Find minimum of semi-infinitely constrained multivariable nonlinear function
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.
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 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 wi∈ℜ1 (or wi∈ℜ2). 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.
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.
Function Arguments contains general descriptions of arguments passed into fseminf. This section provides function-specific details for fun, ntheta, options, seminfcon, and problem:
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 GradObj option is 'on', as set by
options = optimoptions('fseminf','GradObj','on')
then the function fun must return, in the second output argument, the gradient value g, a vector, at x.
The number of semi-infinite constraints.
Options provides the function-specific details for the options values.
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.
|Initial point for x|
|ntheta||Number of semi-infinite constraints|
|seminfcon||Semi-infinite 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|
|Options created with optimoptions|
Function Arguments contains general descriptions of arguments returned by fseminf. This section provides function-specific details for exitflag, lambda, and output:
Integer identifying the reason the algorithm terminated. The following lists the values of exitflag and the corresponding reasons the algorithm terminated.
Function converged to a solution x.
Magnitude of the search direction was less than the specified tolerance and constraint violation was less than options.TolCon.
Magnitude of directional derivative was less than the specified tolerance and constraint violation was less than options.TolCon.
Number of iterations exceeded options.MaxIter or number of function evaluations exceeded options.MaxFunEvals.
Algorithm was terminated by the output function.
No feasible point was found.
Structure containing the Lagrange multipliers at the solution x (separated by constraint type). The fields of the structure are
Lower bounds lb
Upper bounds ub
Structure containing information about the optimization. The fields of the structure are
Number of iterations taken
Number of function evaluations
Size of line search step relative to search direction
Final displacement in x
Optimization algorithm used
Maximum of constraint functions
Measure of first-order optimality
Compare user-supplied derivatives (gradients of objective or constraints) to finite-differencing derivatives. The choices are 'on' or the default 'off'.
Display diagnostic information about the function to be minimized or solved. The choices are 'on' or the default 'off'.
Maximum change in variables for finite-difference gradients (a positive scalar). The default is Inf.
Minimum change in variables for finite-difference gradients (a positive scalar). The default is 0.
Level of display:
Scalar or vector step size factor. When you set FinDiffRelStep to a vector v, forward finite differences delta are
delta = v.*sign(x).*max(abs(x),TypicalX);
and central finite differences are
delta = v.*max(abs(x),TypicalX);
Scalar FinDiffRelStep expands to a vector. The default is sqrt(eps) for forward finite differences, and eps^(1/3) for central finite differences.
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.
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.
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.
Maximum number of function evaluations allowed, a positive integer. The default is 100*numberOfVariables.
Maximum number of iterations allowed, a positive integer. The default is 400.
Maximum number of SQP iterations allowed, a positive integer. The default is 10*max(numberOfVariables, numberOfInequalities + numberOfBounds).
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.
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.
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 ().
Number of iterations for which the bound specified in RelLineSrchBnd should be active (default is 1)
Termination tolerance on the constraint violation, a positive scalar. The default is 1e-6.
Termination tolerance on inner iteration SQP constraint violation, a positive scalar. The default is 1e-6.
Termination tolerance on the function value, a positive scalar. The default is 1e-4.
Termination tolerance on x, a positive scalar. The default value is 1e-4.
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.
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).
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
maxt g(x, t) = (x– 1/2).
maxt 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 are satisfied to within the default value of the constraint tolerance. Active inequalities (to within options.TolCon = 1e-006): lower upper ineqlin ineqnonlin 1 x = 0.5000
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.
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.