# solve

Solve optimization problem or equation problem

## 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

collapse all

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      0.78         0              -              -
18027      1.24         1   2.906000e+03   4.509804e+01
21859      1.52         2   2.073000e+03   2.270974e+01
23546      1.63         3   1.854000e+03   1.180593e+01
24121      1.67         3   1.854000e+03   1.563342e+00
24294      1.68         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      0.42         2   2.154000e+03   2.593968e+01
5844      0.56         3   1.854000e+03   1.180593e+01
6204      0.59         3   1.854000e+03   1.455526e+00
6400      0.59         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 rounding.
Upper bound is -1027.233133.
Relative gap is 0.00%.

Cut Generation:    Applied 1 mir cut, and 2 strong CG cuts.
Lower 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 left side of the first equation. Because this side is not a polynomial or rational function, process this expression into an optimization expression by using fcn2optimexpr.

ls1 = fcn2optimexpr(@(x)exp(-exp(-(x(1)+x(2)))),x);

Create the first equation.

eq1 = ls1 == x(2)*(1 + x(1)^2);

Similarly, create the left side of the second equation by using fcn2optimexpr.

ls2 = fcn2optimexpr(@(x)x(1)*cos(x(2))+x(2)*sin(x(1)),x);

Create the second equation.

eq2 = ls2 == 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:
arg_LHS == (x(2) .* (1 + x(1).^2))

where:

anonymousFunction1 = @(x)exp(-exp(-(x(1)+x(2))));
arg_LHS = anonymousFunction1(x);

eq2:
arg_LHS == 0.5

where:

anonymousFunction2 = @(x)x(1)*cos(x(2))+x(2)*sin(x(1));
arg_LHS = anonymousFunction2(x);

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: -3.8253e-08

exitflag =
EquationSolved

View the solution point.

disp(sol.x)
0.3532
0.6061

## 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 constraintslinprogintlinprog, quadprog, fmincon, fminunc (fminunc is not recommended because unconstrained linear programs are either constant or unbounded)
Linear objective, linear and integer constraintsintlinproglinprog (integer constraints ignored)
Minimize ||C*x - d||^2 subject to linear constraintslsqlin when the objective is a constant plus a sum of squares of linear expressionsquadprog, lsqnonneg (Constraints other than x >= 0 are ignored for lsqnonneg), fmincon, fminunc (with no constraints)
Minimize ||C*x - d||^2 subject to x >= 0lsqlinquadprog, lsqnonneg
Minimize sum(e(i).^2), where e(i) is an optimization expression, subject to bound constraintslsqnonlin when the objective has the form given in Write Objective Function for Problem-Based Least Squareslsqcurvefit, fmincon, fminunc (with no constraints)
Minimize general nonlinear function f(x)fminuncfmincon
Minimize general nonlinear function f(x) subject to some constraints, or minimize any function subject to nonlinear constraintsfmincon(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 Typelsqlinlsqnonnegfzerofsolvelsqnonlin
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 intlinprogNumeric EquivalentMeaning
OptimalWithPoorFeasibility3

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

IntegerFeasible2intlinprog 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-1intlinprog 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 linprogNumeric EquivalentMeaning
OptimalWithPoorFeasibility3

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

OptimalSolution1

The solver converged to a solution x.

SolverLimitExceeded0

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 lsqlinNumeric EquivalentMeaning
FunctionChangeBelowTolerance3

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)

OptimalSolution1

The solver converged to a solution x.

SolverLimitExceeded0

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.

LocalMinimumFound4

Local minimum found; minimum is not unique.

FunctionChangeBelowTolerance3

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)

OptimalSolution1

The solver converged to a solution x.

SolverLimitExceeded0

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 lsqnonlinNumeric EquivalentMeaning
SearchDirectionTooSmall 4

Magnitude of search direction was smaller than options.StepTolerance.

FunctionChangeBelowTolerance3

Change in the residual was less than options.FunctionTolerance.

StepSizeBelowTolerance

2

Step size smaller than options.StepTolerance.

OptimalSolution1

The solver converged to a solution x.

SolverLimitExceeded0

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 fminuncNumeric EquivalentMeaning
NoDecreaseAlongSearchDirection5

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

FunctionChangeBelowTolerance3

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.

OptimalSolution1

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

SolverLimitExceeded0

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 fminconNumeric EquivalentMeaning
NoDecreaseAlongSearchDirection5

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

SearchDirectionTooSmall4

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

FunctionChangeBelowTolerance3

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.

OptimalSolution1

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

SolverLimitExceeded0

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 fsolveNumeric EquivalentMeaning
SearchDirectionTooSmall4

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

FunctionChangeBelowTolerance3

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.

OptimalSolution1

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

SolverLimitExceeded0

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.

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 fzeroNumeric EquivalentMeaning
OptimalSolution1

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-6Did 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:

• linprog for linear objective and linear constraints

• intlinprog for linear objective and linear constraints and integer constraints

• lsqlin or lsqnonneg for linear least-squares with linear constraints

• lsqcurvefit or lsqnonlin for nonlinear least-squares with bound constraints

• fminunc for problems without any constraints (not even variable bounds) and with a general nonlinear objective function

• fmincon for problems with a nonlinear constraint, or with a general nonlinear objective and at least one constraint

• fzero for a scalar nonlinear equation

• lsqlin for systems of linear equations, with or without bounds

• fsolve for systems of nonlinear equations without constraints

• lsqnonlin for systems of nonlinear equations with bounds

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 Least-Squares, Problem-Based.

## Compatibility Considerations

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Errors starting in R2018b