solve
Solve optimization problem or equation problem
Syntax
Description
Use solve
to find the solution of an optimization problem
or equation problem.
Tip
For the full workflow, see ProblemBased Optimization Workflow or ProblemBased Workflow for Solving Equations.
modifies the solution process using one or more namevalue pair arguments in
addition to the input arguments in previous syntaxes.sol
= solve(___,Name,Value
)
Examples
Solve Linear Programming Problem
Solve a linear programming problem defined by an optimization problem.
x = optimvar('x'); y = optimvar('y'); prob = optimproblem; prob.Objective = x  y/3; prob.Constraints.cons1 = x + y <= 2; prob.Constraints.cons2 = x + y/4 <= 1; prob.Constraints.cons3 = x  y <= 2; prob.Constraints.cons4 = x/4 + y >= 1; prob.Constraints.cons5 = x + y >= 1; prob.Constraints.cons6 = x + y <= 2; sol = solve(prob)
Solving problem using linprog. Optimal solution found.
sol = struct with fields:
x: 0.6667
y: 1.3333
Solve Nonlinear Programming Problem Using ProblemBased Approach
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 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
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 nondecreasing 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
Copyright 2018–2020 The MathWorks, Inc.
Solve MixedInteger Linear Program Starting from Initial Point
Compare the number of steps to solve an integer programming problem both with and without an initial feasible point. The problem has eight integer variables and four linear equality constraints, and all variables are restricted to be positive.
prob = optimproblem; x = optimvar('x',8,1,'LowerBound',0,'Type','integer');
Create four linear equality constraints and include them in the problem.
Aeq = [22 13 26 33 21 3 14 26 39 16 22 28 26 30 23 24 18 14 29 27 30 38 26 26 41 26 28 36 18 38 16 26]; beq = [ 7872 10466 11322 12058]; cons = Aeq*x == beq; prob.Constraints.cons = cons;
Create an objective function and include it in the problem.
f = [2 10 13 17 7 5 7 3]; prob.Objective = f*x;
Solve the problem without using an initial point, and examine the display to see the number of branchandbound nodes.
[x1,fval1,exitflag1,output1] = solve(prob);
Solving problem using intlinprog. Running HiGHS 1.7.0: Copyright (c) 2024 HiGHS under MIT licence terms Coefficient ranges: Matrix [3e+00, 4e+01] Cost [2e+00, 2e+01] Bound [0e+00, 0e+00] RHS [8e+03, 1e+04] Presolving model 4 rows, 8 cols, 32 nonzeros 0s 4 rows, 8 cols, 27 nonzeros 0s Objective function is integral with scale 1 Solving MIP model with: 4 rows 8 cols (0 binary, 8 integer, 0 implied int., 0 continuous) 27 nonzeros Nodes  B&B Tree  Objective Bounds  Dynamic Constraints  Work Proc. InQueue  Leaves Expl.  BestBound BestSol Gap  Cuts InLp Confl.  LpIters Time 0 0 0 0.00% 0 inf inf 0 0 0 0 0.0s 0 0 0 0.00% 1554.047531 inf inf 0 0 4 4 0.0s T 20753 210 8189 98.04% 1783.696925 1854 3.79% 30 8 9884 19222 2.9s Solving report Status Optimal Primal bound 1854 Dual bound 1854 Gap 0% (tolerance: 0.01%) Solution status feasible 1854 (objective) 0 (bound viol.) 9.63673585375e14 (int. viol.) 0 (row viol.) Timing 2.98 (total) 0.00 (presolve) 0.00 (postsolve) Nodes 21163 LP iterations 19608 (total) 223 (strong br.) 76 (separation) 1018 (heuristics) Optimal solution found. Intlinprog stopped because the objective value is within a gap tolerance of the optimal value, options.AbsoluteGapTolerance = 1e06. The intcon variables are integer within tolerance, options.ConstraintTolerance = 1e06.
