Optimization Toolbox™

fseminf

Find minimum of semi-infinitely constrained multivariable nonlinear function

Equation

Finds the minimum of a problem specified by

x, b, beq, lb, and ub 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 w₁,w₂,...,w_n. The variables w₁,w₂,...,w_n are vectors of, at most, length two.

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 w_i∈ℜ¹ (or w_i∈ℜ²). 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.

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.

x = fseminf(fun,x0,ntheta,seminfcon,A,b,Aeq,beq,lb,ub,options) minimizes with the optimization options specified in the structure options. Use optimset to set these options.

x = fseminf(problem) finds the minimum for problem, where problem is a structure described in Input Arguments.

Create the structure problem by exporting a problem from Optimization Tool, as described in Exporting to the MATLAB^® Workspace.

[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.

Passing Extra Parameters explains how to parameterize the objective function fun, if necessary.

Note   If the specified input bounds for a problem are inconsistent, the output x is x0 and the output fval is [].

Input Arguments

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

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

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

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

options = optimset('GradObj','on')

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

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

Output Arguments

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

Options

Optimization options used by fseminf. You can use optimset to set or change the values of these fields in the options structure options. See Optimization Options for detailed information.

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 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).

Examples

One-Dimensional Example

Find values of x that minimize

f(x) = (x₁ – 0.5)² + (x₂– 0.5)² + (x₃– 0.5)²

where

for all values of w₁ and w₂ over the ranges

1 ≤ w₁≤ 100,
1 ≤ w₂≤ 100.

Note that the semi-infinite constraints are one-dimensional, that is, vectors. Because the constraints must be in the form K_i(x,w_i) ≤ 0, you need to compute the constraints as

First, write an M-file that computes the objective function.

function f = myfun(x,s)
% Objective function
f = sum((x-0.5).^2);

Second, write an M-file, mycon.m, that computes the nonlinear equality and inequality constraints and the semi-infinite constraints.

function [c,ceq,K1,K2,s] = mycon(X,s)
% Initial sampling interval
if isnan(s(1,1)),
   s = [0.2 0; 0.2 0];
end
% Sample set
w1 = 1:s(1,1):100;
w2 = 1:s(2,1):100;

% Semi-infinite constraints 
K1 = sin(w1*X(1)).*cos(w1*X(2)) - 1/1000*(w1-50).^2 -...
       sin(w1*X(3))-X(3)-1;
K2 = sin(w2*X(2)).*cos(w2*X(1)) - 1/1000*(w2-50).^2 -...
       sin(w2*X(3))-X(3)-1;

% No finite nonlinear constraints
c = []; ceq=[];

% Plot a graph of semi-infinite constraints
plot(w1,K1,'-',w2,K2,':')
title('Semi-infinite constraints')
drawnow

Then, invoke an optimization routine.

x0 = [0.5; 0.2; 0.3];      % Starting guess
[x,fval] = fseminf(@myfun,x0,2,@mycon)

After eight iterations, the solution is

x =
     0.6673
     0.3013
     0.4023

The function value and the maximum values of the semi-infinite constraints at the solution x are

fval =
     0.0770

[c,ceq,K1,K2] = mycon(x,NaN); % Initial sampling interval
max(K1)
ans =
     -0.0017
max(K2)
ans =
     -0.0845

A plot of the semi-infinite constraints is produced.

This plot shows how peaks in both constraints are on the constraint boundary.

The plot command inside 'mycon.m' slows down the computation. Remove this line to improve the speed.

Two-Dimensional Example

Find values of x that minimize

f(x) = (x₁ – 0.2)² + (x₂– 0.2)² + (x₃– 0.2)²,

where

for all values of w₁ and w₂ over the ranges

1 ≤ w₁≤ 100,
1 ≤ w₂≤ 100,

starting at the point x = [0.25,0.25,0.25].

Note that the semi-infinite constraint is two-dimensional, that is, a matrix.

First, write an M-file that computes the objective function.

function f = myfun(x,s)
% Objective function
f = sum((x-0.2).^2);

Second, write an M-file for the constraints, called mycon.m. Include code to draw the surface plot of the semi-infinite constraint each time mycon is called. This enables you to see how the constraint changes as X is being minimized.

function [c,ceq,K1,s] = mycon(X,s)
% Initial sampling interval
if isnan(s(1,1)),
   s = [2 2];
end

% Sampling set
w1x = 1:s(1,1):100;
w1y = 1:s(1,2):100;
[wx,wy] = meshgrid(w1x,w1y);

% Semi-infinite constraint 
K1 = sin(wx*X(1)).*cos(wx*X(2))-1/1000*(wx-50).^2 -...
       sin(wx*X(3))-X(3)+sin(wy*X(2)).*cos(wx*X(1))-...
       1/1000*(wy-50).^2-sin(wy*X(3))-X(3)-1.5;

% No finite nonlinear constraints
c = []; ceq=[];

% Mesh plot
m = surf(wx,wy,K1,'edgecolor','none','facecolor','interp');
camlight headlight
title('Semi-infinite constraint')
drawnow

Next, invoke an optimization routine.

x0 = [0.25, 0.25, 0.25];    % Starting guess
[x,fval] = fseminf(@myfun,x0,1,@mycon)

After nine iterations, the solution is

x =
     0.2926    0.1874    0.2202

and the function value at the solution is

fval = 
     0.0091

The goal was to minimize the objective f(x) such that the semi-infinite constraint satisfied K₁(x,w) ≤ 1.5. Evaluating mycon at the solution x and looking at the maximum element of the matrix K1 shows the constraint is easily satisfied.

[c,ceq,K1] = mycon(x,[0.5,0.5]);  % Sampling interval 0.5
max(max(K1))

ans =
    -0.0027

This call to mycon produces the following surf plot, which shows the semi-infinite constraint at x.

Algorithm

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 function fmincon. 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.

See also SQP Implementation 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 optimset.

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.

See Also

@ (function_handle), fmincon, optimset, optimtool

fminunc

fsolve