Solve multiobjective goal attainment problems
fgoalattain
solves the goal attainment problem, a formulation
for minimizing a multiobjective optimization problem.
fgoalattain
finds the minimum of a problem specified by
$$\underset{x,\gamma}{\text{minimize}}\text{}\gamma \text{suchthat}\{\begin{array}{c}F(x)\text{weight}\cdot \gamma \le \text{goal}\\ c(x)\le 0\\ ceq(x)=0\\ A\cdot x\le b\\ Aeq\cdot x=beq\\ lb\le x\le ub.\end{array}$$
weight
, goal
, b, and
beq are vectors, A and Aeq are
matrices, and F(x),
c(x), and
ceq(x), are functions that return vectors.
F(x), c(x), and
ceq(x) can be nonlinear functions.
x, lb, and ub can be passed as vectors or matrices; see Matrix Arguments.
tries to make the objective functions supplied by x
= fgoalattain(fun
,x0
,goal
,weight
)fun
attain the goals
specified by goal
by varying x
, starting at
x0
, with weight specified by weight
.
Note
Passing Extra Parameters explains how to pass extra parameters to the objective functions and nonlinear constraint functions, if necessary.
solves the goal attainment problem subject to the bounds
x
= fgoalattain(fun
,x0
,goal
,weight
,A
,b
,Aeq
,beq
,lb
,ub
)lb
≤ x
≤ ub
.
If no equalities exist, set Aeq = []
and beq = []
. If
x(i)
is unbounded below, set lb(i) = Inf
; if
x(i)
is unbounded above, set ub(i) = Inf
.
Note
Note
If the specified input bounds for a problem are inconsistent, the output
x
is x0
and the output fval
is []
.
[
additionally returns the attainment factor at the solution x
,fval
,attainfactor
,exitflag
,output
]
= fgoalattain(___)x
, a value
exitflag
that describes the exit condition of
fgoalattain
, and a structure output
with information
about the optimization process.
Consider the twoobjective function
$$F(x)=\left[\genfrac{}{}{0ex}{}{2+(x3{)}^{2}}{5+{x}^{2}/4}\right].$$
This function clearly minimizes $${F}_{1}(x)$$ at $$x=3$$, attaining the value 2, and minimizes $${F}_{2}(x)$$ at $$x=0$$, attaining the value 5.
Set the goal [3,6] and weight [1,1], and solve the goal attainment problem starting at x0
= 1.
fun = @(x)[2+(x3)^2;5+x^2/4]; goal = [3,6]; weight = [1,1]; x0 = 1; x = fgoalattain(fun,x0,goal,weight)
Local minimum possible. Constraints satisfied. fgoalattain stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.
x = 2.0000
Find the value of $$F(x)$$ at the solution.
fun(x)
ans = 2×1
3.0000
6.0000
fgoalattain
achieves the goals exactly.
The objective function is
$$F(x)=\left[\genfrac{}{}{0ex}{}{2+\Vert x{p}_{1}{\Vert}^{2}}{5+\Vert x{p}_{2}{\Vert}^{2}/4}\right].$$
Here, p_1
= [2,3] and p_2
= [4,1]. The goal is [3,6], the weight is [1,1], and the linear constraint is $${x}_{1}+{x}_{2}\le 4$$.
Create the objective function, goal, and weight.
p_1 = [2,3]; p_2 = [4,1]; fun = @(x)[2 + norm(xp_1)^2;5 + norm(xp_2)^2/4]; goal = [3,6]; weight = [1,1];
Create the linear constraint matrices A
and b
representing A*x <= b
.
A = [1,1]; b = 4;
Set an initial point [1,1] and solve the goal attainment problem.
x0 = [1,1]; x = fgoalattain(fun,x0,goal,weight,A,b)
Local minimum possible. Constraints satisfied. fgoalattain stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2
2.0694 1.9306
Find the value of $$F(x)$$ at the solution.
fun(x)
ans = 2×1
3.1484
6.1484
fgoalattain
does not meet the goals. Because the weights are equal, the solver underachieves each goal by the same amount.
The objective function is
$$F(x)=\left[\genfrac{}{}{0ex}{}{2+\Vert x{p}_{1}{\Vert}^{2}}{5+\Vert x{p}_{2}{\Vert}^{2}/4}\right].$$
Here, p_1
= [2,3] and p_2
= [4,1]. The goal is [3,6], the weight is [1,1], and the bounds are $$0\le {x}_{1}\le 3$$, $$2\le {x}_{2}\le 5$$.
Create the objective function, goal, and weight.
p_1 = [2,3]; p_2 = [4,1]; fun = @(x)[2 + norm(xp_1)^2;5 + norm(xp_2)^2/4]; goal = [3,6]; weight = [1,1];
Create the bounds.
lb = [0,2]; ub = [3,5];
Set the initial point to [1,4] and solve the goal attainment problem.
x0 = [1,4];
A = []; % no linear constraints
