Find minimum of semiinfinitely constrained multivariable nonlinear function
Finds the minimum of a problem specified by
$$\underset{x}{\mathrm{min}}f(x)\text{suchthat}\{\begin{array}{c}A\cdot x\le b,\\ Aeq\cdot x=beq,\\ lb\le x\le ub,\\ c(x)\le 0,\\ ceq(x)=0,\\ {K}_{i}(x,{w}_{i})\le 0,\text{}1\le i\le n.\end{array}$$
b and beq are vectors, A and Aeq are matrices, c(x), ceq(x), and K_{i}(x,w_{i}) are functions that return vectors, and f(x) is a function that returns a scalar. f(x), c(x), and ceq(x) can be nonlinear functions. The vectors (or matrices) K_{i}(x,w_{i}) ≤ 0 are continuous functions of both x and an additional set of variables w_{1},w_{2},...,w_{n}. The variables w_{1},w_{2},...,w_{n} are vectors of, at most, length two.
x, lb, and ub can be passed as vectors or matrices; see Matrix Arguments.
x = fseminf(fun,x0,ntheta,seminfcon)
x = fseminf(fun,x0,ntheta,seminfcon,A,b)
x = fseminf(fun,x0,ntheta,seminfcon,A,b,Aeq,beq)
x = fseminf(fun,x0,ntheta,seminfcon,A,b,Aeq,beq,lb,ub)
x = fseminf(fun,x0,ntheta,seminfcon,A,b,Aeq,beq,lb,ub,options)
x = fseminf(problem)
[x,fval] = fseminf(...)
[x,fval,exitflag] = fseminf(...)
[x,fval,exitflag,output] = fseminf(...)
[x,fval,exitflag,output,lambda] = fseminf(...)
fseminf
finds a minimum of a semiinfinitely
constrained scalar function of several variables, starting at an initial
estimate. The aim is to minimize f(x)
so the constraints hold for all possible values of w_{i}∈ℜ^{1} (or w_{i}∈ℜ^{2}).
Because it is impossible to calculate all possible values of K_{i}(x,w_{i}),
a region must be chosen for w_{i} over
which to calculate an appropriately sampled set of values.
Note: Passing Extra Parameters explains how to pass extra parameters to the objective function and nonlinear constraint functions, if necessary. 
x = fseminf(fun,x0,ntheta,seminfcon)
starts
at x0
and finds a minimum of the function fun
constrained
by ntheta
semiinfinite constraints defined in seminfcon
.
x = fseminf(fun,x0,ntheta,seminfcon,A,b)
also
tries to satisfy the linear inequalities A*x ≤ b
.
x = fseminf(fun,x0,ntheta,seminfcon,A,b,Aeq,beq)
minimizes
subject to the linear equalities Aeq*x = beq
as well. Set A = []
and b = []
if
no inequalities exist.
x = fseminf(fun,x0,ntheta,seminfcon,A,b,Aeq,beq,lb,ub)
defines
a set of lower and upper bounds on the design variables in x
,
so that the solution is always in the range lb
≤ x
≤ ub
.
Note: See Iterations Can Violate Constraints. 
x = fseminf(fun,x0,ntheta,seminfcon,A,b,Aeq,beq,lb,ub,options)
minimizes
with the optimization options specified in options
.
Use optimoptions
to set these
options.
x = fseminf(problem)
finds the minimum
for problem
, where problem
is
a structure described in Input Arguments.
Create the problem
structure by exporting
a problem from Optimization app, as described in Exporting Your Work.
[x,fval] = fseminf(...)
returns
the value of the objective function fun
at the
solution x
.
[x,fval,exitflag] = fseminf(...)
returns
a value exitflag
that describes the exit condition.
[x,fval,exitflag,output] = fseminf(...)
returns
a structure output
that contains information about
the optimization.
[x,fval,exitflag,output,lambda] = fseminf(...)
returns
a structure lambda
whose fields contain the Lagrange
multipliers at the solution x
.