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. Running HiGHS 1.7.0: Copyright (c) 2024 HiGHS under MIT licence terms Coefficient ranges: Matrix [3e+00, 4e+01] Cost [2e+00, 2e+01] Bound [0e+00, 0e+00] RHS [8e+03, 1e+04] Assessing feasibility of MIP using primal feasibility and integrality tolerance of 1e06 Solution has num max sum Col infeasibilities 0 0 0 Integer infeasibilities 0 0 0 Row infeasibilities 0 0 0 Row residuals 0 0 0 Presolving model 4 rows, 8 cols, 32 nonzeros 0s 4 rows, 8 cols, 27 nonzeros 0s MIP start solution is feasible, objective value is 3901 Objective function is integral with scale 1 Solving MIP model with: 4 rows 8 cols (0 binary, 8 integer, 0 implied int., 0 continuous) 27 nonzeros Nodes  B&B Tree  Objective Bounds  Dynamic Constraints  Work Proc. InQueue  Leaves Expl.  BestBound BestSol Gap  Cuts InLp Confl.  LpIters Time 0 0 0 0.00% 0 3901 100.00% 0 0 0 0 0.0s 0 0 0 0.00% 1554.047531 3901 60.16% 0 0 4 4 0.0s T 6266 708 2644 73.61% 1662.791423 3301 49.63% 20 6 9746 10699 1.5s T 9340 919 3970 80.72% 1692.410008 2687 37.01% 29 6 9995 16120 2.5s T 21750 192 9514 96.83% 1791.542628 1854 3.37% 20 6 9984 40278 5.7s Solving report Status Optimal Primal bound 1854 Dual bound 1854 Gap 0% (tolerance: 0.01%) Solution status feasible 1854 (objective) 0 (bound viol.) 1.42108547152e13 (int. viol.) 0 (row viol.) Timing 5.83 (total) 0.00 (presolve) 0.00 (postsolve) Nodes 22163 LP iterations 40863 (total) 538 (strong br.) 64 (separation) 2782 (heuristics) Optimal solution found. Intlinprog stopped because the objective value is within a gap tolerance of the optimal value, options.AbsoluteGapTolerance = 1e06. The intcon variables are integer within tolerance, options.ConstraintTolerance = 1e06.
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 21163 steps. With an initial point, solve took 22163 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.
Specify Starting Points and Values for surrogateopt
, ProblemBased
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: 0.2279
y: 1.6258
fval = 6.5511
eflag = SolverLimitExceeded
output = struct with fields:
elapsedtime: 20.4544
funccount: 200
constrviolation: 0
ineq: [1x1 struct]
rngstate: [1x1 struct]
message: 'surrogateopt stopped because it exceeded the function evaluation limit set by ...'
solver: 'surrogateopt'
Minimize Nonlinear Function Using MultipleStart Solver, ProblemBased
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 = 1.1224e07
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 Integer Programming Problem with Nondefault Options
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: 0
Examine the solution.
sol.x
ans = 2×1
0
6
sol.x3
ans = 0
Use intlinprog
to Solve a Linear Program
Force solve
to use intlinprog
as the solver for a linear programming problem.
x = optimvar('x'); y = optimvar('y'); prob = optimproblem; prob.Objective = x  y/3; prob.Constraints.cons1 = x + y <= 2; prob.Constraints.cons2 = x + y/4 <= 1; prob.Constraints.cons3 = x  y <= 2; prob.Constraints.cons4 = x/4 + y >= 1; prob.Constraints.cons5 = x + y >= 1; prob.Constraints.cons6 = x + y <= 2; sol = solve(prob,'Solver', 'intlinprog')