b = [];
Aeq = [];
beq = [];
x = fgoalattain(fun,x0,goal,weight,A,b,Aeq,beq,lb,ub)
Local minimum possible. Constraints satisfied. fgoalattain stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2
2.6667 2.3333
Find the value of $$F(x)$$ at the solution.
fun(x)
ans = 2×1
2.8889
5.8889
fgoalattain
more than meets the goals. Because the weights are equal, the solver overachieves each goal by the same amount.
The objective function is
$$F(x)=\left[\genfrac{}{}{0ex}{}{2+\Vert x{p}_{1}{\Vert}^{2}}{5+\Vert x{p}_{2}{\Vert}^{2}/4}\right].$$
Here, p_1
= [2,3] and p_2
= [4,1]. The goal is [3,6], the weight is [1,1], and the nonlinear constraint is $$\Vert x{\Vert}^{2}\le 4$$.
Create the objective function, goal, and weight.
p_1 = [2,3]; p_2 = [4,1]; fun = @(x)[2 + norm(xp_1)^2;5 + norm(xp_2)^2/4]; goal = [3,6]; weight = [1,1];
The nonlinear constraint function is in the norm4.m
file.
type norm4
function [c,ceq] = norm4(x) ceq = []; c = norm(x)^2  4;
Create empty input arguments for the linear constraints and bounds.
A = []; Aeq = []; b = []; beq = []; lb = []; ub = [];
Set the initial point to [1,1] and solve the goal attainment problem.
x0 = [1,1]; x = fgoalattain(fun,x0,goal,weight,A,b,Aeq,beq,lb,ub,@norm4)
Local minimum possible. Constraints satisfied. fgoalattain stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2
1.1094 1.6641
Find the value of $$F(x)$$ at the solution.
fun(x)
ans = 2×1
4.5778
7.1991
fgoalattain
does not meet the goals. Despite the equal weights, $${F}_{1}(x)$$ is about 1.58 from its goal of 3, and $${F}_{2}(x)$$ is about 1.2 from its goal of 6. The nonlinear constraint prevents the solution x
from achieving the goals equally.
Monitor a goal attainment solution process by setting options to return iterative display.
options = optimoptions('fgoalattain','Display','iter');
The objective function is
$$F(x)=\left[\genfrac{}{}{0ex}{}{2+\Vert x{p}_{1}{\Vert}^{2}}{5+\Vert x{p}_{2}{\Vert}^{2}/4}\right].$$
Here, p_1
= [2,3] and p_2
= [4,1]. The goal is [3,6], the weight is [1,1], and the linear constraint is $${x}_{1}+{x}_{2}\le 4$$.
Create the objective function, goal, and weight.
p_1 = [2,3]; p_2 = [4,1]; fun = @(x)[2 + norm(xp_1)^2;5 + norm(xp_2)^2/4]; goal = [3,6]; weight = [1,1];
Create the linear constraint matrices A
and b
representing A*x <= b
.
A = [1,1]; b = 4;
Create empty input arguments for the linear equality constraints, bounds, and nonlinear constraints.
Aeq = []; beq = []; lb = []; ub = []; nonlcon = [];
Set an initial point [1,1] and solve the goal attainment problem.
x0 = [1,1]; x = fgoalattain(fun,x0,goal,weight,A,b,Aeq,beq,lb,ub,nonlcon,options)
Attainment Max Line search Directional Iter Fcount factor constraint steplength derivative Procedure 0 4 0 4 1 9 1 2.5 1 0.535 2 14 1.235e08 0.2813 1 0.883 3 19 0.1452 0.005926 1 0.883 4 24 0.1484 2.868e06 1 0.883 5 29 0.1484 6.757e13 1 0.883 Hessian modified Local minimum possible. Constraints satisfied. fgoalattain stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2
2.0694 1.9306
The positive value of the reported attainment factor indicates that fgoalattain
does not find a solution satisfying the goals.
The objective function is
$$F(x)=\left[\genfrac{}{}{0ex}{}{2+\Vert x{p}_{1}{\Vert}^{2}}{5+\Vert x{p}_{2}{\Vert}^{2}/4}\right].$$
Here, p_1
= [2,3] and p_2
= [4,1]. The goal is [3,6], the weight is [1,1], and the linear constraint is $${x}_{1}+{x}_{2}\le 4$$.
Create the objective function, goal, and weight.
p_1 = [2,3]; p_2 = [4,1]; fun = @(x)[2 + norm(xp_1)^2;5 + norm(xp_2)^2/4]; goal = [3,6]; weight = [1,1];
Create the linear constraint matrices A
and b
representing A*x <= b
.
A = [1,1]; b = 4;
Set an initial point [1,1] and solve the goal attainment problem. Request the value of the objective function.
x0 = [1,1]; [x,fval] = fgoalattain(fun,x0,goal,weight,A,b)
Local minimum possible. Constraints satisfied. fgoalattain stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2
