# solve

Solve optimization problem or equation problem

## Description

Use solve to find the solution of an optimization problem or equation problem.

example

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

example

sol = solve(prob,x0) solves prob starting from the point or set of values x0.

example

sol = solve(prob,x0,ms) solves prob using the ms multiple-start solver. Use this syntax to search for a better solution than you obtain when not using the ms argument.

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, 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 for 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

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      0.42         0              -              -
18025      0.71         1   2.906000e+03   4.509804e+01
21857      0.90         2   2.073000e+03   2.270974e+01
23544      0.97         3   1.854000e+03   1.180593e+01
24097      1.00         3   1.854000e+03   1.617251e+00
24293      1.01         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      0.22         2   2.154000e+03   2.593968e+01
5844      0.32         3   1.854000e+03   1.180593e+01
6204      0.34         3   1.854000e+03   1.455526e+00
6400      0.35         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 24293 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.

For some solvers, you can pass the objective and constraint function values, if any, to solve in the x0 argument. This can save time in the solver. Pass a vector of OptimizationValues objects. Create this vector using the optimvalues function.

The solvers that can use the objective function values are:

• ga

• gamultiobj

• paretosearch

• surrogateopt

The solvers that can use nonlinear constraint function values are:

• paretosearch

• surrogateopt

For example, minimize the peaks function using surrogateopt, starting with values from a grid of initial points. Create a grid from -10 to 10 in the x variable, and –5/2 to 5/2 in the y variable with spacing 1/2. Compute the objective function values at the initial points.

x = optimvar("x",LowerBound=-10,UpperBound=10);
y = optimvar("y",LowerBound=-5/2,UpperBound=5/2);
prob = optimproblem("Objective",peaks(x,y));
xval = -10:10;
yval = (-5:5)/2;
[x0x,x0y] = meshgrid(xval,yval);
peaksvals = peaks(x0x,x0y);

Pass the values in the x0 argument by using optimvalues. This saves time for solve, as solve does not need to compute the values. Pass the values as row vectors.

x0 = optimvalues(prob,'x',x0x(:)','y',x0y(:)',...
"Objective",peaksvals(:)');

Solve the problem using surrogateopt with the initial values.

[sol,fval,eflag,output] = solve(prob,x0,Solver="surrogateopt")
Solving problem using surrogateopt.

surrogateopt stopped because it exceeded the function evaluation limit set by
'options.MaxFunctionEvaluations'.
sol = struct with fields:
x: 0.2283
y: -1.6256

fval = -6.5511
eflag =
SolverLimitExceeded

output = struct with fields:
elapsedtime: 45.5839
funccount: 200
constrviolation: 0
ineq: [1x1 struct]
rngstate: [1x1 struct]
message: 'surrogateopt stopped because it exceeded the function evaluation limit set by ...'
solver: 'surrogateopt'

Find a local minimum of the peaks function on the range $-5\le x,y\le 5$ starting from the point [–1,2].

x = optimvar("x",LowerBound=-5,UpperBound=5);
y = optimvar("y",LowerBound=-5,UpperBound=5);
x0.x = -1;
x0.y = 2;
prob = optimproblem(Objective=peaks(x,y));
opts = optimoptions("fmincon",Display="none");
[sol,fval] = solve(prob,x0,Options=opts)
sol = struct with fields:
x: -3.3867
y: 3.6341

fval = 1.1224e-07

Try to find a better solution by using the GlobalSearch solver. This solver runs fmincon multiple times, which potentially yields a better solution.

ms = GlobalSearch;
[sol2,fval2] = solve(prob,x0,ms)
Solving problem using GlobalSearch.

GlobalSearch stopped because it analyzed all the trial points.

All 15 local solver runs converged with a positive local solver exit flag.
sol2 = struct with fields:
x: 0.2283
y: -1.6255

fval2 = -6.5511

GlobalSearch finds a solution with a better (lower) objective function value. The exit message shows that fmincon, the local solver, runs 15 times. The returned solution has an objective function value of about –6.5511, which is lower than the value at the first solution, 1.1224e–07.

