# fminimax

Solve minimax constraint problem

## Description

fminimax seeks a point that minimizes the maximum of a set of objective functions.

The problem includes any type of constraint. In detail, fminimax seeks the minimum of a problem specified by

where b and beq are vectors, A and Aeq are matrices, and c(x), ceq(x), and F(x) are functions that return vectors. F(x), c(x), and ceq(x) can be nonlinear functions.

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

You can also solve max-min problems with fminimax, using the identity

$\underset{x}{\mathrm{max}}\underset{i}{\mathrm{min}}{F}_{i}\left(x\right)=-\underset{x}{\mathrm{min}}\underset{i}{\mathrm{max}}\left(-{F}_{i}\left(x\right)\right).$

You can solve problems of the form

$\underset{x}{\mathrm{min}}\underset{i}{\mathrm{max}}|{F}_{i}\left(x\right)|$

by using the AbsoluteMaxObjectiveCount option; see Solve Minimax Problem Using Absolute Value of One Objective.

example

x = fminimax(fun,x0) starts at x0 and finds a minimax solution x to the functions described in fun.

Note

Passing Extra Parameters explains how to pass extra parameters to the objective functions and nonlinear constraint functions, if necessary.

example

x = fminimax(fun,x0,A,b) solves the minimax problem subject to the linear inequalities A*x ≤ b.

x = fminimax(fun,x0,A,b,Aeq,beq) solves the minimax problem subject to the linear equalities Aeq*x = beq as well. If no inequalities exist, set A = [] and b = [].

example

x = fminimax(fun,x0,A,b,Aeq,beq,lb,ub) solves the minimax problem subject to the bounds lb  x  ub. If no equalities exist, set Aeq = [] and beq = []. If x(i) is unbounded below, set lb(i) = –Inf; if x(i) is unbounded above, set ub(i) = Inf.

Note

Note

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

example

x = fminimax(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon) solves the minimax problem subject to the nonlinear inequalities c(x) or equalities ceq(x) defined in nonlcon. The function optimizes such that c(x) ≤ 0 and ceq(x) = 0. If no bounds exist, set lb = [] or ub = [], or both.

example

x = fminimax(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options) solves the minimax problem with the optimization options specified in options. Use optimoptions to set these options.

x = fminimax(problem) solves the minimax problem for problem, a structure described in problem.

example

[x,fval] = fminimax(___), for any syntax, returns the values of the objective functions computed in fun at the solution x.

example

[x,fval,maxfval,exitflag,output] = fminimax(___) additionally returns the maximum value of the objective functions at the solution x, a value exitflag that describes the exit condition of fminimax, and a structure output with information about the optimization process.

example

[x,fval,maxfval,exitflag,output,lambda] = fminimax(___) additionally returns a structure lambda whose fields contain the Lagrange multipliers at the solution x.

## Examples

collapse all

Create a plot of the sin and cos functions and their maximum over the interval [–pi,pi].

t = linspace(-pi,pi);
plot(t,sin(t),'r-')
hold on
plot(t,cos(t),'b-');
plot(t,max(sin(t),cos(t)),'ko')
legend('sin(t)','cos(t)','max(sin(t),cos(t))','Location','NorthWest')

The plot shows two local minima of the maximum, one near 1, and the other near –2. Find the minimum near 1.

fun = @(x)[sin(x);cos(x)];
x0 = 1;
x1 = fminimax(fun,x0)
Local minimum possible. Constraints satisfied.

fminimax stopped because the size of the current search direction is less than
twice the value of the step size tolerance and constraints are
satisfied to within the value of the constraint tolerance.
x1 = 0.7854

Find the minimum near –2.

x0 = -2;
x2 = fminimax(fun,x0)
Local minimum possible. Constraints satisfied.

fminimax stopped because the size of the current search direction is less than
twice the value of the step size tolerance and constraints are
satisfied to within the value of the constraint tolerance.
x2 = -2.3562

The objective functions for this example are linear plus constants. For a description and plot of the objective functions, see Compare fminimax and fminunc.