2.0694 1.9306
fval = 2×1
3.1484
6.1484
The objective function values are higher than the goal, meaning fgoalattain
does not satisfy the goal.
The objective function is
$$F(x)=\left[\genfrac{}{}{0ex}{}{2+\Vert x{p}_{1}{\Vert}^{2}}{5+\Vert x{p}_{2}{\Vert}^{2}/4}\right].$$
Here, p_1
= [2,3] and p_2
= [4,1]. The goal is [3,6], the weight is [1,1], and the linear constraint is $${x}_{1}+{x}_{2}\le 4$$.
Create the objective function, goal, and weight.
p_1 = [2,3]; p_2 = [4,1]; fun = @(x)[2 + norm(xp_1)^2;5 + norm(xp_2)^2/4]; goal = [3,6]; weight = [1,1];
Create the linear constraint matrices A
and b
representing A*x <= b
.
A = [1,1]; b = 4;
Set an initial point [1,1] and solve the goal attainment problem. Request the value of the objective function, attainment factor, exit flag, output structure, and Lagrange multipliers.
x0 = [1,1]; [x,fval,attainfactor,exitflag,output,lambda] = fgoalattain(fun,x0,goal,weight,A,b)
Local minimum possible. Constraints satisfied. fgoalattain stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2
2.0694 1.9306
fval = 2×1
3.1484
6.1484
attainfactor = 0.1484
exitflag = 4
output = struct with fields:
iterations: 6
funcCount: 29
lssteplength: 1
stepsize: 4.0824e13
algorithm: 'activeset'
firstorderopt: []
constrviolation: 6.7568e13
message: '...'
lambda = struct with fields:
lower: [2x1 double]
upper: [2x1 double]
eqlin: [0x1 double]
eqnonlin: [0x1 double]
ineqlin: 0.5394
ineqnonlin: [0x1 double]
The positive value of attainfactor
indicates that the goals are not attained; you can also see this by comparing fval
with goal
.
The lambda.ineqlin
value is nonzero, indicating that the linear inequality constrains the solution.
The objective function is
$$F(x)=\left[\genfrac{}{}{0ex}{}{2+\Vert x{p}_{1}{\Vert}^{2}}{5+\Vert x{p}_{2}{\Vert}^{2}/4}\right].$$
Here, p_1
= [2,3] and p_2
= [4,1]. The goal is [3,6], and the initial weight is [1,1].
Create the objective function, goal, and initial weight.
p_1 = [2,3]; p_2 = [4,1]; fun = @(x)[2 + norm(xp_1)^2;5 + norm(xp_2)^2/4]; goal = [3,6]; weight = [1,1];
Set the linear constraint $${x}_{1}+{x}_{2}\le 4$$.
A = [1 1]; b = 4;
Solve the goal attainment problem starting from the point x0 = [1 1]
.
x0 = [1 1]; [x,fval] = fgoalattain(fun,x0,goal,weight,A,b)
Local minimum possible. Constraints satisfied. fgoalattain stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2
2.0694 1.9306
fval = 2×1
3.1484
6.1484
Each component of fval
is above the corresponding component of goal
, indicating that the goals are not attained.
Increase the importance of satisfying the first goal by setting weight(1)
to a smaller value.
weight(1) = 1/10; [x,fval] = fgoalattain(fun,x0,goal,weight,A,b)
Local minimum possible. Constraints satisfied. fgoalattain stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2
2.0115 1.9885
fval = 2×1
3.0233
6.2328
Now the value of fval(1)
is much closer to goal(1)
, whereas fval(2)
is farther from goal(2)
.
Change goal(2)
to 7, which is above the current solution. The solution changes.
goal(2) = 7; [x,fval] = fgoalattain(fun,x0,goal,weight,A,b)
Local minimum possible. Constraints satisfied. fgoalattain stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2
1.9639 2.0361
fval = 2×1
2.9305
6.3047
Both components of fval
are less than the corresponding components of goal
. But fval(1)
is much closer to goal(1)
than fval(2)
is to goal(2)
. A smaller weight is more likely to make its component nearly satisfied when the goals cannot be achieved, but makes the degree of overachievement less when the goal can be achieved.
Change the weights to be equal. The fval
results have equal distance from their goals.
weight(2) = 1/10; [x,fval] = fgoalattain(fun,x0,goal,weight,A,b)
Local minimum possible. Constraints satisfied. fgoalattain stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2
1.7613 2.2387
fval = 2×1
2.6365
6.6365
Constraints can keep the resulting fval