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: -2.4070e-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

collapse all

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

• ga (Global Optimization Toolbox), gamultiobj (Global Optimization Toolbox), paretosearch (Global Optimization Toolbox) and particleswarm (Global Optimization Toolbox). These solvers accept multiple starting points as members of the initial population.

• MultiStart (Global Optimization Toolbox). This solver accepts multiple initial points for a local solver such as fmincon.

• surrogateopt (Global Optimization Toolbox). This solver accepts multiple initial points to help create an initial surrogate.

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

Multiple start solver, specified as a MultiStart (Global Optimization Toolbox) object or a GlobalSearch (Global Optimization Toolbox) object. Create ms using the MultiStart or GlobalSearch commands.

Currently, GlobalSearch supports only the fmincon local solver, and MultiStart supports only the fmincon, fminunc, and lsqnonlin local solvers.

Example: ms = MultiStart;

Example: ms = GlobalSearch(FunctionTolerance=1e-4);

### Name-Value Arguments

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

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

Example: solve(prob,'Options',opts)

Minimum number of start points for MultiStart (Global Optimization Toolbox), specified as a positive integer. This argument applies only when you call solve using the ms argument. solve uses all of the values in x0 as start points. If MinNumStartPoints is greater than the number of values in x0, then solve generates more start points uniformly at random within the problem bounds. If a component is unbounded, solve generates points using the default artificial bounds for MultiStart.

Example: solve(prob,x0,ms,MinNumStartPoints=50)

Data Types: double

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

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

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

Problem TypeDefault Solver
Linear Programming (LP)linprog
Mixed-Integer Linear Programming (MILP)intlinprog
Second-Order Cone Programming (SOCP)coneprog
Linear Least Squareslsqlin
Nonlinear Least Squareslsqnonlin
Nonlinear Programming (NLP)

fminunc for problems with no constraints, otherwise fmincon

Mixed-Integer Nonlinear Programming (MINLP)ga (Global Optimization Toolbox)
Multiobjectivegamultiobj (Global Optimization Toolbox)

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

Problem Type

LPMILPQPSOCPLinear Least SquaresNonlinear Least SquaresNLPMINLP
Solver
linprog

xxxxxxx
intlinprog

xxxxxx

x

xxx
coneprog

xx

xxxx
lsqlinxxxx

xxx
lsqnonnegxxxx

xxx
lsqnonlinxxxx

xx
fminunc

x

x

x
fmincon

x

x
patternsearch (Global Optimization Toolbox)

x

x
ga (Global Optimization Toolbox)

particleswarm (Global Optimization Toolbox)

x

x

x
simulannealbnd (Global Optimization Toolbox)

x

x

x
surrogateopt (Global Optimization Toolbox)

gamultiobj (Global Optimization Toolbox)

paretosearch (Global Optimization Toolbox)

x

x

Note

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

Caution

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

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

• * indicates the default solver for the problem type.

• Y indicates an available solver.

• N indicates an unavailable solver.

Supported Solvers for Equations

Equation 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

Indication to use automatic differentiation (AD) for nonlinear objective function, specified as '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 for 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

Indication to use automatic differentiation (AD) for nonlinear constraint functions, specified as '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 for 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

Indication to use automatic differentiation (AD) for nonlinear constraint functions, specified as '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 for 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

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Solution, returned as a structure or an OptimizationValues vector. sol is an OptimizationValues vector when the problem is multiobjective. For single-objective problems, the fields of the returned structure are the names of the optimization variables in the problem. See optimvar.

Objective function value at the solution, returned as one of the following:

Problem TypeReturned Value(s)
Optimize scalar objective function f(x)Real number f(sol)
Least squaresReal number, the sum of squares of the residuals at the solution
Solve equationIf prob.Equations is a single entry: Real vector of function values at the solution, meaning the left side minus the right side of the equations
If prob.Equations has multiple named fields: Structure with same names as prob.Equations, where each field value is the left side minus the right side of the named equations
MultiobjectiveMatrix with one row for each objective function component, and one column for each solution point.