Solving problem using intlinprog. Running HiGHS 1.7.0: Copyright (c) 2024 HiGHS under MIT licence terms Coefficient ranges: Matrix [2e01, 1e+00] Cost [3e01, 1e+00] Bound [0e+00, 0e+00] RHS [1e+00, 2e+00] Presolving model 6 rows, 2 cols, 12 nonzeros 0s 4 rows, 2 cols, 8 nonzeros 0s 4 rows, 2 cols, 8 nonzeros 0s Presolve : Reductions: rows 4(2); columns 2(0); elements 8(4) Solving the presolved LP Using EKK dual simplex solver  serial Iteration Objective Infeasibilities num(sum) 0 1.3333333333e+03 Ph1: 3(4499); Du: 2(1.33333) 0s 3 1.1111111111e+00 Pr: 0(0) 0s Solving the original LP from the solution after postsolve Model status : Optimal Simplex iterations: 3 Objective value : 1.1111111111e+00 HiGHS run time : 0.00 Optimal solution found. No integer variables specified. Intlinprog solved the linear problem.
sol = struct with fields:
x: 0.6667
y: 1.3333
Return All Outputs
Solve the 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)
Solving problem using intlinprog. Running HiGHS 1.7.0: Copyright (c) 2024 HiGHS under MIT licence terms Coefficient ranges: Matrix [1e+00, 4e+00] Cost [1e+00, 3e+00] Bound [1e+00, 1e+00] RHS [7e+00, 1e+01] Presolving model 2 rows, 3 cols, 6 nonzeros 0s 0 rows, 0 cols, 0 nonzeros 0s Presolve: Optimal Solving report Status Optimal Primal bound 12 Dual bound 12 Gap 0% (tolerance: 0.01%) Solution status feasible 12 (objective) 0 (bound viol.) 0 (int. viol.) 0 (row viol.) Timing 0.00 (total) 0.00 (presolve) 0.00 (postsolve) Nodes 0 LP iterations 0 (total) 0 (strong br.) 0 (separation) 0 (heuristics) Optimal solution found. Intlinprog stopped at the root node because the objective value is within a gap tolerance of the optimal value, options.AbsoluteGapTolerance = 1e06. The intcon variables are integer within tolerance, options.ConstraintTolerance = 1e06.
sol = struct with fields:
x: [2x1 double]
x3: 0
fval = 12
exitflag = OptimalSolution
output = struct with fields:
relativegap: 0
absolutegap: 0
numfeaspoints: 0
numnodes: 0
constrviolation: 0
algorithm: 'highs'
message: 'Optimal solution found....'
solver: 'intlinprog'
For a problem without any integer constraints, you can also obtain a nonempty Lagrange multiplier structure as the fifth output.
View Solution with Index Variables
Create and solve an optimization problem using named index variables. The problem is to maximize the 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);
Solving problem using intlinprog. Running HiGHS 1.7.0: Copyright (c) 2024 HiGHS under MIT licence terms Coefficient ranges: Matrix [4e02, 1e+00] Cost [1e01, 1e+00] Bound [0e+00, 0e+00] RHS [1e+01, 4e+01] Presolving model 3 rows, 12 cols, 12 nonzeros 0s 3 rows, 12 cols, 12 nonzeros 0s Solving MIP model with: 3 rows 12 cols (0 binary, 12 integer, 0 implied int., 0 continuous) 12 nonzeros Nodes  B&B Tree  Objective Bounds  Dynamic Constraints  Work Proc. InQueue  Leaves Expl.  BestBound BestSol Gap  Cuts InLp Confl.  LpIters Time 0 0 0 0.00% 1160.150059 inf inf 0 0 0 0 0.0s S 0 0 0 0.00% 1160.150059 1027.233133 12.94% 0 0 0 0 0.0s Solving report Status Optimal Primal bound 1027.23313332 Dual bound 1027.23313332 Gap 0% (tolerance: 0.01%) Solution status feasible 1027.23313332 (objective) 0 (bound viol.) 0 (int. viol.) 0 (row viol.) Timing 0.01 (total) 0.00 (presolve) 0.00 (postsolve) Nodes 1 LP iterations 3 (total) 0 (strong br.) 0 (separation) 0 (heuristics) Optimal solution found. Intlinprog stopped at the root node because the objective value is within a gap tolerance of the optimal value, options.AbsoluteGapTolerance = 1e06. The intcon variables are integer within tolerance, options.ConstraintTolerance = 1e06.