from being equally close to the goals. For example, set an upper bound of 2 on x(2)
.
ub = [Inf,2]; lb = []; Aeq = []; beq = []; [x,fval] = fgoalattain(fun,x0,goal,weight,A,b,Aeq,beq,lb,ub)
Local minimum possible. Constraints satisfied. fgoalattain stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2
2.0000 2.0000
fval = 2×1
3.0000
6.2500
In this case, fval(1)
meets its goal exactly, but fval(2)
is less than its goal.
fun
— Objective functionsObjective functions, specified as a function handle or function name.
fun
is a function that accepts a vector x
and
returns a vector F
, the objective functions evaluated at
x
. You can specify the function fun
as a
function handle for a function file:
x = fgoalattain(@myfun,x0,goal,weight)
where myfun
is a MATLAB^{®} function such as
function F = myfun(x) F = ... % Compute function values at x.
fun
can also be a function handle for an anonymous
function:
x = fgoalattain(@(x)sin(x.*x),x0,goal,weight);
fgoalattain
passes x
to
your objective function and any nonlinear constraint functions in the shape of the
x0
argument. For example, if x0
is a 5by3 array,
then fgoalattain
passes x
to fun
as a
5by3 array. However, fgoalattain
multiplies linear constraint matrices
A
or Aeq
with x
after
converting x
to the column vector x(:)
.
To make an objective function as near as possible to a goal value (that is, neither
greater than nor less than), use optimoptions
to set the
EqualityGoalCount
option to the number of objectives required to be
in the neighborhood of the goal values. Such objectives must be
partitioned into the first elements of the vector F
returned by
fun
.
Suppose that the gradient of the objective function can also be computed
and the SpecifyObjectiveGradient
option is
true
, as set by:
options = optimoptions('fgoalattain','SpecifyObjectiveGradient',true)
In this case, the function fun
must return, in the second output
argument, the gradient value G
(a matrix) at x
.
The gradient consists of the partial derivative dF/dx of each
F
at the point x
. If F
is a
vector of length m
and x
has length
n
, where n
is the length of
x0
, then the gradient G
of
F(x)
is an n
bym
matrix
where G(i,j)
is the partial derivative of F(j)
with respect to x(i)
(that is, the j
th column of
G
is the gradient of the j
th objective function
F(j)
).
Note
Setting SpecifyObjectiveGradient
to true
is
effective only when the problem has no nonlinear constraints, or the problem has a
nonlinear constraint with SpecifyConstraintGradient
set to
true
. Internally, the objective is folded into the constraints,
so the solver needs both gradients (objective and constraint) supplied in order to
avoid estimating a gradient.
Data Types: char
 string
 function_handle
x0
— Initial pointInitial point, specified as a real vector or real array. Solvers use the number of elements in
x0
and the size of x0
to determine the number
and size of variables that fun
accepts.
Example: x0 = [1,2,3,4]
Data Types: double
goal
— Goal to attainGoal to attain, specified as a real vector. fgoalattain
attempts
to find the smallest multiplier γ that makes these inequalities hold
for all values of i at the solution x:
$${F}_{i}(x){\text{goal}}_{i}\le {\text{weight}}_{i}\text{\hspace{0.17em}}\gamma .$$
Assuming that weight
is a positive vector:
If the solver finds a point x
that simultaneously achieves
all the goals, then the attainment factor γ is negative, and the
goals are overachieved.
If the solver cannot find a point x
that simultaneously
achieves all the goals, then the attainment factor γ is positive,
and the goals are underachieved.
Example: [1 3 6]
Data Types: double
weight
— Relative attainment factorRelative attainment factor, specified as a real vector.
fgoalattain
attempts to find the smallest multiplier
γ that makes these inequalities hold for all values of
i at the solution x:
$${F}_{i}(x){\text{goal}}_{i}\le {\text{weight}}_{i}\text{\hspace{0.17em}}\gamma .$$