Tip

If you neglect to ask for fval for an objective defined as an optimization expression or equation expression, you can calculate it using

fval = evaluate(prob.Objective,sol)

If the objective is defined as a structure with only one field,

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

If the objective is a structure with multiple fields, write a loop.

fnames = fields(prob.Equations);
for i = 1:length(fnames)
fval.(fnames{i}) = evaluate(prob.Equations.(fnames{i}),sol);
end

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 coneprog solver.

Exit Flag for coneprogNumeric EquivalentMeaning
OptimalSolution1

The solver converged to a solution x.

SolverLimitExceeded0

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 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.

This table describes the exit flags for the patternsearch solver.

Exit Flag for patternsearchNumeric EquivalentMeaning
SearchDirectionTooSmall4

The magnitude of the step is smaller than machine precision, and the constraint violation is less than ConstraintTolerance.

FunctionChangeBelowTolerance3

The change in fval and the mesh size are both less than the specified tolerance, and the constraint violation is less than ConstraintTolerance.

StepSizeBelowTolerance

2

Change in x and the mesh size are both smaller than StepTolerance, and the constraint violation is less than ConstraintTolerance.

SolverConvergedSuccessfully1

Without nonlinear constraints — The magnitude of the mesh size is less than the specified tolerance, and the constraint violation is less than ConstraintTolerance.

With nonlinear constraints — The magnitude of the complementarity measure (defined after this table) is less than sqrt(ConstraintTolerance), the subproblem is solved using a mesh finer than MeshTolerance, and the constraint violation is less than ConstraintTolerance.

SolverLimitExceeded0

The maximum number of function evaluations or iterations is reached.

OutputFcnStop-1

Stopped by an output function or plot function.

NoFeasiblePointFound-2

No feasible point found.

In the nonlinear constraint solver, the complementarity measure is the norm of the vector whose elements are ciλi, where ci is the nonlinear inequality constraint violation, and λi is the corresponding Lagrange multiplier.

This table describes the exit flags for the ga solver.

Exit Flag for gaNumeric EquivalentMeaning
MinimumFitnessLimitReached5

Minimum fitness limit FitnessLimit reached and the constraint violation is less than ConstraintTolerance.

SearchDirectionTooSmall4

The magnitude of the step is smaller than machine precision, and the constraint violation is less than ConstraintTolerance.

FunctionChangeBelowTolerance3

Value of the fitness function did not change in MaxStallGenerations generations and the constraint violation is less than ConstraintTolerance.

SolverConvergedSuccessfully1

Without nonlinear constraints — Average cumulative change in value of the fitness function over MaxStallGenerations generations is less than FunctionTolerance, and the constraint violation is less than ConstraintTolerance.

With nonlinear constraints — Magnitude of the complementarity measure (see Complementarity Measure (Global Optimization Toolbox)) is less than sqrt(ConstraintTolerance), the subproblem is solved using a tolerance less than FunctionTolerance, and the constraint violation is less than ConstraintTolerance.

SolverLimitExceeded0

Maximum number of generations MaxGenerations exceeded.

OutputFcnStop-1

Stopped by an output function or plot function.

NoFeasiblePointFound-2

No feasible point found.

StallTimeLimitExceeded-4

Stall time limit MaxStallTime exceeded.

TimeLimitExceeded-5

Time limit MaxTime exceeded.

This table describes the exit flags for the particleswarm solver.

Exit Flag for particleswarmNumeric EquivalentMeaning
SolverConvergedSuccessfully1

Relative change in the objective value over the last options.MaxStallIterations iterations is less than options.FunctionTolerance.

SolverLimitExceeded0

Number of iterations exceeded options.MaxIterations.

OutputFcnStop-1

Iterations stopped by output function or plot function.

NoFeasiblePointFound-2

Bounds are inconsistent: for some i, lb(i) > ub(i).