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
70 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 28 980
This display means that 70 berries are going to NYC
, 28 apples are going to BOS
, and 980 oranges are going to LAX
.
Solve Nonlinear System of Equations, ProblemBased
To solve the nonlinear system of equations
$$\begin{array}{l}\mathrm{exp}(\mathrm{exp}(({x}_{1}+{x}_{2})))={x}_{2}(1+{x}_{1}^{2})\\ {x}_{1}\mathrm{cos}({x}_{2})+{x}_{2}\mathrm{sin}({x}_{1})=\frac{1}{2}\end{array}$$
using the problembased approach, first define x
as a twoelement 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 problembased 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: 2.4070e07
eq2: 3.8255e08
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;
See Supported Operations for Optimization Variables and Expressions and Convert Nonlinear Function to Optimization Expression.
Input Arguments
prob
— Optimization problem or equation problem
OptimizationProblem
object  EquationProblem
object
Optimization problem or equation problem, specified as an OptimizationProblem
object or an EquationProblem
object. Create an optimization problem by using optimproblem
; create an equation problem by using eqnproblem
.
Warning
The problembased approach does not support complex values in the following: an objective function, nonlinear equalities, and nonlinear inequalities. If a function calculation has a complex value, even as an intermediate value, the final result might be incorrect.
Example: prob = optimproblem; prob.Objective = obj; prob.Constraints.cons1 =
cons1;
Example: prob = eqnproblem; prob.Equations = eqs;
x0
— Initial point
structure  vector of OptimizationValues
objects
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) andparticleswarm
(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 asfmincon
.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
ms
— Multiple start solver
MultiStart
object  GlobalSearch
object
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=1e4);
NameValue Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Namevalue 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)
MinNumStartPoints
— Minimum number of start points for MultiStart
20 (default)  positive integer
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
Options
— Optimization options
object created by optimoptions
 options structure
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 '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 Optimization Options in Common Use: Tuning and Troubleshooting and Improve Results.
Example: options =
optimoptions('intlinprog','Display','none')
Solver
— Optimization solver
'intlinprog'
 'linprog'
 'lsqlin'
 'lsqcurvefit'
 'lsqnonlin'
 'lsqnonneg'
 'quadprog'
 'fminbnd'
 'fminunc'
 'fmincon'
 'fminsearch'
 'fzero'
 'fsolve'
 'coneprog'
 'ga'
 'gamultiobj'
 'paretosearch'
 'patternsearch'
 'particleswarm'
 'surrogateopt'
 'simulannealbnd'
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 ProblemBased Optimization.
The default solver for each optimization problem type is listed here.
Problem Type  Default Solver 