Note:
If the specified input bounds for a problem are inconsistent,
the output 
Function Input Arguments contains
general descriptions of arguments passed into fseminf
.
This section provides functionspecific details for fun
, ntheta
, options
, seminfcon
,
and problem
:
 The
function to be minimized. x = fseminf(@myfun,x0,ntheta,seminfcon) where function f = myfun(x) f = ... % Compute function value at x
fun = @(x)sin(x''*x); If
the gradient of options = optimoptions('fseminf','SpecifyObjectiveGradient',true) then
the function  
ntheta  The number of semiinfinite constraints.  
options  Options provides the functionspecific
details for the  
 The function that computes the vector of nonlinear
inequality constraints, x = fseminf(@myfun,x0,ntheta,@myinfcon) where function [c,ceq,K1,K2,...,Kntheta,S] = myinfcon(x,S) % Initial sampling interval if isnan(S(1,1)), S = ...% S has ntheta rows and 2 columns end w1 = ...% Compute sample set w2 = ...% Compute sample set ... wntheta = ... % Compute sample set K1 = ... % 1st semiinfinite constraint at x and w K2 = ... % 2nd semiinfinite constraint at x and w ... Kntheta = ...% Last semiinfinite constraint at x and w c = ... % Compute nonlinear inequalities at x ceq = ... % Compute the nonlinear equalities at x
The vectors or matrices
Passing Extra Parameters explains
how to parameterize  
problem 
 Objective function  
 Initial point for x  
ntheta  Number of semiinfinite constraints  
seminfcon  Semiinfinite constraint function  
 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  
 'fseminf'  
 Options created with optimoptions 
Function Input Arguments contains
general descriptions of arguments returned by fseminf
.
This section provides functionspecific details for exitflag
, lambda
,
and output
:
 Integer identifying the
reason the algorithm terminated. The following lists the values of  
 Function converged to a solution  
 Magnitude of the search direction was less than the specified
tolerance and constraint violation was less than  
 Magnitude of directional derivative was less than the
specified tolerance and constraint violation was less than  
 Number of iterations exceeded  
 Algorithm was terminated by the output function.  
 No feasible point was found.  
 Structure containing the
Lagrange multipliers at the solution  
lower  Lower bounds  
upper  Upper bounds  
ineqlin  Linear inequalities  
eqlin  Linear equalities  
ineqnonlin  Nonlinear inequalities  
eqnonlin  Nonlinear equalities  
 Structure containing information about the optimization. The fields of the structure are  
iterations  Number of iterations taken  
funcCount  Number of function evaluations  
lssteplength  Size of line search step relative to search direction  
stepsize  Final displacement in  
algorithm  Optimization algorithm used  
constrviolation  Maximum of constraint functions  
firstorderopt  Measure of firstorder optimality  
message  Exit message 
Optimization options used by fseminf
. Use optimoptions
to set or change options
.
See Optimization Options Reference for detailed
information.
Some options are absent from the optimoptions
display.
These options are listed in italics. For details, see View Options.
 Compare usersupplied derivatives
(gradients of objective or constraints) to finitedifferencing derivatives.
The choices are 
ConstraintTolerance  Termination tolerance on the constraint
violation, a positive scalar. The default is 
Diagnostics  Display 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 
Display  Level of display (see Iterative Display):

FiniteDifferenceStepSize  Scalar or vector step size factor for finite differences. When
you set
where
Scalar 
FiniteDifferenceType  Finite differences, used to estimate
gradients, are either The algorithm is careful to obey bounds when estimating both types of finite differences. So, for example, it could take a backward, rather than a forward, difference to avoid evaluating at a point outside bounds. 
FunctionTolerance  Termination tolerance on the function
value, a positive scalar. The default is 
FunValCheck  Check whether objective function
and constraints values are valid. 