Unbounded-3

Best objective function value is below options.ObjectiveLimit.

StallTimeLimitExceeded-4

Best objective function value did not change within options.MaxStallTime seconds.

TimeLimitExceeded-5

Run time exceeded options.MaxTime seconds.

This table describes the exit flags for the simulannealbnd solver.

Exit Flag for simulannealbndNumeric EquivalentMeaning
ObjectiveValueBelowLimit5

Objective function value is less than options.ObjectiveLimit.

SolverConvergedSuccessfully1

Average change in the value of the objective function over options.MaxStallIterations iterations is less than options.FunctionTolerance.

SolverLimitExceeded0

Maximum number of generations MaxGenerations exceeded.

OutputFcnStop-1

Optimization terminated by an output function or plot function.

NoFeasiblePointFound-2

No feasible point found.

TimeLimitExceeded-5

Time limit exceeded.

This table describes the exit flags for the surrogateopt solver.

Exit Flag for surrogateoptNumeric EquivalentMeaning
BoundsEqual10

Problem has a unique feasible solution due to one of the following:

• All upper bounds ub (Global Optimization Toolbox) are equal to the lower bounds lb (Global Optimization Toolbox).

• The linear equality constraints Aeq*x = beq and the bounds have a unique solution point.

surrogateopt returns the feasible point and function value without performing any optimization.

FeasiblePointFound3Feasible point found. Solver stopped because too few new feasible points were found to continue.
ObjectiveLimitAttained1

The objective function value is less than options.ObjectiveLimit. This exit flag takes precedence over exit flag 10 when both apply.

SolverLimitExceeded0

The number of function evaluations exceeds options.MaxFunctionEvaluations or the elapsed time exceeds options.MaxTime. If the problem has nonlinear inequalities, the solution is feasible.

OutputFcnStop-1

The optimization is terminated by an output function or plot function.

NoFeasiblePointFound-2

No feasible point is found due to one of the following:

• A lower bound lb(i) exceeds a corresponding upper bound ub(i). Or one or more ceil(lb(i)) exceeds a corresponding floor(ub(i)) for i in intcon (Global Optimization Toolbox). In this case, solve returns x = [] and fval = [].

• lb = ub and the point lb is infeasible. In this case, x = lb, and fval = objconstr(x).Fval.

• The linear and, if present, integer constraints are infeasible together with the bounds. In this case, solve returns x = [] and fval = [].

• The bounds, integer, and linear constraints are feasible, but no feasible solution is found with nonlinear constraints. In this case, x is the point of least maximum infeasibility of nonlinear constraints, and fval = objconstr(x).Fval.

This table describes the exit flags for the MultiStart and GlobalSearch solvers.

Exit Flag for MultiStart or GlobalSearchNumeric EquivalentMeaning
LocalMinimumFoundSomeConverged2At least one local minimum found. Some runs of the local solver converged.
LocalMinimumFoundAllConverged1At least one local minimum found. All runs of the local solver converged.
SolverLimitExceeded0No local minimum found. Local solver called at least once and at least one local solver call ran out of iterations.
OutputFcnStop–1Stopped by an output function or plot function.
NoFeasibleLocalMinimumFound–2No feasible local minimum found.
TimeLimitExceeded–5MaxTime limit exceeded.
NoSolutionFound–8No solution found. All runs had local solver exit flag –2 or smaller, not all equal –2.
FailureInSuppliedFcn–10Encountered failures in the objective or nonlinear constraint functions.

This table describes the exit flags for the paretosearch solver.

Exit Flag for paretosearchNumeric EquivalentMeaning
SolverConvergedSuccessfully1

One of the following conditions is met:

• Mesh size of all incumbents is less than options.MeshTolerance and constraints (if any) are satisfied to within options.ConstraintTolerance.

• Relative change in the spread of the Pareto set is less than options.ParetoSetChangeTolerance and constraints (if any) are satisfied to within options.ConstraintTolerance.