Linear Programming (LP)  linprog 
MixedInteger Linear Programming (MILP)  intlinprog 
Quadratic Programming (QP)  quadprog 
SecondOrder Cone Programming (SOCP)  coneprog 
Linear Least Squares  lsqlin 
Nonlinear Least Squares  lsqnonlin 
Nonlinear Programming (NLP)  
MixedInteger Nonlinear Programming (MINLP)  ga (Global Optimization Toolbox) 
Multiobjective  gamultiobj (Global Optimization Toolbox) 
In this table, means the solver is available for the problem type, x means the solver is not available.
Problem Type  LP  MILP  QP  SOCP  Linear Least Squares  Nonlinear Least Squares  NLP  MINLP 

Solver  
linprog 
 x  x  x  x  x  x  x 
intlinprog 

 x  x  x  x  x  x 
quadprog 
 x 


 x  x  x 
coneprog 
 x  x 
 x  x  x  x 
lsqlin  x  x  x  x 
 x  x  x 
lsqnonneg  x  x  x  x 
 x  x  x 
lsqnonlin  x  x  x  x 

 x  x 
fminunc 
 x 
 x 


 x 
fmincon 
 x 




 x 
fminbnd  x  x  x  x 


 x 
fminsearch  x  x  x  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 leastsquares
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
leastsquares solver (one with a name beginning lsq
). If you do,
solve
throws an error, because these solvers cannot
maximize.
For equation solving, this table contains the available solvers for each problem type. In the table,
* indicates the default solver for the problem type.
Y indicates an available solver.
N indicates an unavailable solver.
Supported Solvers for Equations
Equation Type  lsqlin  lsqnonneg  fzero  fsolve  lsqnonlin 

Linear  *  N  Y (scalar only)  Y  Y 
Linear plus bounds  *  Y  N  N  Y 
Scalar nonlinear  N  N  *  Y  Y 
Nonlinear system  N  N  N  *  Y 
Nonlinear system plus bounds  N  N  N  N  * 
Example: 'intlinprog'
Data Types: char
 string
ObjectiveDerivative
— Indication to use automatic differentiation for objective function
'auto'
(default)  'autoforward'
 'autoreverse'
 'finitedifferences'
Indication to use automatic differentiation (AD) for nonlinear
objective function, specified as 'auto'
(use AD if
possible), 'autoforward'
(use forward AD if
possible), 'autoreverse'
(use reverse AD if
possible), or 'finitedifferences'
(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 ProblemBased 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 leastsquares objective function,
fmincon
andfminunc
default to forward AD for the objective function. For the definition of a problembased leastsquares objective function, see Write Objective Function for ProblemBased 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: 'finitedifferences'
Data Types: char
 string
ConstraintDerivative
— Indication to use automatic differentiation for constraint functions
'auto'
(default)  'autoforward'
 'autoreverse'
 'finitedifferences'
Indication to use automatic differentiation (AD) for nonlinear
constraint functions, specified as 'auto'
(use AD if
possible), 'autoforward'
(use forward AD if
possible), 'autoreverse'
(use reverse AD if
possible), or 'finitedifferences'
(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 ProblemBased 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 leastsquares objective function,
fmincon
andfminunc
default to forward AD for the objective function. For the definition of a problembased leastsquares objective function, see Write Objective Function for ProblemBased 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: 'finitedifferences'
Data Types: char
 string
EquationDerivative
— Indication to use automatic differentiation for equations
'auto'
(default)  'autoforward'
 'autoreverse'
 'finitedifferences'
Indication to use automatic differentiation (AD) for nonlinear
constraint functions, specified as 'auto'
(use AD if
possible), 'autoforward'
(use forward AD if
possible), 'autoreverse'
(use reverse AD if
possible), or 'finitedifferences'
(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 ProblemBased 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 leastsquares objective function,
fmincon
andfminunc
default to forward AD for the objective function. For the definition of a problembased leastsquares objective function, see Write Objective Function for ProblemBased 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: 'finitedifferences'
Data Types: char
 string
Output Arguments
sol
— Solution
structure  OptimizationValues
vector
Solution, returned as a structure or an OptimizationValues
vector. sol
is an
OptimizationValues
vector when the problem is multiobjective. For
singleobjective problems, the fields of the returned structure are the names of the
optimization variables in the problem. See optimvar
.
fval
— Objective function value at the solution
real number  real vector  real matrix  structure
Objective function value at the solution, returned as one of the following:
Problem Type  Returned Value(s) 

Optimize scalar objective function f(x)  Real number f(sol) 
Least squares  Real number, the sum of squares of the residuals at the solution 
Solve equation  If 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  
Multiobjective  Matrix 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
exitflag
— Reason solver stopped
enumeration variable
Reason the solver stopped, returned as an enumeration variable. You can convert
exitflag
to its numeric equivalent using
double(exitflag)
, and to its string equivalent using
string(exitflag)
.
This table describes the exit flags for the intlinprog
solver.
Exit Flag for intlinprog  Numeric Equivalent  Meaning 

