObjectiveSense', 'maximize'); flow = optimvar('flow', ... {'apples', 'oranges', 'bananas', 'berries'}, {'NYC', 'BOS', 'LAX'}, ... 'LowerBound',0,'Type','integer'); p.Objective = sum(sum(rand(4,3).*flow)); p.Constraints.NYC = rand(1,4)*flow(:,'NYC') <= 10; p.Constraints.BOS = rand(1,4)*flow(:,'BOS') <= 12; p.Constraints.LAX = rand(1,4)*flow(:,'LAX') <= 35; sol = solve(p);```
```Solving problem using intlinprog. LP: Optimal objective value is -1027.472366. Heuristics: Found 1 solution using ZI round. Upper bound is -1027.233133. Relative gap is 0.00%. Optimal solution found. Intlinprog stopped at the root node because the objective value is within a gap tolerance of the optimal value, options.AbsoluteGapTolerance = 0 (the default value). The intcon variables are integer within tolerance, options.IntegerTolerance = 1e-05 (the default value). ```

Find the optimal flow of oranges and berries to New York and Los Angeles.

`[idxFruit,idxAirports] = findindex(flow, {'oranges','berries'}, {'NYC', 'LAX'})`
```idxFruit = 1×2 2 4 ```
```idxAirports = 1×2 1 3 ```
`orangeBerries = sol.flow(idxFruit, idxAirports)`
```orangeBerries = 2×2 0 980.0000 70.0000 0 ```

This display means that no oranges are going to `NYC`, 70 berries are going to `NYC`, 980 oranges are going to `LAX`, and no berries are going to `LAX`.

List the optimal flow of the following:

`Fruit Airports`

` ----- --------`

` Berries NYC`

` Apples BOS`

` Oranges LAX`

`idx = findindex(flow, {'berries', 'apples', 'oranges'}, {'NYC', 'BOS', 'LAX'})`
```idx = 1×3 4 5 10 ```
`optimalFlow = sol.flow(idx)`
```optimalFlow = 1×3 70.0000 28.0000 980.0000 ```

This display means that 70 berries are going to `NYC`, 28 apples are going to `BOS`, and 980 oranges are going to `LAX`.

To solve the nonlinear system of equations

`$\begin{array}{l}\mathrm{exp}\left(-\mathrm{exp}\left(-\left({x}_{1}+{x}_{2}\right)\right)\right)={x}_{2}\left(1+{x}_{1}^{2}\right)\\ {x}_{1}\mathrm{cos}\left({x}_{2}\right)+{x}_{2}\mathrm{sin}\left({x}_{1}\right)=\frac{1}{2}\end{array}$`

using the problem-based approach, first define `x` as a two-element optimization variable.

`x = optimvar('x',2);`

Create the first equation as an optimization equality expression.

`eq1 = exp(-exp(-(x(1) + x(2)))) == x(2)*(1 + x(1)^2);`

Similarly, create the second equation as an optimization equality expression.

`eq2 = x(1)*cos(x(2)) + x(2)*sin(x(1)) == 1/2;`

Create an equation problem, and place the equations in the problem.

```prob = eqnproblem; prob.Equations.eq1 = eq1; prob.Equations.eq2 = eq2;```

Review the problem.

`show(prob)`
``` EquationProblem : Solve for: x eq1: exp(-exp(-(x(1) + x(2)))) == (x(2) .* (1 + x(1).^2)) eq2: ((x(1) .* cos(x(2))) + (x(2) .* sin(x(1)))) == 0.5 ```

Solve the problem starting from the point `[0,0]`. For the problem-based approach, specify the initial point as a structure, with the variable names as the fields of the structure. For this problem, there is only one variable, `x`.

```x0.x = [0 0]; [sol,fval,exitflag] = solve(prob,x0)```
```Solving problem using fsolve. Equation solved. fsolve completed because the vector of function values is near zero as measured by the value of the function tolerance, and the problem appears regular as measured by the gradient. ```
```sol = struct with fields: x: [2x1 double] ```
```fval = struct with fields: eq1: -2.4069e-07 eq2: -3.8255e-08 ```
```exitflag = EquationSolved ```

View the solution point.

`disp(sol.x)`
``` 0.3532 0.6061 ```

