Documentation

fseminf

Find minimum of semi-infinitely constrained multivariable nonlinear function

Equation

Finds the minimum of a problem specified by

b and beq are vectors, A and Aeq are matrices, c(x), ceq(x), and Ki(x,wi) 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) Ki(x,wi) ≤ 0 are continuous functions of both x and an additional set of variables w1,w2,...,wn. The variables w1,w2,...,wn are vectors of, at most, length two.

x, lb, and ub can be passed as vectors or matrices; see Matrix Arguments.

Syntax

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(...)

Description

fseminf finds a minimum of a semi-infinitely 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 wi1 (or wi2). Because it is impossible to calculate all possible values of Ki(x,wi), a region must be chosen for wi 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 semi-infinite 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

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 x is x0 and the output fval is [].

Input Arguments

Function Input Arguments contains general descriptions of arguments passed into fseminf. This section provides function-specific details for fun, ntheta, options, seminfcon, and problem:

fun

The function to be minimized. fun is a function that accepts a vector x and returns a scalar f, the objective function evaluated at x. The function fun can be specified as a function handle for a file

x = fseminf(@myfun,x0,ntheta,seminfcon)

where myfun is a MATLAB® function such as

function f = myfun(x)
f = ...            % Compute function value at x

fun can also be a function handle for an anonymous function.

fun = @(x)sin(x''*x);

If the gradient of fun can also be computed and the SpecifyObjectiveGradient option is true, as set by

then the function fun must return, in the second output argument, the gradient value g, a vector, at x.

ntheta

The number of semi-infinite constraints.

options

Options provides the function-specific details for the options values.

seminfcon

The function that computes the vector of nonlinear inequality constraints, c, a vector of nonlinear equality constraints, ceq, and ntheta semi-infinite constraints (vectors or matrices) K1, K2,..., Kntheta evaluated over an interval S at the point x. The function seminfcon can be specified as a function handle.

x = fseminf(@myfun,x0,ntheta,@myinfcon)

where myinfcon is a MATLAB function such as

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 semi-infinite constraint at x and w
K2 = ... % 2nd semi-infinite constraint at x and w
...
Kntheta = ...% Last semi-infinite constraint at x and w
c = ...      % Compute nonlinear inequalities at x
ceq = ...    % Compute the nonlinear equalities at x

S is a recommended sampling interval, which might or might not be used. Return [] for c and ceq if no such constraints exist.

The vectors or matrices K1, K2, ..., Kntheta contain the semi-infinite constraints evaluated for a sampled set of values for the independent variables w1, w2, ..., wntheta, respectively. The two-column matrix, S, contains a recommended sampling interval for values of w1, w2, ..., wntheta, which are used to evaluate K1, K2, ..., Kntheta. The ith row of S contains the recommended sampling interval for evaluating Ki. When Ki is a vector, use only S(i,1) (the second column can be all zeros). When Ki is a matrix, S(i,2) is used for the sampling of the rows in Ki, S(i,1) is used for the sampling interval of the columns of Ki (see Two-Dimensional Semi-Infinite Constraint). On the first iteration S is NaN, so that some initial sampling interval must be determined by seminfcon.

Note

Because Optimization Toolbox™ functions only accept inputs of type double, user-supplied objective and nonlinear constraint functions must return outputs of type double.

Passing Extra Parameters explains how to parametrize seminfcon, if necessary. Example of Creating Sampling Points contains an example of both one- and two-dimensional sampling points.

problem

objective

Objective function

x0

Initial point for x
nthetaNumber of semi-infinite constraints
seminfconSemi-infinite constraint function

Aineq

Matrix for linear inequality constraints

bineq

Vector for linear inequality constraints

Aeq

Matrix for linear equality constraints

beq

Vector for linear equality constraints
lbVector of lower bounds
ubVector of upper bounds

solver

'fseminf'

options

Options created with optimoptions

Output Arguments

Function Input Arguments contains general descriptions of arguments returned by fseminf. This section provides function-specific details for exitflag, lambda, and output:

 exitflag Integer identifying the reason the algorithm terminated. The following lists the values of exitflag and the corresponding reasons the algorithm terminated. 1 Function converged to a solution x. 4 Magnitude of the search direction was less than the specified tolerance and constraint violation was less than options.ConstraintTolerance. 5 Magnitude of directional derivative was less than the specified tolerance and constraint violation was less than options.ConstraintTolerance. 0 Number of iterations exceeded options.MaxIterations or number of function evaluations exceeded options.MaxFunctionEvaluations. -1 Algorithm was terminated by the output function. -2 No feasible point was found. lambda Structure containing the Lagrange multipliers at the solution x (separated by constraint type). The fields of the structure are lower Lower bounds lb upper Upper bounds ub ineqlin Linear inequalities eqlin Linear equalities ineqnonlin Nonlinear inequalities eqnonlin Nonlinear equalities output 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 x algorithm Optimization algorithm used constrviolation Maximum of constraint functions firstorderopt Measure of first-order optimality message Exit message

Options

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 appear in italics in the following table. For details, see View Options.

Notes

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 wi, over which Ki(x,wi) 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 Ki(x,wi).

Examples

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(xt) ≤ 0 for all 0 ≤ t ≤ 1,

implies x ≤ 1/2. You can see this by noticing that (t – 1/2)2 ≥ 0, so

maxt g(x, t) = (x– 1/2).

Therefore

maxt g(x, t) ≤ 0 when x ≤ 1/2.

To solve this problem using fseminf:

1. Write the objective function as an anonymous function:

objfun = @(x)(x-1)^2;
2. Write the semi-infinite 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 semi-infinite constraint function g(xt):

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 semi-infinite constraint
K1 = (x - 0.5) - (t - 0.5).^2;
3. 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
non-decreasing 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 = 1e-006):
lower      upper     ineqlin   ineqnonlin
1

x =
0.5000

Limitations

The function to be minimized, the constraints, and semi-infinite 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.

Algorithms

fseminf uses cubic and quadratic interpolation techniques to estimate peak values in the semi-infinite 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.