OptimalWithPoorFeasibility  3  The solution is feasible with respect to the relative

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  For optimization problems, the problem is infeasible. Or, for
the For equation problems, no solution found. 
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 coneprog
solver.
Exit Flag for coneprog  Numeric Equivalent  Meaning 

OptimalSolution  1  The solver converged to a solution

SolverLimitExceeded  0  The number of iterations exceeds

NoFeasiblePointFound  2  The problem is infeasible. 
Unbounded  3  The problem is unbounded. 
DirectionTooSmall 
 The search direction became too small. No further progress could be made. 
Unstable  10  The problem is numerically unstable. 
This table describes the exit flags for the lsqcurvefit
or
lsqnonlin
solver.
Exit Flag for lsqnonlin  Numeric Equivalent  Meaning 

SearchDirectionTooSmall  4  Magnitude of search direction was smaller than

FunctionChangeBelowTolerance  3  Change in the residual was less than

StepSizeBelowTolerance 
 Step size smaller than

OptimalSolution  1  The solver converged to a solution

SolverLimitExceeded  0  Number of iterations exceeded

OutputFcnStop  1  Stopped by an output function or plot function. 
NoFeasiblePointFound  2  For optimization problems, problem is infeasible: the bounds
For equation problems, no solution found. 
This table describes the exit flags for the fminunc
solver.
Exit Flag for fminunc  Numeric Equivalent  Meaning 

NoDecreaseAlongSearchDirection  5  Predicted decrease in the objective function is less than the

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

This table describes the exit flags for the fsolve
solver.
Exit Flag for fsolve  Numeric Equivalent  Meaning 

SearchDirectionTooSmall  4  Magnitude of the search direction is less than

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  Converged to a point that is not a root. 
TrustRegionRadiusTooSmall  3  Equation not solved. Trust region radius became too small
( 
This table describes the exit flags for the fzero
solver.
Exit Flag for fzero  Numeric Equivalent  Meaning 

OptimalSolution  1  Equation solved. 
OutputFcnStop  1  Stopped by an output function or plot function. 
FoundNaNInfOrComplex  4 

SingularPoint  5  Might have converged to a singular point. 
CannotDetectSignChange  6  Did not find two points with opposite signs of function value. 
This table describes the exit flags for the patternsearch
solver.
Exit Flag for patternsearch  Numeric Equivalent  Meaning 

SearchDirectionTooSmall  4  The magnitude of the step is smaller than machine precision,
and the constraint violation is less than

FunctionChangeBelowTolerance  3  The change in 
StepSizeBelowTolerance 
 Change in 
SolverConvergedSuccessfully  1  Without nonlinear constraints
— The magnitude of the mesh size is less than the specified
tolerance, and the constraint violation is less than

With nonlinear constraints
— The magnitude of the complementarity
measure (defined after this table) is less than
 
SolverLimitExceeded  0  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 c_{i}λ_{i}, where c_{i} is the nonlinear inequality constraint violation, and λ_{i} is the corresponding Lagrange multiplier.
This table describes the exit flags for the ga
solver.
Exit Flag for ga  Numeric Equivalent  Meaning 

MinimumFitnessLimitReached  5  Minimum fitness limit 
SearchDirectionTooSmall  4  The magnitude of the step is smaller than machine precision,
and the constraint violation is less than