Unsupported Functions Require `fcn2optimexpr`

If your equation functions are not composed of elementary functions, you must convert the functions to optimization expressions using `fcn2optimexpr`. For the present example:

```ls1 = fcn2optimexpr(@(x)exp(-exp(-(x(1)+x(2)))),x); eq1 = ls1 == x(2)*(1 + x(1)^2); ls2 = fcn2optimexpr(@(x)x(1)*cos(x(2))+x(2)*sin(x(1)),x); eq2 = ls2 == 1/2;```

For the list of supported functions, see Supported Operations on Optimization Variables and Expressions.

## 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 an objective function, nonlinear equalities, or nonlinear inequalities. If a function calculation has a complex value, even as an intermediate value, the final result can 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 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 Pair Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside quotes. You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Example: `solve(prob,'options',opts)`

Optimization options, specified as the comma-separated pair consisting of `'options'` and an object created by `optimoptions` or an options structure such as created by `optimset`.

Internally, the `solve` function calls a relevant solver as detailed in the `'solver'` argument reference. Ensure that `options` is compatible with the solver. For example, `intlinprog` does not allow options to be a structure, and `lsqnonneg` does not allow options to be an object.

For suggestions on options settings to improve an `intlinprog` solution or the speed of a solution, see Tuning Integer Linear Programming. For `linprog`, the default `'dual-simplex'` algorithm is generally memory-efficient and speedy. Occasionally, `linprog` solves a large problem faster when the `Algorithm` option is `'interior-point'`. For suggestions on options settings to improve a nonlinear problem's solution, see Options in Common Use: Tuning and Troubleshooting and Improve Results.

Example: ```options = optimoptions('intlinprog','Display','none')```

Optimization solver, specified as the comma-separated pair consisting of `'solver'` and the name of a listed solver. For optimization problems, this table contains the available solvers for each problem type.

Problem TypeDefault SolverOther Allowed Solvers
Linear objective, linear constraints`linprog``intlinprog`, `quadprog`, `fmincon`, `fminunc` (`fminunc` is not recommended because unconstrained linear programs are either constant or unbounded)
Linear objective, linear and integer constraints`intlinprog``linprog` (integer constraints ignored)
Quadratic objective, linear constraints`quadprog``fmincon`, `fminunc` (with no constraints)
Minimize ||C*x - d||^2 subject to linear constraints`lsqlin` when the objective is a constant plus a sum of squares of linear expressions`quadprog`, `lsqnonneg` (Constraints other than x >= 0 are ignored for `lsqnonneg`), `fmincon`, `fminunc` (with no constraints)
Minimize ||C*x - d||^2 subject to x >= 0`lsqlin``quadprog`, `lsqnonneg`
Minimize `sum(e(i).^2)`, where `e(i)` is an optimization expression, subject to bound constraints`lsqnonlin` when the objective has the form given in Write Objective Function for Problem-Based Least Squares`lsqcurvefit`, `fmincon`, `fminunc` (with no constraints)
Minimize general nonlinear function f(x)`fminunc``fmincon`
Minimize general nonlinear function f(x) subject to some constraints, or minimize any function subject to nonlinear constraints`fmincon`(none)

### 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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Solution, returned as a structure. The fields of the structure are the names of the optimization variables. See `optimvar`.

Objective function value at the solution, returned as a real number, or, for systems of equations, a real vector. For least-squares problems, `fval` is the sum of squares of the residuals at the solution. For equation-solving problems, `fval` is the function value at the solution, meaning the left-hand side minus the right-hand side of the equations.

### Tip

If you neglect to ask for `fval` for an optimization problem, you can calculate it using:

`fval = evaluate(prob.Objective,sol)`

Reason the solver stopped, returned as an enumeration variable. You can convert `exitflag` to its numeric equivalent using `double(exitflag)`, and to its string equivalent using `string(exitflag)`.

This table describes the exit flags for the `intlinprog` solver.

Exit Flag for `intlinprog`Numeric EquivalentMeaning
