fminimax - Solve minimax constraint problem

fun

The function to be minimized. fun is a function that accepts a vector x and returns a vector F, the objective functions evaluated at x. The function fun can be specified as a function handle for an M-file function

x = fminimax(@myfun,x0)

where myfun is a MATLAB^® function such as

function F = myfun(x)
F = ...            % Compute function values at x

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

x = fminimax(@(x)sin(x.*x),x0);

If the user-defined values for x and F are matrices, they are converted to a vector using linear indexing.

To minimize the worst case absolute values of any of the elements of the vector F(x) (i.e., min{max abs{F(x)} } ), partition those objectives into the first elements of F and use optimset to set the MinAbsMax option to be the number of such objectives.

If the gradient of the objective function 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 matrix, at x. Note that by checking the value of nargout, the function can avoid computing G when myfun is called with only one output argument (in the case where the optimization algorithm only needs the value of F but not G).

function [F,G] = myfun(x)
F = ...          % Compute the function values at x
if nargout > 1   % Two output arguments
   G = ...       % Gradients evaluated at x
end

nonlcon

The gradient consists of the partial derivative dF/dx of each F at the point x. If F is a vector of length m and x has length n, where n is the length of x0, then the gradient G of F(x) is an n-by-m matrix where G(i,j) is the partial derivative of F(j) with respect to x(i) (i.e., the jth column of G is the gradient of the jth objective function F(j)).

The function that computes the nonlinear inequality constraints c(x) ≤ 0 and nonlinear equality constraints ceq(x) = 0. The function nonlcon accepts a vector x and returns two vectors c and ceq. The vector c contains the nonlinear inequalities evaluated at x, and ceq contains the nonlinear equalities evaluated at x. The function nonlcon can be specified as a function handle.

x = fminimax(@myfun,x0,A,b,Aeq,beq,lb,ub,@mycon)

where mycon is a MATLAB function such as

function [c,ceq] = mycon(x)
c = ...     % Compute nonlinear inequalities at x
ceq = ...   % Compute nonlinear equalities at x

If the gradients of the constraints can also be computed and the GradConstr option is 'on', as set by

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

then the function nonlcon must also return, in the third and fourth output arguments, GC, the gradient of c(x), and GCeq, the gradient of ceq(x). Nonlinear Constraints explains how to "conditionalize" the gradients for use in solvers that do not accept supplied gradients, and explains the syntax of gradients.

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 the nonlinear constraint function nonlcon, if necessary.

objective

x0

Aineq

bineq

Aeq

beq

lb

ub

nonlcon

solver

options

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 less than the specified tolerance and constraint violation less than options.TolCon.

5

Magnitude of directional derivative less than the specified tolerance and constraint violation 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

maxfval

Maximum of the function values evaluated at the solution x, that is, maxfval = max{fun(x)}.

output

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.

Measure of first-order optimality

Maximum of nonlinear constraint functions

Exit message

DerivativeCheck

Compare user-supplied derivatives (gradients of the objective or constraints) to finite-differencing derivatives.

Diagnostics

Display diagnostic information about the function to be minimized or solved.

DiffMaxChange

Maximum change in variables for finite-difference gradients.

DiffMinChange

Minimum change in variables for finite-difference gradients.

Display

Level of display. 'off' displays no output; 'iter' displays output at each iteration; 'notify' displays output only if the function does not converge;'final' (default) displays just the final output.

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. 'off' displays no error.

GradConstr

Gradient for the user-defined constraints. See the preceding description of nonlcon to see how to define the gradient in nonlcon.

GradObj

Gradient for the user-defined objective function. See the preceding description of fun to see how to define the gradient in fun.

MaxFunEvals

Maximum number of function evaluations allowed.

MaxIter

Maximum number of iterations allowed.

Maximum number of SQP iterations allowed.

MeritFunction

Use the goal attainment/minimax merit function if set to 'multiobj'. Use the fmincon merit function if set to 'singleobj'.

MinAbsMax

Number of F(x) to minimize the worst case absolute values.

Specify one or more user-defined functions that are called after each iteration of an optimization (medium scale algorithm only). See Output Function.

PlotFcns

Plots various measures of progress while the algorithm executes, select from predefined plots or write your own. Specifying @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 of optimality.

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 are considered too large.

Number of iterations for which the bound specified in RelLineSrchBnd should be active (default is 1).

TolCon

Termination tolerance on the constraint violation.

TolConSQP

Termination tolerance on inner iteration SQP constraint violation.

TolFun

Termination tolerance on the function value.

TolX

Termination tolerance on x.

When 'always', estimate gradients in parallel. Disable by setting to 'never'.