Set the objective functions as three linear functions of the form $dot\left(x,v\right)+{v}_{0}$ for three vectors $v$ and three constants ${v}_{0}$.

a = [1;1];
b = [-1;1];
c = [0;-1];
a0 = 2;
b0 = -3;
c0 = 4;
fun = @(x)[x*a+a0,x*b+b0,x*c+c0];

Find the minimax point subject to the inequality x(1) + 3*x(2) <= –4.

A = [1,3];
b = -4;
x0 = [-1,-2];
x = fminimax(fun,x0,A,b)
Local minimum possible. Constraints satisfied.

fminimax stopped because the size of the current search direction is less than
twice the value of the step size tolerance and constraints are
satisfied to within the value of the constraint tolerance.
x = 1×2

-5.8000    0.6000

The objective functions for this example are linear plus constants. For a description and plot of the objective functions, see Compare fminimax and fminunc.

Set the objective functions as three linear functions of the form $dot\left(x,v\right)+{v}_{0}$ for three vectors $v$ and three constants ${v}_{0}$.

a = [1;1];
b = [-1;1];
c = [0;-1];
a0 = 2;
b0 = -3;
c0 = 4;
fun = @(x)[x*a+a0,x*b+b0,x*c+c0];

Set bounds that –2 <= x(1) <= 2 and –1 <= x(2) <= 1 and solve the minimax problem starting from [0,0].

lb = [-2,-1];
ub = [2,1];
x0 = [0,0];
A = []; % No linear constraints
b = [];
Aeq = [];
beq = [];
[x,fval] = fminimax(fun,x0,A,b,Aeq,beq,lb,ub)
Local minimum possible. Constraints satisfied.

fminimax stopped because the size of the current search direction is less than
twice the value of the step size tolerance and constraints are
satisfied to within the value of the constraint tolerance.
x = 1×2

-0.0000    1.0000

fval = 1×3

3.0000   -2.0000    3.0000

In this case, the solution is not unique. Many points satisfy the constraints and have the same minimax value. Plot the surface representing the maximum of the three objective functions, and plot a red line showing the points that have the same minimax value.

[X,Y] = meshgrid(linspace(-2,2),linspace(-1,1));
Z = max(fun([X(:),Y(:)]),[],2);
Z = reshape(Z,size(X));
surf(X,Y,Z,'LineStyle','none')
view(-118,28)
hold on
line([-2,0],[1,1],[3,3],'Color','r','LineWidth',8)
hold off

The objective functions for this example are linear plus constants. For a description and plot of the objective functions, see Compare fminimax and fminunc.

Set the objective functions as three linear functions of the form $dot\left(x,v\right)+{v}_{0}$ for three vectors $v$ and three constants ${v}_{0}$.

a = [1;1];
b = [-1;1];
c = [0;-1];
a0 = 2;
b0 = -3;
c0 = 4;
fun = @(x)[x*a+a0,x*b+b0,x*c+c0];

The unitdisk function represents the nonlinear inequality constraint $‖x{‖}^{2}\le 1$.

type unitdisk
function [c,ceq] = unitdisk(x)
c = x(1)^2 + x(2)^2 - 1;
ceq = [];

Solve the minimax problem subject to the unitdisk constraint, starting from x0 = [0,0].

x0 = [0,0];
A = []; % No other constraints
b = [];
Aeq = [];
beq = [];
lb = [];
ub = [];
nonlcon = @unitdisk;
x = fminimax(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)
Local minimum possible. Constraints satisfied.

fminimax stopped because the size of the current search direction is less than
twice the value of the step size tolerance and constraints are
satisfied to within the value of the constraint tolerance.
x = 1×2

-0.0000    1.0000

fminimax can minimize the maximum of either ${F}_{i}\left(x\right)$ or $|{F}_{i}\left(x\right)|$ for the first several values of $i$ by using the AbsoluteMaxObjectiveCount option. To minimize the absolute values of $k$ of the objectives, arrange the objective function values so that ${F}_{1}\left(x\right)$ through ${F}_{k}\left(x\right)$ are the objectives for absolute minimization, and set the AbsoluteMaxObjectiveCount option to k.