MaxFunctionEvaluations  Maximum number of function evaluations
allowed, a positive integer. The default is 
MaxIterations  Maximum number of iterations allowed,
a positive integer. The default is 
MaxSQPIter  Maximum number of SQP iterations
allowed, a positive integer. The default is 
OptimalityTolerance  Termination tolerance on the firstorder optimality, a positive
scalar. The default is 
OutputFcn  Specify one or more userdefined
functions that an optimization function calls at each iteration, either
as a function handle or as a cell array of function handles. The default
is none ( 
PlotFcn  Plots various measures of progress
while the algorithm executes, select from predefined plots or write
your own. Pass a function handle or a cell array of function handles.
The default is none (
For information on writing a custom plot function, see Plot Functions. 
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 
SpecifyObjectiveGradient  Gradient for the objective function
defined by the user. See the preceding description of 
StepTolerance  Termination tolerance on 
TolConSQP  Termination tolerance on inner
iteration SQP constraint violation, a positive scalar. The default
is 
TypicalX  Typical 
The
optimization routine fseminf
might vary the recommended
sampling interval, S
, set in seminfcon
,
during the computation because values other than the recommended interval
might be more appropriate for efficiency or robustness. Also, the
finite region w_{i}, over which K_{i}(x,w_{i}) is
calculated, is allowed to vary during the optimization, provided that
it does not result in significant changes in the number of local minima
in K_{i}(x,w_{i}).
This example minimizes the function
(x – 1)^{2},
subject to the constraints
0 ≤ x ≤ 2
g(x, t)
= (x – 1/2) – (t –
1/2)^{2} ≤ 0 for all 0 ≤ t ≤
1.
The unconstrained objective function is minimized at x = 1. However, the constraint,
g(x, t) ≤ 0 for all 0 ≤ t ≤ 1,
implies x ≤ 1/2. You can see this by noticing that (t – 1/2)^{2} ≥ 0, so
max_{t} g(x, t) = (x– 1/2).
Therefore
max_{t} g(x, t) ≤ 0 when x ≤ 1/2.
To solve this problem using fseminf
:
Write the objective function as an anonymous function:
objfun = @(x)(x1)^2;
Write the semiinfinite constraint function, which includes the nonlinear constraints ([ ] in this case), initial sampling interval for t (0 to 1 in steps of 0.01 in this case), and the semiinfinite constraint function g(x, t):
function [c, ceq, K1, s] = seminfcon(x,s) % No finite nonlinear inequality and equality constraints c = []; ceq = []; % Sample set if isnan(s) % Initial sampling interval s = [0.01 0]; end t = 0:s(1):1; % Evaluate the semiinfinite constraint K1 = (x  0.5)  (t  0.5).^2;
Call fseminf
with initial point
0.2, and view the result:
x = fseminf(objfun,0.2,1,@seminfcon) Local minimum found that satisfies the constraints. Optimization completed because the objective function is nondecreasing in feasible directions, to within the default value of the function tolerance, and constraints are satisfied to within the default value of the constraint tolerance. Active inequalities (to within options.ConstraintTolerance = 1e006): lower upper ineqlin ineqnonlin 1 x = 0.5000
The function to be minimized, the constraints, and semiinfinite
constraints, must be continuous functions of x
and w
. fseminf
might
only give local solutions.
When the problem is not feasible, fseminf
attempts
to minimize the maximum constraint value.
fseminf
uses cubic and quadratic interpolation
techniques to estimate peak values in the semiinfinite constraints.
The peak values are used to form a set of constraints that are supplied
to an SQP method as in the fmincon
function.
When the number of constraints changes, Lagrange multipliers are reallocated
to the new set of constraints.
The recommended sampling interval calculation uses the difference between the interpolated peak values and peak values appearing in the data set to estimate whether the function needs to take more or fewer points. The function also evaluates the effectiveness of the interpolation by extrapolating the curve and comparing it to other points in the curve. The recommended sampling interval is decreased when the peak values are close to constraint boundaries, i.e., zero.
For more details on the algorithm used and the types of procedures
displayed under the Procedures
heading when the Display
option
is set to 'iter'
with optimoptions
,
see also SQP Implementation. For more
details on the fseminf
algorithm, see fseminf Problem Formulation and Algorithm.