When the values of goal
are all nonzero, to
ensure the same percentage of underachievement or overattainment of the active
objectives, set weight
to abs(goal)
. (The active
objectives are the set of objectives that are barriers to further improvement of the
goals at the solution.)
Note
Setting a component of the weight
vector to zero causes the
corresponding goal constraint to be treated as a hard constraint rather than a goal
constraint. An alternative method to setting a hard constraint is to use the input
argument nonlcon
.
When weight
is positive, fgoalattain
attempts
to make the objective functions less than the goal values. To make the objective
functions greater than the goal values, set weight
to be negative
rather than positive. To see some effects of weights on a solution, see Effects of Weights, Goals, and Constraints in Goal Attainment.
To make an objective function as near as possible to a goal value, use the
EqualityGoalCount
option and specify the objective as the first
element of the vector returned by fun
(see fun
and options
). For an example, see MultiObjective Goal Attainment Optimization.
Example: abs(goal)
Data Types: double
A
— Linear inequality constraintsLinear inequality constraints, specified as a real matrix. A
is
an M
byN
matrix, where M
is
the number of inequalities, and N
is the number
of variables (number of elements in x0
). For
large problems, pass A
as a sparse matrix.
A
encodes the M
linear
inequalities
A*x <= b
,
where x
is the column vector of N
variables x(:)
,
and b
is a column vector with M
elements.
For example, consider these inequalities:
x_{1} + 2x_{2} ≤
10
3x_{1} +
4x_{2} ≤ 20
5x_{1} +
6x_{2} ≤ 30,
Specify the inequalities by entering the following constraints.
A = [1,2;3,4;5,6]; b = [10;20;30];
Example: To specify that the x components sum to 1 or less, use A =
ones(1,N)
and b = 1
.
Data Types: double
b
— Linear inequality constraintsLinear inequality constraints, specified as a real vector. b
is
an M
element vector related to the A
matrix.
If you pass b
as a row vector, solvers internally
convert b
to the column vector b(:)
.
For large problems, pass b
as a sparse vector.
b
encodes the M
linear
inequalities
A*x <= b
,
where x
is the column vector of N
variables x(:)
,
and A
is a matrix of size M
byN
.
For example, consider these inequalities:
x_{1}
+ 2x_{2} ≤
10
3x_{1}
+ 4x_{2} ≤
20
5x_{1}
+ 6x_{2} ≤
30.
Specify the inequalities by entering the following constraints.
A = [1,2;3,4;5,6]; b = [10;20;30];
Example: To specify that the x components sum to 1 or less, use A =
ones(1,N)
and b = 1
.
Data Types: double
Aeq
— Linear equality constraintsLinear equality constraints, specified as a real matrix. Aeq
is
an Me
byN
matrix, where Me
is
the number of equalities, and N
is the number of
variables (number of elements in x0
). For large
problems, pass Aeq
as a sparse matrix.
Aeq
encodes the Me
linear
equalities
Aeq*x = beq
,
where x
is the column vector of N
variables x(:)
,
and beq
is a column vector with Me
elements.
For example, consider these inequalities:
x_{1} + 2x_{2} +
3x_{3} = 10
2x_{1} +
4x_{2} + x_{3} =
20,
Specify the inequalities by entering the following constraints.
Aeq = [1,2,3;2,4,1]; beq = [10;20];
Example: To specify that the x components sum to 1, use Aeq = ones(1,N)
and
beq = 1
.
Data Types: double
beq
— Linear equality constraintsLinear equality constraints, specified as a real vector. beq
is
an Me
element vector related to the Aeq
matrix.
If you pass beq
as a row vector, solvers internally
convert beq
to the column vector beq(:)
.
For large problems, pass beq
as a sparse vector.
beq
encodes the Me
linear
equalities
Aeq*x = beq
,
where x
is the column vector of N
variables
x(:)
, and Aeq
is a matrix of size
Me
byN
.
For example, consider these equalities:
x_{1}
+ 2x_{2} +
3x_{3} =
10
2x_{1}
+ 4x_{2} +
x_{3} =
20.