`OptimalWithPoorFeasibility``3`

The solution is feasible with respect to the relative `ConstraintTolerance` tolerance, but is not feasible with respect to the absolute tolerance.

`IntegerFeasible`2`intlinprog` stopped prematurely, and found an integer feasible point.
`OptimalSolution`

`1`

The solver converged to a solution `x`.

`SolverLimitExceeded`

`0`

`intlinprog` exceeds one of the following tolerances:

• `LPMaxIterations`

• `MaxNodes`

• `MaxTime`

• `RootLPMaxIterations`

See Tolerances and Stopping Criteria. `solve` also returns this exit flag when it runs out of memory at the root node.

`OutputFcnStop``-1``intlinprog` stopped by an output function or plot function.
`NoFeasiblePointFound`

`-2`

No feasible point found.

`Unbounded`

`-3`

The problem is unbounded.

`FeasibilityLost`

`-9`

Solver lost feasibility.

Exitflags `3` and `-9` relate to solutions that have large infeasibilities. These usually arise from linear constraint matrices that have large condition number, or problems that have large solution components. To correct these issues, try to scale the coefficient matrices, eliminate redundant linear constraints, or give tighter bounds on the variables.

This table describes the exit flags for the `linprog` solver.

Exit Flag for `linprog`Numeric EquivalentMeaning
`OptimalWithPoorFeasibility``3`

The solution is feasible with respect to the relative `ConstraintTolerance` tolerance, but is not feasible with respect to the absolute tolerance.

`OptimalSolution``1`

The solver converged to a solution `x`.

`SolverLimitExceeded``0`

The number of iterations exceeds `options.MaxIterations`.

`NoFeasiblePointFound``-2`

No feasible point found.

`Unbounded``-3`

The problem is unbounded.

`FoundNaN``-4`

`NaN` value encountered during execution of the algorithm.

`PrimalDualInfeasible``-5`

Both primal and dual problems are infeasible.

`DirectionTooSmall``-7`

The search direction is too small. No further progress can be made.

`FeasibilityLost``-9`

Solver lost feasibility.

Exitflags `3` and `-9` relate to solutions that have large infeasibilities. These usually arise from linear constraint matrices that have large condition number, or problems that have large solution components. To correct these issues, try to scale the coefficient matrices, eliminate redundant linear constraints, or give tighter bounds on the variables.

This table describes the exit flags for the `lsqlin` solver.

Exit Flag for `lsqlin`Numeric EquivalentMeaning
`FunctionChangeBelowTolerance``3`

Change in the residual is smaller than the specified tolerance `options.FunctionTolerance`. (`trust-region-reflective` algorithm)

`StepSizeBelowTolerance`

`2`

Step size smaller than `options.StepTolerance`, constraints satisfied. (`interior-point` algorithm)

`OptimalSolution``1`

The solver converged to a solution `x`.

`SolverLimitExceeded``0`

The number of iterations exceeds `options.MaxIterations`.

`NoFeasiblePointFound``-2`

For optimization problems, the problem is infeasible. Or, for the `interior-point` algorithm, step size smaller than `options.StepTolerance`, but constraints are not satisfied.

For equation problems, no solution found.

`IllConditioned``-4`

Ill-conditioning prevents further optimization.

`NoDescentDirectionFound``-8`

The search direction is too small. No further progress can be made. (`interior-point` algorithm)

This table describes the exit flags for the `quadprog` solver.

Exit Flag for `quadprog`Numeric EquivalentMeaning
`LocalMinimumFound``4`

Local minimum found; minimum is not unique.

`FunctionChangeBelowTolerance``3`

Change in the objective function value is smaller than the specified tolerance `options.FunctionTolerance`. (`trust-region-reflective` algorithm)

`StepSizeBelowTolerance`

`2`

Step size smaller than `options.StepTolerance`, constraints satisfied. (`interior-point-convex` algorithm)

`OptimalSolution``1`

The solver converged to a solution `x`.

`SolverLimitExceeded``0`

The number of iterations exceeds `options.MaxIterations`.

`NoFeasiblePointFound``-2`

The problem is infeasible. Or, for the `interior-point` algorithm, step size smaller than `options.StepTolerance`, but constraints are not satisfied.

`IllConditioned``-4`

Ill-conditioning prevents further optimization.

`Nonconvex`

`-6`

Nonconvex problem detected. (`interior-point-convex` algorithm)

`NoDescentDirectionFound``-8`

Unable to compute a step direction. (`interior-point-convex` algorithm)

This table describes the exit flags for the `lsqcurvefit` or `lsqnonlin` solver.