In this example, minimize the maximum of sin and cos, specify sin as the first objective, and set AbsoluteMaxObjectiveCount to 1.

fun = @(x)[sin(x),cos(x)];
options = optimoptions('fminimax','AbsoluteMaxObjectiveCount',1);
x0 = 1;
A = []; % No constraints
b = [];
Aeq = [];
beq = [];
lb = [];
ub = [];
nonlcon = [];
x1 = fminimax(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)
Local minimum possible. Constraints satisfied.

fminimax stopped because the size of the current search direction is less than
twice the value of the step size tolerance and constraints are
satisfied to within the value of the constraint tolerance.
x1 = 0.7854

Try starting from x0 = –2.

x0 = -2;
x2 = fminimax(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)
Local minimum possible. Constraints satisfied.

fminimax stopped because the size of the current search direction is less than
twice the value of the step size tolerance and constraints are
satisfied to within the value of the constraint tolerance.
x2 = -3.1416

Plot the function.

t = linspace(-pi,pi);
plot(t,max(abs(sin(t)),cos(t)))

To see the effect of the AbsoluteMaxObjectiveCount option, compare this plot to the plot in the example Minimize Maximum of sin and cos.

Obtain both the location of the minimax point and the value of the objective functions. For a description and plot of the objective functions, see Compare fminimax and fminunc.

Set the objective functions as three linear functions of the form $dot\left(x,v\right)+{v}_{0}$ for three vectors $v$ and three constants ${v}_{0}$.

a = [1;1];
b = [-1;1];
c = [0;-1];
a0 = 2;
b0 = -3;
c0 = 4;
fun = @(x)[x*a+a0,x*b+b0,x*c+c0];

Set the initial point to [0,0] and find the minimax point and value.

x0 = [0,0];
[x,fval] = fminimax(fun,x0)
Local minimum possible. Constraints satisfied.

fminimax stopped because the size of the current search direction is less than
twice the value of the step size tolerance and constraints are
satisfied to within the value of the constraint tolerance.
x = 1×2

-2.5000    2.2500

fval = 1×3

1.7500    1.7500    1.7500

All three objective functions have the same value at the minimax point. Unconstrained problems typically have at least two objectives that are equal at the solution, because if a point is not a local minimum for any objective and only one objective has the maximum value, then the maximum objective can be lowered.

The objective functions for this example are linear plus constants. For a description and plot of the objective functions, see Compare fminimax and fminunc.

Set the objective functions as three linear functions of the form $dot\left(x,v\right)+{v}_{0}$ for three vectors $v$ and three constants ${v}_{0}$.

a = [1;1];
b = [-1;1];
c = [0;-1];
a0 = 2;
b0 = -3;
c0 = 4;
fun = @(x)[x*a+a0,x*b+b0,x*c+c0];

Find the minimax point subject to the inequality x(1) + 3*x(2) <= –4.

A = [1,3];
b = -4;
x0 = [-1,-2];

Set options for iterative display, and obtain all solver outputs.

options = optimoptions('fminimax','Display','iter');
Aeq = []; % No other constraints
beq = [];
lb = [];
ub = [];
nonlcon = [];
[x,fval,maxfval,exitflag,output,lambda] =...
fminimax(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)
Objective        Max     Line search     Directional
Iter F-count         value    constraint   steplength      derivative   Procedure
0      4              0             6
1      9              5             0            1           0.981
2     14          4.889             0            1          -0.302    Hessian modified twice
3     19            3.4     8.132e-09            1          -0.302    Hessian modified twice

Local minimum possible. Constraints satisfied.

fminimax stopped because the size of the current search direction is less than
twice the value of the step size tolerance and constraints are
satisfied to within the value of the constraint tolerance.
x = 1×2