Specify the equalities by entering the following constraints.
Aeq = [1,2,3;2,4,1]; beq = [10;20];
Example: To specify that the x components sum to 1, use Aeq = ones(1,N)
and
beq = 1
.
Data Types: double
lb
— Lower boundsLower bounds, specified as a real vector or real array. If the number of elements in
x0
is equal to the number of elements in lb
,
then lb
specifies that
x(i) >= lb(i)
for all i
.
If numel(lb) < numel(x0)
, then lb
specifies
that
x(i) >= lb(i)
for 1 <=
i <= numel(lb)
.
If lb
has fewer elements than x0
, solvers issue a
warning.
Example: To specify that all x components are positive, use lb =
zeros(size(x0))
.
Data Types: double
ub
— Upper boundsUpper bounds, specified as a real vector or real array. If the number of elements in
x0
is equal to the number of elements in ub
,
then ub
specifies that
x(i) <= ub(i)
for all i
.
If numel(ub) < numel(x0)
, then ub
specifies
that
x(i) <= ub(i)
for 1 <=
i <= numel(ub)
.
If ub
has fewer elements than x0
, solvers issue
a warning.
Example: To specify that all x components are less than 1, use ub =
ones(size(x0))
.
Data Types: double
nonlcon
— Nonlinear constraintsNonlinear constraints, specified as a function handle or function name. nonlcon
is a function that accepts a vector or array x
and returns two arrays, c(x)
and ceq(x)
.
c(x)
is the array of nonlinear inequality constraints at x
. fgoalattain
attempts to satisfy
c(x) <= 0
for all entries of
c
.
ceq(x)
is the array of nonlinear equality constraints at x
. fgoalattain
attempts to satisfy
ceq(x) = 0
for all entries of
ceq
.
For example,
x = fgoalattain(@myfun,x0,...,@mycon)
where mycon
is a MATLAB function such as the
following:
function [c,ceq] = mycon(x) c = ... % Compute nonlinear inequalities at x. ceq = ... % Compute nonlinear equalities at x.
Suppose that the gradients of the constraints can also be computed and
the SpecifyConstraintGradient
option is true
, as
set by:
options = optimoptions('fgoalattain','SpecifyConstraintGradient',true)
In this case, the function nonlcon
must also return, in the third and
fourth output arguments, GC
, the gradient of c(x)
,
and GCeq
, the gradient of ceq(x)
. See Nonlinear Constraints for an explanation of how to “conditionalize” the
gradients for use in solvers that do not accept supplied gradients.
If nonlcon
returns a vector c
of m
components and x
has length n
, where
n
is the length of x0
, then the gradient
GC
of c(x)
is an
n
bym
matrix, where
GC(i,j)
is the partial derivative of c(j)
with
respect to x(i)
(that is, the j
th column of
GC
is the gradient of the j
th inequality
constraint c(j)
). Likewise, if ceq
has
p
components, the gradient GCeq
of
ceq(x)
is an n
byp
matrix,
where GCeq(i,j)
is the partial derivative of
ceq(j)
with respect to x(i)
(that is, the
j
th column of GCeq
is the gradient of the
j
th equality constraint ceq(j)
).
Note
Setting SpecifyConstraintGradient
to true
is
effective only when SpecifyObjectiveGradient
is set to
true
. Internally, the objective is folded into the
constraint, so the solver needs both gradients (objective and constraint) supplied
in order to avoid estimating a gradient.
Note
Because Optimization Toolbox™ functions accept only inputs of type double
,
usersupplied objective and nonlinear constraint functions must return outputs of
type double
.
See Passing Extra Parameters for an explanation of how to parameterize the
nonlinear constraint function nonlcon
, if necessary.
Data Types: char
 function_handle
 string
options
— Optimization optionsoptimoptions
 structure such as optimset
returnsOptimization options, specified as the output of optimoptions
or a structure such as optimset
returns.
Some options are absent from the
optimoptions
display. These options appear in italics in the following
table. For details, see View Options.
For details about options that have different names for
optimset
, see Current and Legacy Option Names.
Option  Description 