• Relative change in the volume of the Pareto set is less than options.ParetoSetChangeTolerance and constraints (if any) are satisfied to within options.ConstraintTolerance.

SolverLimitExceeded0Number of iterations exceeds options.MaxIterations, or the number of function evaluations exceeds options.MaxFunctionEvaluations.
OutputFcnStop–1Stopped by an output function or plot function.
NoFeasiblePointFound–2Solver cannot find a point satisfying all the constraints.
TimeLimitExceeded–5Optimization time exceeds options.MaxTime.

This table describes the exit flags for the gamultiobj solver.

Exit Flag for paretosearchNumeric EquivalentMeaning
SolverConvergedSuccessfully1Geometric average of the relative change in value of the spread over options.MaxStallGenerations generations is less than options.FunctionTolerance, and the final spread is less than the mean spread over the past options.MaxStallGenerations generations.
SolverLimitExceeded0Number of generations exceeds options.MaxGenerations.
OutputFcnStop–1Stopped by an output function or plot function.
NoFeasiblePointFound–2Solver cannot find a point satisfying all the constraints.
TimeLimitExceeded–5Optimization time exceeds options.MaxTime.

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:

• 'ga' output (Global Optimization Toolbox)

• 'gamultiobj' output (Global Optimization Toolbox)

• 'paretosearch' output (Global Optimization Toolbox)

• 'particleswarm' output (Global Optimization Toolbox)

• 'patternsearch' output (Global Optimization Toolbox)

• 'simulannealbnd' output (Global Optimization Toolbox)

• 'surrogateopt' output (Global Optimization Toolbox)

• 'MultiStart' and 'GlobalSearch' return the output structure from the local solver. In addition, the output structure contains the following fields:

• globalSolver — Either 'MultiStart' or 'GlobalSearch'.

• objectiveDerivative — Takes the values described at the end of this section.

• constraintDerivative — Takes the values described at the end of this section, or "auto" when prob has no nonlinear constraint.

• solver — The local solver, such as 'fmincon'.

• local — Structure containing extra information about the optimization.

• sol — Local solutions, returned as a vector of OptimizationValues objects.

• x0 — Initial points for the local solver, returned as a cell array.

• exitflag — Exit flags of local solutions, returned as an integer vector.

• output — Structure array, with one row for each local solution. Each row is the local output structure corresponding to one local solution.

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

When Solver is a nonlinear Optimization Toolbox™ 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

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

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### Conversion to Solver Form

Internally, the solve function solves optimization problems by calling a solver. For the default solver for the problem and supported solvers for the problem, see the 'solver' argument.

Before solve can call a solver, 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 nonlinear optimization problems, solve uses automatic differentiation to compute the gradients of the objective function and nonlinear constraint functions. These derivatives apply when the objective and constraint functions are composed of Supported Operations for Optimization Variables and Expressions and do not use the fcn2optimexpr function. When automatic differentiation does not apply, solvers estimate derivatives using finite differences. For details of automatic differentiation, see Automatic Differentiation Background.

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 either norm(expr)^2 or sum(expr.^2), and not as expr'*expr or any other form. The internal parser recognizes a sum of squares only when represented as a square of a norm or an 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.

### Automatic Differentiation

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

• The objective and constraint functions are supported, as described in Supported Operations for Optimization Variables and Expressions. They do not require use of the fcn2optimexpr function.

• The solver called by solve is fmincon, fminunc, fsolve, or lsqnonlin.

• For optimization problems, the 'ObjectiveDerivative' and 'ConstraintDerivative' name-value pair arguments for solve or prob2struct are set to 'auto' (default), 'auto-forward', or 'auto-reverse'.

• For equation problems, the 'EquationDerivative' option is set to 'auto' (default), 'auto-forward', or 'auto-reverse'.

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

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

Solvers choose the following type of AD by default:

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

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

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

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

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

Note

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

problem.options = options;

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

## Version History

Introduced in R2017b

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Errors starting in R2018b