-5.8000    0.6000

fval = 1×3

-3.2000    3.4000    3.4000

maxfval = 3.4000
exitflag = 4
output = struct with fields:
iterations: 4
funcCount: 19
lssteplength: 1
stepsize: 6.0684e-10
algorithm: 'active-set'
firstorderopt: []
constrviolation: 8.1323e-09
message: 'Local minimum possible. Constraints satisfied....'

lambda = struct with fields:
lower: [2x1 double]
upper: [2x1 double]
eqlin: [0x1 double]
eqnonlin: [0x1 double]
ineqlin: 0.2000
ineqnonlin: [0x1 double]

Examine the returned information:

• Two objective function values are equal at the solution.

• The solver converges in 4 iterations and 19 function evaluations.

• The lambda.ineqlin value is nonzero, indicating that the linear constraint is active at the solution.

## Input Arguments

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Objective functions, specified as a function handle or function name. fun is a function that accepts a vector x and returns a vector F, the objective functions evaluated at x. You can specify the function fun as a function handle for a function file:

x = fminimax(@myfun,x0,goal,weight)

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,goal,weight);

fminimax passes x to your objective function and any nonlinear constraint functions in the shape of the x0 argument. For example, if x0 is a 5-by-3 array, then fminimax passes x to fun as a 5-by-3 array. However, fminimax multiplies linear constraint matrices A or Aeq with x after converting x to the column vector x(:).

To minimize the worst-case absolute values of some elements of the vector F(x) (that is, min{max abs{F(x)} } ), partition those objectives into the first elements of F and use optimoptions to set the AbsoluteMaxObjectiveCount option to the number of these objectives. These objectives must be partitioned into the first elements of the vector F returned by fun. For an example, see Solve Minimax Problem Using Absolute Value of One Objective.

Assume that the gradients of the objective functions can also be computed and the SpecifyObjectiveGradient option is true, as set by:

In this case, the function fun must return, in the second output argument, the gradient values G (a matrix) at x. 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) (that is, the jth column of G is the gradient of the jth objective function F(j)). If you define F as an array, then the preceding discussion applies to F(:), the linear ordering of the F array. In any case, G is a 2-D matrix.

Note

Setting SpecifyObjectiveGradient to true is effective only when the problem has no nonlinear constraint, or when the problem has a nonlinear constraint with SpecifyConstraintGradient set to true. Internally, the objective is folded into the constraints, so the solver needs both gradients (objective and constraint) supplied in order to avoid estimating a gradient.

Data Types: char | string | function_handle

Initial point, specified as a real vector or real array. Solvers use the number of elements in x0 and the size of x0 to determine the number and size of variables that fun accepts.

Example: x0 = [1,2,3,4]

Data Types: double

Linear inequality constraints, specified as a real matrix. A is an M-by-N matrix, where M is the number of inequalities, and N is the number of variables (number of elements in x0). For large problems, pass A as a sparse matrix.

A encodes the M linear inequalities

A*x <= b,

where x is the column vector of N variables x(:), and b is a column vector with M elements.

For example, consider these inequalities:

x1 + 2x2 ≤ 10
3x1 + 4x2 ≤ 20
5x1 + 6x2 ≤ 30,

Specify the inequalities by entering the following constraints.

A = [1,2;3,4;5,6];
b = [10;20;30];

Example: To specify that the x components sum to 1 or less, use A = ones(1,N) and b = 1.

Data Types: double

Linear inequality constraints, specified as a real vector. b is an M-element vector related to the A matrix. If you pass b as a row vector, solvers internally convert b to the column vector b(:). For large problems, pass b as a sparse vector.

b encodes the M linear inequalities

A*x <= b,

where x is the column vector of N variables x(:), and A is a matrix of size M-by-N.

For example, consider these inequalities:

x1 + 2x2 ≤ 10
3x1 + 4x2 ≤ 20
5x1 + 6x2 ≤ 30.

Specify the inequalities by entering the following constraints.