ConstraintTolerance  Termination tolerance on the constraint violation, a positive scalar.
The default is For

Diagnostics  Display of diagnostic information about the function to be minimized
or solved. The choices are 
DiffMaxChange  Maximum change in variables for finitedifference gradients (a
positive scalar). The default is 
DiffMinChange  Minimum change in variables for finitedifference gradients (a
positive scalar). The default is 
 Level of display (see Iterative Display):

EqualityGoalCount  Number of objectives required for the objective
For 
FiniteDifferenceStepSize 
Scalar or vector step size factor for finite differences. When
you set
sign′(x) = sign(x) except sign′(0) = 1 .
Central finite differences are
FiniteDifferenceStepSize expands to a vector. The default
is sqrt(eps) for forward finite differences, and eps^(1/3)
for central finite differences.
For 
FiniteDifferenceType  Type of finite differences used to estimate gradients, either
The algorithm is careful to obey bounds when estimating both types of finite differences. For example, it might take a backward step, rather than a forward step, to avoid evaluating at a point outside the bounds. For

FunctionTolerance  Termination tolerance on the function value (a positive scalar). The
default is For

FunValCheck  Check that signifies whether the objective function and constraint
values are valid. 
MaxFunctionEvaluations  Maximum number of function evaluations allowed (a positive integer).
The default is For

