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.
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
.
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
.
If the specified input bounds for a problem are inconsistent, the output
x
is x0
and the output
fval
is []
.
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 NoteBecause Optimization
Toolbox™ functions only accept inputs of type
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
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.

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,
max_{t} g(x, t) = (x– 1/2).
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.