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.

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.

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.

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:

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 an M-file function

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 GradObj option is 'on', as set by

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

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 Example: 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 parameterize 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

ntheta Number of semi-infinite constraints

seminfcon Semi-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

lb Vector of lower bounds

ub Vector of upper bounds

solver

'fseminf'

options

Options structure created with optimset

Output Arguments

Function 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.TolCon`.
	`5`	Magnitude of directional derivative was less than the specified tolerance and constraint violation was less than `options.TolCon`.
	`0`	Number of iterations exceeded `options.MaxIter` or number of function evaluations exceeded `options.FunEvals`.
	`-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. You can use optimset to set or change the values of these fields in the options structure options. See Optimization Options for detailed information.

`DerivativeCheck`	Compare user-supplied derivatives (gradients of objective or constraints) to finite-differencing derivatives. The choices are `'on'` or the default `'off'`.
`Diagnostics`	Display diagnostic information about the function to be minimized or solved. The choices are `'on'` or the default `'off'`.
`DiffMaxChange`	Maximum change in variables for finite-difference gradients (a positive scalar). The default is `0.1`.
`DiffMinChange`	Minimum change in variables for finite-difference gradients (a positive scalar). The default is `1e-8`.
`Display`	Level of display: `'off'` displays no output. `'iter'` displays output at each iteration, and gives the default exit message. `'iter-detailed'` displays output at each iteration, and gives the technical exit message. `'notify'` displays output only if the function does not converge, and gives the default exit message. `'notify-detailed'` displays output only if the function does not converge, and gives the technical exit message. `'final'` (default) displays just the final output, and gives the default exit message. `'final-detailed'` displays just the final output, and gives the technical exit message.
`FinDiffType`	Finite differences, used to estimate gradients, are either `'forward'` (the default), or `'central'` (centered). `'central'` takes twice as many function evaluations, but should be more accurate. 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.
`FunValCheck`	Check whether objective function and constraints values are valid. `'on'` displays an error when the objective function or constraints return a value that is `complex`, `Inf`, or `NaN`. The default `'off'` displays no error.
`GradObj`	Gradient for the objective function defined by the user. See the preceding description of `fun` to see how to define the gradient in `fun`. Set to `'on'` to have `fseminf` use a user-defined gradient of the objective function. The default `'off'` causes `fseminf` to estimate gradients using finite differences.
`MaxFunEvals`	Maximum number of function evaluations allowed, a positive integer. The default is `100*numberOfVariables`.
`MaxIter`	Maximum number of iterations allowed, a positive integer. The default is `400`.
`MaxSQPIter`	Maximum number of SQP iterations allowed, a positive integer. The default is `10*max(numberOfVariables, numberOfInequalities + numberOfBounds)`.
`OutputFcn`	Specify one or more user-defined 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 (`[]`). See Output Function.
`PlotFcns`	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 (`[]`): `@optimplotx` plots the current point. `@optimplotfunccount` plots the function count. `@optimplotfval` plots the function value. `@optimplotconstrviolation` plots the maximum constraint violation. `@optimplotstepsize` plots the step size. `@optimplotfirstorderopt` plots the first-order optimality measure.
`RelLineSrchBnd`	Relative bound (a real nonnegative scalar value) on the line search step length such that the total displacement in `x` satisfies \|Δx(i)\| ≤ relLineSrchBnd· max(\|x(i)\|,\|typicalx(i)\|). This option provides control over the magnitude of the displacements in `x` for cases in which the solver takes steps that `fseminf` considers too large. The default is no bounds (`[]`).
`RelLineSrchBndDuration`	Number of iterations for which the bound specified in `RelLineSrchBnd` should be active (default is `1`)
`TolCon`	Termination tolerance on the constraint violation, a positive scalar. The default is `1e-6`.
`TolConSQP`	Termination tolerance on inner iteration SQP constraint violation, a positive scalar. The default is `1e-6`.
`TolFun`	Termination tolerance on the function value, a positive scalar. The default is `1e-4`.
`TolX`	Termination tolerance on `x`, a positive scalar. The default value is `1e-4`.
`TypicalX`	Typical `x` values. The number of elements in `TypicalX` is equal to the number of elements in `x0`, the starting point. The default value is `ones(numberofvariables,1)`. `fseminf` uses `TypicalX` for scaling finite differences for gradient estimation.

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).

Example

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)(x-1)^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(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 semi-infinite 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
non-decreasing in feasible directions, to within the 
default value of the function tolerance, and constraints
were satisfiedto within the default value of the
constraint tolerance.

Active inequalities (to within options.TolCon = 1e-006):
  lower      upper     ineqlin   ineqnonlin
                                     1

x =
    0.5000

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 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 optimset, see also SQP Implementation. For more details on the fseminf algorithm, see fseminf Problem Formulation and Algorithm.

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.