MaxIterations  Maximum number of iterations allowed (a positive integer). The
default is For

MaxSQPIter  Maximum number of SQP iterations allowed (a positive integer). The
default is 
MeritFunction  If this option is set to 
OptimalityTolerance 
Termination tolerance on the firstorder optimality (a positive
scalar). The default is For 
OutputFcn  One or more userdefined functions that an optimization function
calls at each iteration. Pass a function handle or a cell array of function
handles. The default is none ( 
PlotFcn  Plots showing various measures of progress while the algorithm
executes. Select from predefined plots or write your own. Pass a name,
function handle, or cell array of names or function handles. For custom plot
functions, pass function handles. The default is none
(
Custom plot functions use the same syntax as output functions. See Output Functions for Optimization Toolbox™ and Output Function and Plot Function Syntax. For 
RelLineSrchBnd  Relative bound (a real nonnegative scalar value) on the line search
step length such that the total displacement in 
RelLineSrchBndDuration  Number of iterations for which the bound specified in

SpecifyConstraintGradient  Gradient for nonlinear constraint functions defined by the user. When
this option is set to For

SpecifyObjectiveGradient  Gradient for the objective function defined by the user. Refer to the
description of For 
StepTolerance  Termination tolerance on For

TolConSQP  Termination tolerance on the inner iteration SQP constraint violation
(a positive scalar). The default is 
TypicalX  Typical 
UseParallel  Indication of parallel computing. When 
Example: optimoptions('fgoalattain','PlotFcn','optimplotfval')
problem
— Problem structureProblem structure, specified as a structure with the fields in this table.
Field Name  Entry 

 Objective function fun 
 Initial point for x 
 Goals to attain 
 Relative importance factors of goals 
 Matrix for linear inequality constraints 
 Vector for linear inequality constraints 
 Matrix for linear equality constraints 
 Vector for linear equality constraints 
lb  Vector of lower bounds 
ub  Vector of upper bounds 
 Nonlinear constraint function 
 'fgoalattain' 
 Options created with optimoptions 
You must supply at least the objective
, x0
,
goal
, weight
, solver
, and
options
fields in the problem
structure.
Data Types: struct
x
— SolutionSolution, returned as a real vector or real array. The size
of x
is the same as the size of x0
.
Typically, x
is a local solution to the problem
when exitflag
is positive. For information on
the quality of the solution, see When the Solver Succeeds.
fval
— Objective function values at solutionObjective function values at the solution, returned as a real array. Generally,
fval
= fun(x)
.
attainfactor
— Attainment factorAttainment factor, returned as a real number. attainfactor
contains the value of γ at the solution. If
attainfactor
is negative, the goals have been overachieved; if
attainfactor
is positive, the goals have been underachieved. See
goal
.
exitflag
— Reason fgoalattain
stoppedReason fgoalattain
stopped, returned as an integer.
 Function converged to a solution

 Magnitude of the search direction was less than the specified
tolerance, and the constraint violation was less than

 Magnitude of the directional derivative was less than the
specified tolerance, and the constraint violation was less than

 Number of iterations exceeded

 Stopped by an output function or plot function 
 No feasible point was found. 
output
— Information about optimization processInformation about the optimization process, returned as a structure with the fields in this table.
iterations  Number of iterations taken 
funcCount  Number of function evaluations 
lssteplength  Size of the line search step relative to the search direction 
constrviolation  Maximum of the constraint functions 
stepsize  Length of the last displacement in

algorithm  Optimization algorithm used 
firstorderopt  Measure of firstorder optimality 
message  Exit message 
lambda
— Lagrange multipliers at solutionLagrange multipliers at the solution, returned as a structure with the fields in this table.
For a description of the fgoalattain
algorithm and a discussion of goal
attainment concepts, see Algorithms.
The Optimize Live Editor task provides a visual interface for fgoalattain
.
To run in parallel, set the 'UseParallel'
option to true
.
options = optimoptions('
solvername
','UseParallel',true)
For more information, see Using Parallel Computing in Optimization Toolbox.
fmincon
 fminimax
 optimoptions
 Optimize
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