Exit Flag for `lsqnonlin`Numeric EquivalentMeaning
`SearchDirectionTooSmall ``4`

Magnitude of search direction was smaller than `options.StepTolerance`.

`FunctionChangeBelowTolerance``3`

Change in the residual was less than `options.FunctionTolerance`.

`StepSizeBelowTolerance`

`2`

Step size smaller than `options.StepTolerance`.

`OptimalSolution``1`

The solver converged to a solution `x`.

`SolverLimitExceeded``0`

Number of iterations exceeded `options.MaxIterations` or number of function evaluations exceeded `options.MaxFunctionEvaluations`.

`OutputFcnStop``-1`

Stopped by an output function or plot function.

`NoFeasiblePointFound``-2`

For optimization problems, problem is infeasible: the bounds `lb` and `ub` are inconsistent.

For equation problems, no solution found.

This table describes the exit flags for the `fminunc` solver.

Exit Flag for `fminunc`Numeric EquivalentMeaning
`NoDecreaseAlongSearchDirection``5`

Predicted decrease in the objective function is less than the `options.FunctionTolerance` tolerance.

`FunctionChangeBelowTolerance``3`

Change in the objective function value is less than the `options.FunctionTolerance` tolerance.

`StepSizeBelowTolerance`

`2`

Change in `x` is smaller than the `options.StepTolerance` tolerance.

`OptimalSolution``1`

Magnitude of gradient is smaller than the `options.OptimalityTolerance` tolerance.

`SolverLimitExceeded``0`

Number of iterations exceeds `options.MaxIterations` or number of function evaluations exceeds `options.MaxFunctionEvaluations`.

`OutputFcnStop``-1`

Stopped by an output function or plot function.

`Unbounded``-3`

Objective function at current iteration is below `options.ObjectiveLimit`.

This table describes the exit flags for the `fmincon` solver.

Exit Flag for `fmincon`Numeric EquivalentMeaning
`NoDecreaseAlongSearchDirection``5`

Magnitude of directional derivative in search direction is less than 2*`options.OptimalityTolerance` and maximum constraint violation is less than `options.ConstraintTolerance`.

`SearchDirectionTooSmall``4`

Magnitude of the search direction is less than 2*`options.StepTolerance` and maximum constraint violation is less than `options.ConstraintTolerance`.

`FunctionChangeBelowTolerance``3`

Change in the objective function value is less than `options.FunctionTolerance` and maximum constraint violation is less than `options.ConstraintTolerance`.

`StepSizeBelowTolerance`

`2`

Change in `x` is less than `options.StepTolerance` and maximum constraint violation is less than `options.ConstraintTolerance`.

`OptimalSolution``1`

First-order optimality measure is less than `options.OptimalityTolerance`, and maximum constraint violation is less than `options.ConstraintTolerance`.

`SolverLimitExceeded``0`

Number of iterations exceeds `options.MaxIterations` or number of function evaluations exceeds `options.MaxFunctionEvaluations`.

`OutputFcnStop``-1`

Stopped by an output function or plot function.

`NoFeasiblePointFound``-2`

No feasible point found.

`Unbounded``-3`

Objective function at current iteration is below `options.ObjectiveLimit` and maximum constraint violation is less than `options.ConstraintTolerance`.

This table describes the exit flags for the `fsolve` solver.

Exit Flag for `fsolve`Numeric EquivalentMeaning
`SearchDirectionTooSmall``4`

Magnitude of the search direction is less than `options.StepTolerance`, equation solved.

`FunctionChangeBelowTolerance``3`

Change in the objective function value is less than `options.FunctionTolerance`, equation solved.

`StepSizeBelowTolerance`

`2`

Change in `x` is less than `options.StepTolerance`, equation solved.

`OptimalSolution``1`

First-order optimality measure is less than `options.OptimalityTolerance`, equation solved.

`SolverLimitExceeded``0`

Number of iterations exceeds `options.MaxIterations` or number of function evaluations exceeds `options.MaxFunctionEvaluations`.

`OutputFcnStop``-1`

Stopped by an output function or plot function.

`NoFeasiblePointFound``-2`

Converged to a point that is not a root.

`TrustRegionRadiusTooSmall``-3`

Equation not solved. Trust region radius became too small (`trust-region-dogleg` algorithm).

This table describes the exit flags for the `fzero` solver.

Exit Flag for `fzero`Numeric EquivalentMeaning
`OptimalSolution``1`

Equation solved.

`OutputFcnStop``-1`

Stopped by an output function or plot function.

`FoundNaNInfOrComplex``-4`

`NaN`, `Inf`, or complex value encountered during search for an interval containing a sign change.