FunctionChangeBelowTolerance  3  Value of the fitness function did not change in

SolverConvergedSuccessfully  1  Without nonlinear constraints
— Average cumulative change in value of the fitness function
over 
With nonlinear constraints
— Magnitude of the complementarity measure (see Complementarity Measure (Global Optimization Toolbox)) is
less than  
SolverLimitExceeded  0  Maximum number of generations 
OutputFcnStop  1  Stopped by an output function or plot function. 
NoFeasiblePointFound  2  No feasible point found. 
StallTimeLimitExceeded  4  Stall time limit 
TimeLimitExceeded  5  Time limit 
This table describes the exit flags for the particleswarm
solver.
Exit Flag for particleswarm  Numeric Equivalent  Meaning 

SolverConvergedSuccessfully  1  Relative change in the objective value over the last

SolverLimitExceeded  0  Number of iterations exceeded

OutputFcnStop  1  Iterations stopped by output function or plot function. 
NoFeasiblePointFound  2  Bounds are inconsistent: for some 
Unbounded  3  Best objective function value is below

StallTimeLimitExceeded  4  Best objective function value did not change within

TimeLimitExceeded  5  Run time exceeded 
This table describes the exit flags for the simulannealbnd
solver.
Exit Flag for simulannealbnd  Numeric Equivalent  Meaning 

ObjectiveValueBelowLimit  5  Objective function value is less than

SolverConvergedSuccessfully  1  Average change in the value of the objective function over

SolverLimitExceeded  0  Maximum number of generations 
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 surrogateopt  Numeric Equivalent  Meaning 

BoundsEqual  10  Problem has a unique feasible solution due to one of the following:

FeasiblePointFound  3  Feasible point found. Solver stopped because too few new feasible points were found to continue. 
ObjectiveLimitAttained  1  The objective function value is less than

SolverLimitExceeded  0  The number of function evaluations exceeds

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:

This table describes the exit flags for the MultiStart
and
GlobalSearch
solvers.
Exit Flag for MultiStart or
GlobalSearch  Numeric Equivalent  Meaning 

LocalMinimumFoundSomeConverged  2  At least one local minimum found. Some runs of the local solver converged. 
LocalMinimumFoundAllConverged  1  At least one local minimum found. All runs of the local solver converged. 
SolverLimitExceeded  0  No local minimum found. Local solver called at least once and at least one local solver call ran out of iterations. 
OutputFcnStop  –1  Stopped by an output function or plot function. 
NoFeasibleLocalMinimumFound  –2  No feasible local minimum found. 
TimeLimitExceeded  –5  MaxTime limit exceeded. 
NoSolutionFound  –8  No solution found. All runs had local solver exit flag –2 or smaller, not all equal –2. 
FailureInSuppliedFcn  –10  Encountered failures in the objective or nonlinear constraint functions. 
This table describes the exit flags for the paretosearch
solver.
Exit Flag for paretosearch  Numeric Equivalent  Meaning 

SolverConvergedSuccessfully  1  One of the following conditions is met:

SolverLimitExceeded  0  Number of iterations exceeds
options.MaxIterations , or the number of function
evaluations exceeds
options.MaxFunctionEvaluations . 
OutputFcnStop  –1  Stopped by an output function or plot function. 
NoFeasiblePointFound  –2  Solver cannot find a point satisfying all the constraints. 
TimeLimitExceeded  –5  Optimization time exceeds options.MaxTime . 
This table describes the exit flags for the gamultiobj
solver.
Exit Flag for paretosearch  Numeric Equivalent  Meaning 

