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Solve optimization problem
sol = solve(prob)
sol = solve(prob,x0)
sol = solve(___,Name,Value)
[sol,fval] = solve(___)
[sol,fval,exitflag,output,lambda] = solve(___)
modifies the solution process using one or more namevalue pair arguments in
addition to the input arguments in previous syntaxes.sol
= solve(___,Name,Value
)
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)
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)
Local minimum found that satisfies the constraints. Optimization completed because the objective function is nondecreasing 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 branchandbound nodes.
[x1,fval1,exitflag1,output1] = solve(prob);
LP: Optimal objective value is 1554.047531. Cut Generation: Applied 8 strong CG cuts. Lower bound is 1591.000000. Branch and Bound: nodes total num int integer relative explored time (s) solution fval gap (%) 10000 1.13 0   18188 1.85 1 2.906000e+03 4.509804e+01 22039 2.30 2 2.073000e+03 2.270974e+01 24105 2.51 3 1.854000e+03 9.973046e+00 24531 2.56 3 1.854000e+03 1.347709e+00 24701 2.58 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 = 1e05 (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);
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 0.64 2 2.154000e+03 2.593968e+01 5844 0.86 3 1.854000e+03 1.180593e+01 6204 0.91 3 1.854000e+03 1.455526e+00 6400 0.92 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 = 1e05 (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 24701 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
$$\underset{x}{\mathrm{min}}\left(3{x}_{1}2{x}_{2}{x}_{3}\right)\phantom{\rule{0.2777777777777778em}{0ex}}subject\phantom{\rule{0.2777777777777778em}{0ex}}to\{\begin{array}{l}{x}_{3}\phantom{\rule{0.2777777777777778em}{0ex}}binary\\ {x}_{1},{x}_{2}\ge 0\\ {x}_{1}+{x}_{2}+{x}_{3}\le 7\\ 4{x}_{1}+2{x}_{2}+{x}_{3}=12\end{array}$$
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
intlinprog
to Solve a Linear ProgramForce 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')
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 mixedinteger 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)
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 = 1e05 (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 profitweighted 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);
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 = 1e05 (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
.
prob
— Optimization problemOptimizationProblem
objectOptimization problem, specified as an OptimizationProblem
object. Create a problem by using optimproblem
.
Example: prob = optimproblem; prob.Objective = obj; prob.Constraints.cons1 =
cons1;
x0
— Initial pointInitial 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
Specify optional
commaseparated 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
.
solve(prob,'options',opts)
'options'
— Optimization optionsoptimoptions
 options structureOptimization options, specified as the commaseparated pair consisting
of 'options'
and an object created by optimoptions
or an options
structure such as created by optimset
.
Internally, the solve
function calls
linprog
, intlinprog
,
quadprog
, lsqlin
,
lsqnonneg
, fminunc
, or
fmincon
. For the default solver, see 'solver'
.
Ensure that options
are 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 'dualsimplex'
algorithm is generally
memoryefficient and speedy. Occasionally, linprog
solves a large problem faster when the Algorithm
option is 'interiorpoint'
. 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'
 'lsqnonneg'
 'quadprog'
 'fminunc'
 'fmincon'
Optimization solver, specified as the commaseparated pair consisting
of 'solver'
and the name of a listed solver.
Problem Type  Default Solver  Other Allowed Solvers 

Linear objective, linear constraints  linprog  intlinprog ,
quadprog 
Linear objective, linear and integer constraints  intlinprog  linprog ,
quadprog
(integer constraints ignored) 
Quadratic objective, linear constraints  quadprog  lsqlin ,
lsqnonneg
(if objective cannot be converted to minimize C*x 
d^2 then solve throws an error for
these solvers) 
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 ) 
Minimize C*x  d^2 subject to x >= 0  lsqlin  quadprog ,
lsqnonneg 
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) 
For maximization problems, do not specify the
lsqlin
and lsqnonneg
solvers. If you do, solve
throws an error,
because these solvers cannot maximize.
Example: 'intlinprog'
Data Types: char
 string
sol
— SolutionSolution, returned as a structure. The fields of the structure are the names of the optimization variables. See optimvar
.
fval
— Objective function value at the solutionObjective function value at the solution, returned as a real number.
If you neglect to ask for fval
, you can calculate
it using:
fval = evaluate(prob.Objective,sol)
exitflag
— Reason solver stoppedReason the solver stopped, returned as a categorical variable. 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