`SingularPoint``-5`

Might have converged to a singular point.

`CannotDetectSignChange``-6`Did not find two points with opposite signs of function value.

Information about the optimization process, returned as a structure. The output structure contains the fields in the relevant underlying solver output field, depending on which solver `solve` called:

`solve` includes the additional field `Solver` in the `output` structure to identify the solver used, such as `'intlinprog'`.

Lagrange multipliers at the solution, returned as a structure.

### Note

`solve` does not return `lambda` for equation-solving problems.

For the `intlinprog` and `fminunc` solvers, `lambda` is empty, `[]`. For the other solvers, `lambda` has these fields:

• `Variables` – Contains fields for each problem variable. Each problem variable name is a structure with two fields:

• `Lower` – Lagrange multipliers associated with the variable `LowerBound` property, returned as an array of the same size as the variable. Nonzero entries mean that the solution is at the lower bound. These multipliers are in the structure `lambda.Variables.variablename.Lower`.

• `Upper` – Lagrange multipliers associated with the variable `UpperBound` property, returned as an array of the same size as the variable. Nonzero entries mean that the solution is at the upper bound. These multipliers are in the structure `lambda.Variables.variablename.Upper`.

• `Constraints` – Contains a field for each problem constraint. Each problem constraint is in a structure whose name is the constraint name, and whose value is a numeric array of the same size as the constraint. Nonzero entries mean that the constraint is active at the solution. These multipliers are in the structure `lambda.Constraints.constraintname`.

### Note

Elements of a constraint array all have the same comparison (`<=`, `==`, or `>=`) and are all of the same type (linear, quadratic, or nonlinear).

## Algorithms

Internally, the `solve` function solves optimization problems by calling a solver:

Before `solve` can call these functions, the problems must be converted to solver form, either by `solve` or some other associated functions or objects. This conversion entails, for example, linear constraints having a matrix representation rather than an optimization variable expression.

The first step in the algorithm occurs as you place optimization expressions into the problem. An `OptimizationProblem` object has an internal list of the variables used in its expressions. Each variable has a linear index in the expression, and a size. Therefore, the problem variables have an implied matrix form. The `prob2struct` function performs the conversion from problem form to solver form. For an example, see Convert Problem to Structure.

For the default and allowed solvers that `solve` calls, depending on the problem objective and constraints, see `'solver'`. You can override the default by using the `'solver'` name-value pair argument when calling `solve`.

For the algorithm that `intlinprog` uses to solve MILP problems, see intlinprog Algorithm. For the algorithms that `linprog` uses to solve linear programming problems, see Linear Programming Algorithms. For the algorithms that `quadprog` uses to solve quadratic programming problems, see Quadratic Programming Algorithms. For linear or nonlinear least-squares solver algorithms, see Least-Squares (Model Fitting) Algorithms. For nonlinear solver algorithms, see Unconstrained Nonlinear Optimization Algorithms and Constrained Nonlinear Optimization Algorithms.

For nonlinear equation solving, `solve` internally represents each equation as the difference between the left and right sides. Then `solve` attempts to minimize the sum of squares of the equation components. For the algorithms for solving nonlinear systems of equations, see Equation Solving Algorithms. When the problem also has bounds, `solve` calls `lsqnonlin` to minimize the sum of squares of equation components. See Least-Squares (Model Fitting) Algorithms.

### Note

If your objective function is a sum of squares, and you want `solve` to recognize it as such, write it as `sum(expr.^2)`, and not as `expr'*expr` or any other form. The internal parser recognizes only explicit sums of squares. For details, see Write Objective Function for Problem-Based Least Squares. For an example, see Nonnegative Linear Least Squares, Problem-Based.

## Compatibility Considerations

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Errors starting in R2018b