SolverConvergedSuccessfully  1  Geometric 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. 
SolverLimitExceeded  0  Number of generations exceeds
options.MaxGenerations . 
OutputFcnStop  –1  Stopped by an output function or plot function. 
NoFeasiblePointFound  –2  Solver cannot find a point satisfying all the constraints. 
TimeLimitExceeded  –5  Optimization time exceeds options.MaxTime . 
output
— Information about optimization process
structure
Information about the optimization process, returned as a structure. The output
structure contains the fields in the relevant underlying solver output field, depending
on which solver solve
called:
'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"
whenprob
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 ofOptimizationValues
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:
"reverseAD"
for reverse automatic differentiation"forwardAD"
for forward automatic differentiation"finitedifferences"
for finite difference estimation"closedform"
for linear or quadratic functions
For details, see Automatic Differentiation Background.
lambda
— Lagrange multipliers at the solution
structure
Lagrange multipliers at the solution, returned as a structure.
Note
solve
does not return lambda
for
equationsolving 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 variableLowerBound
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 structurelambda.Variables.
.variablename
.LowerUpper
– Lagrange multipliers associated with the variableUpperBound
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 structurelambda.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 structurelambda.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
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 solvers
function. You can override the default by using the 'solver'
namevalue pair argument when calling
solve
.
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. When automatic
differentiation does not apply, solvers estimate derivatives using finite differences. For
details of automatic differentiation, see Automatic Differentiation Background. You can control how
solve
uses automatic differentiation with the ObjectiveDerivative
namevalue argument.
For the algorithm that
intlinprog
uses to solve MILP problems, see Legacy 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 leastsquares solver
algorithms, see LeastSquares (Model Fitting) Algorithms. For nonlinear solver algorithms, see Unconstrained Nonlinear Optimization Algorithms and
Constrained Nonlinear Optimization Algorithms.
For Global Optimization Toolbox solver algorithms, see Global Optimization Toolbox documentation.
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 LeastSquares (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 ProblemBased Least Squares. For an example, see
Nonnegative Linear Least Squares, ProblemBased.
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
isfmincon
,fminunc
,fsolve
, orlsqnonlin
.For optimization problems, the
'ObjectiveDerivative'
and'ConstraintDerivative'
namevalue pair arguments forsolve
orprob2struct
are set to'auto'
(default),'autoforward'
, or'autoreverse'
.For equation problems, the
'EquationDerivative'
option is set to'auto'
(default),'autoforward'
, or'autoreverse'
.
When AD Applies  All Constraint Functions Supported  One or More Constraints Not Supported 

Objective Function Supported  AD used for objective and constraints  AD used for objective only 
Objective Function Not Supported  AD used for constraints only  AD not used 
Note
For linear or quadratic objective or constraint functions, applicable solvers always use explicit function gradients. These gradients are not produced using AD. See Closed Form.
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 leastsquares objective function,
fmincon
andfminunc
default to forward AD for the objective function. For the definition of a problembased leastsquares objective function, see Write Objective Function for ProblemBased 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.
options = optimoptions('fmincon','SpecifyObjectiveGradient',true,... 'SpecifyConstraintGradient',true); 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 ProblemBased Workflow.
Extended Capabilities
Automatic Parallel Support
Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.
solve
estimates derivatives in parallel for nonlinear solvers
when the UseParallel
option for the solver is
true
. For example,
options = optimoptions('fminunc','UseParallel',true); [sol,fval] = solve(prob,x0,'Options',options)
solve
does not use parallel derivative estimation when all
objective and nonlinear constraint functions consist only of supported operations,
as described in Supported Operations for Optimization Variables and Expressions. In this case,
solve
uses automatic differentiation for calculating
derivatives. See Automatic Differentiation.
You can override automatic differentiation and use finite difference estimates in
parallel by setting the 'ObjectiveDerivative'
and 'ConstraintDerivative'
arguments to
'finitedifferences'
.
When you specify a Global Optimization Toolbox solver that support parallel computation (ga
(Global Optimization Toolbox), particleswarm
(Global Optimization Toolbox), patternsearch
(Global Optimization Toolbox), and surrogateopt
(Global Optimization Toolbox)), solve
compute in parallel when
the UseParallel
option for the solver is true
.
For example,
options = optimoptions("patternsearch","UseParallel",true); [sol,fval] = solve(prob,x0,"Options",options,"Solver","patternsearch")
Version History
Introduced in R2017bR2018b: solve(prob,solver)
, solve(prob,options)
, and solve(prob,solver,options)
syntaxes have been removed
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.
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