IntegerFeasible  2  intlinprog stopped prematurely, and found an
integer feasible point. 
OptimalSolution 
 The solver converged to a solution

SolverLimitExceeded 

See Tolerances and Stopping Criteria. 
OutputFcnStop  1  intlinprog stopped by an output function or plot
function. 
NoFeasiblePointFound 
 No feasible point found. 
Unbounded 
 The problem is unbounded. 
FeasibilityLost 
 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

OptimalSolution  1  The solver converged to a solution

SolverLimitExceeded  0  The number of iterations exceeds

NoFeasiblePointFound  2  No feasible point found. 
Unbounded  3  The problem is unbounded. 
FoundNaN  4 

PrimalDualInfeasible  5  Both primal and dual problems are infeasible. 
DirectionTooSmall  7  The search direction is too small. No further progress can be made. 
FeasibilityLost  9  Solver lost feasibility. 
Exitflags 3
and 9
relate
to solutions that have large infeasibilities. These usually arise from linear constraint
matrices that have large condition number, or problems that have large solution components. To
correct these issues, try to scale the coefficient matrices, eliminate redundant linear
constraints, or give tighter bounds on the variables.
This table describes the exit flags for the lsqlin
solver.
Exit Flag for lsqlin  Numeric Equivalent  Meaning 

FunctionChangeBelowTolerance  3  Change in the residual is smaller than the specified tolerance

StepSizeBelowTolerance 
 Step size smaller than

OptimalSolution  1  The solver converged to a solution

SolverLimitExceeded  0  The number of iterations exceeds

NoFeasiblePointFound  2  The problem is infeasible. Or, for the

IllConditioned  4  Illconditioning prevents further optimization. 
NoDescentDirectionFound  8  The search direction is too small. No further progress can be
made. ( 
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 
StepSizeBelowTolerance 
 Step size smaller than

OptimalSolution  1  The solver converged to a solution

SolverLimitExceeded  0  The number of iterations exceeds

NoFeasiblePointFound  2  The problem is infeasible. Or, for the

IllConditioned  4  Illconditioning prevents further optimization. 
Nonconvex 
 Nonconvex problem detected.
( 
NoDescentDirectionFound  8  Unable to compute a step direction.
( 
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

FunctionChangeBelowTolerance  3  Change in the objective function value is less than the

StepSizeBelowTolerance 
 Change in 
OptimalSolution  1  Magnitude of gradient is smaller than the

SolverLimitExceeded  0  Number of iterations exceeds

OutputFcnStop  1  Stopped by an output function or plot function. 
Unbounded  3  Objective function at current iteration is below

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* 
SearchDirectionTooSmall  4  Magnitude of the search direction is less than
2* 
FunctionChangeBelowTolerance  3  Change in the objective function value is less than

StepSizeBelowTolerance 
 Change in 
OptimalSolution  1  Firstorder optimality measure is less than

SolverLimitExceeded  0  Number of iterations exceeds

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

output
— Information about optimization processInformation 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'
.
lambda
— Lagrange multipliers at the solutionLagrange multipliers at the solution, returned as a structure. For the
intlinprog
solver, 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
Elements of a constraint array all have the same comparison
(<=
, ==
, or
>=
) and are all of the same type (linear, quadratic,
or nonlinear).
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
quadprog
for quadratic objective and linear constraints
lsqlin
or lsqnonneg
for linear leastsquares with linear 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
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'
namevalue 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 the algorithms that lsqlin
uses
to solve linear leastsquares problems, see LeastSquares (Model Fitting) Algorithms. For
nonlinear solver algorithms, see Unconstrained Nonlinear Optimization Algorithms and Constrained Nonlinear Optimization Algorithms.
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
. The internal parser recognizes only explicit sums of
squares. For an example, see Nonnegative LeastSquares, ProblemBased.
Errors starting in R2018b
To choose options or the underlying solver for solve
, use
namevalue pairs. For example,
sol = solve(prob,'options',opts,'solver','quadprog');
The previous syntaxes were not as flexible, standard, or extensible as namevalue pairs.
OptimizationProblem
 evaluate
 findindex
 fmincon
 fminunc
 intlinprog
 linprog
 lsqlin
 lsqnonneg
 optimoptions
 prob2struct
 quadprog
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