Documentation

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

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

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.

[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

Solve Linear Programming Problem

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

Solve Nonlinear Programming Problem Using Problem-Based Approach

Find a minimum of the peaks function, which is included in MATLAB®, in the region $x^{2} + y^{2} \leq 4$ . To do so, create optimization variables x and y.

x = optimvar('x');
y = optimvar('y');

Create an optimization problem having peaks as the objective function.

prob = optimproblem("Objective",peaks(x,y));

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

Unsupported Functions Require fcn2optimexpr

If your objective or nonlinear constraint functions are not entirely composed of elementary functions, you must convert the functions to optimization expressions using fcn2optimexpr. See Convert Nonlinear Function to Optimization Expression and Supported Operations on Optimization Variables and Expressions.

To convert the present example:

convpeaks = fcn2optimexpr(@peaks,x,y);
prob.Objective = convpeaks;
sol2 = 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.

sol2 = struct with fields:
    x: 0.2283
    y: -1.6255

Solve Mixed-Integer Linear Program Starting from Initial Point

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      0.76         0              -              -                                          
   18027      1.40         1   2.906000e+03   4.509804e+01                                          
   21859      1.80         2   2.073000e+03   2.270974e+01                                          
   23546      1.96         3   1.854000e+03   1.180593e+01                                          
   24121      2.01         3   1.854000e+03   1.563342e+00                                          
   24294      2.02         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      0.39         2   2.154000e+03   2.593968e+01                                          
    5844      0.60         3   1.854000e+03   1.180593e+01                                          
    6204      0.64         3   1.854000e+03   1.455526e+00                                          
    6400      0.65         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 Integer Programming Problem with Nondefault Options

Solve the problem

$\min_{x} (- 3 x_{1} - 2 x_{2} - x_{3}) s u b j e c t t o {\begin{array}{llllllllllllllllllll} x_{3} b i n a r y \\ x_{1}, x_{2} \geq 0 \\ x_{1} + x_{2} + x_{3} \leq 7 \\ 4 x_{1} + 2 x_{2} + x_{3} = 12 \end{array}$

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

sol.x3

ans = 1

Use `intlinprog` to Solve a Linear Program

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

Return All Outputs

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.

View Solution with Index Variables

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.

Solve Nonlinear System of Equations, Problem-Based

See Supported Operations on Optimization Variables and Expressions and Convert Nonlinear Function to Optimization Expression.

To solve the nonlinear system of equations

$\begin{array}{l} \exp (- \exp (- (x_{1} + x_{2}))) = x_{2} (1 + x_{1}^{2}) \\ x_{1} \cos (x_{2}) + x_{2} \sin (x_{1}) = \frac{1}{2} \end{array}$

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: -2.4070e-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;

Input Arguments

`prob` — Optimization problem or equation problem
`OptimizationProblem` object | `EquationProblem` object

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;

`x0` — Initial point
structure

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)

`'options'` — Optimization options
object created by `optimoptions` | options structure

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')

`'solver'` — Optimization solver
`'intlinprog'` | `'linprog'` | `'lsqlin'` | `'lsqcurvefit'` | `'lsqnonlin'` | `'lsqnonneg'` | `'quadprog'` | `'fminunc'` | `'fmincon'` | `'fzero'` | `'fsolve'`

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 Type	Default Solver	Other 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)
Linear objective, optional linear constraints, and cone constraints of the form `norm(linear expression) + constant <= linear expression` or `sqrt(sum of squares) + constant <= linear expression`	`coneprog`	`fmincon`
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	*	N	Y (scalar only)	Y	Y
Linear plus bounds	*	Y	N	N	Y
Scalar nonlinear	N	N	*	Y	Y
Nonlinear system	N	N	N	*	Y
Nonlinear system plus bounds	N	N	N	N	*

Example: 'intlinprog'

Data Types: char | string

`'ObjectiveDerivative'` — Indication to use automatic differentiation for objective function
`'auto'` (default) | `'auto-forward'` | `'auto-reverse'` | `'finite-differences'`

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 underlying solver to use gradient information when solving the problem provided that the objective function is supported, as described in Supported Operations on Optimization Variables and Expressions. For an example, see Effect of Automatic Differentiation in Problem-Based Optimization.

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.

Example: 'finite-differences'

Data Types: char | string

`'ConstraintDerivative'` — Indication to use automatic differentiation for constraint functions
`'auto'` (default) | `'auto-forward'` | `'auto-reverse'` | `'finite-differences'`

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 underlying solver to use gradient information when solving the problem provided that the constraint functions are supported, as described in Supported Operations on Optimization Variables and Expressions. For an example, see Effect of Automatic Differentiation in Problem-Based Optimization.

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.

Example: 'finite-differences'

Data Types: char | string

`'EquationDerivative'` — Indication to use automatic differentiation for equations
`'auto'` (default) | `'auto-forward'` | `'auto-reverse'` | `'finite-differences'`

Indication to use automatic differentiation (AD) for nonlinear constraint functions, specified as the comma-separated pair consisting of 'EquationDerivative' 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 underlying solver to use gradient information when solving the problem provided that the equation functions are supported, as described in Supported Operations on Optimization Variables and Expressions. For an example, see Effect of Automatic Differentiation in Problem-Based Optimization.

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.

Example: 'finite-differences'

Data Types: char | string

Output Arguments

`sol` — Solution
structure

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

`fval` — Objective function value at the solution
real number | real vector

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)

`exitflag` — Reason solver stopped
enumeration variable

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 Equivalent	Meaning
`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 Equivalent	Meaning
`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.

This table describes the exit flags for the lsqlin solver.

Exit Flag for `lsqlin`	Numeric Equivalent	Meaning
`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 Equivalent	Meaning
`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 coneprog solver.

Exit Flag for `coneprog`	Numeric Equivalent	Meaning
`OptimalSolution`	`1`	The solver converged to a solution `x`.
`SolverLimitExceeded`	`0`	The number of iterations exceeds `options.MaxIterations`, or the solution time in seconds exceeded `options.MaxTime`.
`NoFeasiblePointFound`	`-2`	The problem is infeasible.
`Unbounded`	`-3`	The problem is unbounded.
`DirectionTooSmall`	`-7`	The search direction became too small. No further progress could be made.
`Unstable`	`-10`	The problem is numerically unstable.

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

Exit Flag for `lsqnonlin`	Numeric Equivalent	Meaning
`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 Equivalent	Meaning
`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 Equivalent	Meaning
`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 Equivalent	Meaning
`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 Equivalent	Meaning
`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.

`output` — Information about optimization process
structure

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:

'fmincon' output
'fminunc' output
'fsolve' output
'fzero' output
'intlinprog' output
'linprog' output
'lsqcurvefit' or 'lsqnonlin' output
'lsqlin' output
'lsqnonneg' output
'quadprog' output

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

When Solver is a nonlinear solver, solve includes one or two extra fields describing the derivative estimation type. The objectivederivative and, if appropriate, constraintderivative fields can take the following values:

"reverse-AD" for reverse automatic differentiation
"forward-AD" for forward automatic differentiation
"finite-differences" for finite difference estimation
"closed-form" for linear or quadratic functions

`lambda` — Lagrange multipliers at the solution
structure

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