A = [1,2;3,4;5,6];
b = [10;20;30];

Example: To specify that the x components sum to 1 or less, use A = ones(1,N) and b = 1.

Data Types: double

Linear equality constraints, specified as a real matrix. Aeq is an Me-by-N matrix, where Me is the number of equalities, and N is the number of variables (number of elements in x0). For large problems, pass Aeq as a sparse matrix.

Aeq encodes the Me linear equalities

Aeq*x = beq,

where x is the column vector of N variables x(:), and beq is a column vector with Me elements.

For example, consider these inequalities:

x1 + 2x2 + 3x3 = 10
2x1 + 4x2 + x3 = 20,

Specify the inequalities by entering the following constraints.

Aeq = [1,2,3;2,4,1];
beq = [10;20];

Example: To specify that the x components sum to 1, use Aeq = ones(1,N) and beq = 1.

Data Types: double

Linear equality constraints, specified as a real vector. beq is an Me-element vector related to the Aeq matrix. If you pass beq as a row vector, solvers internally convert beq to the column vector beq(:). For large problems, pass beq as a sparse vector.

beq encodes the Me linear equalities

Aeq*x = beq,

where x is the column vector of N variables x(:), and Aeq is a matrix of size Me-by-N.

For example, consider these equalities:

x1 + 2x2 + 3x3 = 10
2x1 + 4x2 + x3 = 20.

Specify the equalities by entering the following constraints.

Aeq = [1,2,3;2,4,1];
beq = [10;20];

Example: To specify that the x components sum to 1, use Aeq = ones(1,N) and beq = 1.

Data Types: double

Lower bounds, specified as a real vector or real array. If the number of elements in x0 is equal to the number of elements in lb, then lb specifies that

x(i) >= lb(i) for all i.

If numel(lb) < numel(x0), then lb specifies that

x(i) >= lb(i) for 1 <= i <= numel(lb).

If lb has fewer elements than x0, solvers issue a warning.

Example: To specify that all x components are positive, use lb = zeros(size(x0)).

Data Types: double

Upper bounds, specified as a real vector or real array. If the number of elements in x0 is equal to the number of elements in ub, then ub specifies that

x(i) <= ub(i) for all i.

If numel(ub) < numel(x0), then ub specifies that

x(i) <= ub(i) for 1 <= i <= numel(ub).

If ub has fewer elements than x0, solvers issue a warning.

Example: To specify that all x components are less than 1, use ub = ones(size(x0)).

Data Types: double

Nonlinear constraints, specified as a function handle or function name. nonlcon is a function that accepts a vector or array x and returns two arrays, c(x) and ceq(x).

• c(x) is the array of nonlinear inequality constraints at x. fminimax attempts to satisfy

c(x) <= 0 for all entries of c.

• ceq(x) is the array of nonlinear equality constraints at x. fminimax attempts to satisfy

ceq(x) = 0 for all entries of ceq.

For example,

x = fminimax(@myfun,x0,...,@mycon)

where mycon is a MATLAB function such as the following:

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

Suppose that the gradients of the constraints can also be computed and the SpecifyConstraintGradient option is true, as set by:

In this case, 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). See Nonlinear Constraints for an explanation of how to “conditionalize” the gradients for use in solvers that do not accept supplied gradients.

If nonlcon returns a vector c of m components and x has length n, where n is the length of x0, then the gradient GC of c(x) is an n-by-m matrix, where GC(i,j) is the partial derivative of c(j) with respect to x(i) (that is, the jth column of GC is the gradient of the jth inequality constraint c(j)). Likewise, if ceq has p components, the gradient GCeq of ceq(x) is an n-by-p matrix, where GCeq(i,j) is the partial derivative of ceq(j) with respect to x(i) (that is, the jth column of GCeq is the gradient of the jth equality constraint ceq(j)).

Note

Setting SpecifyConstraintGradient to true is effective only when SpecifyObjectiveGradient is set to true. Internally, the objective is folded into the constraint, so the solver needs both gradients (objective and constraint) supplied in order to avoid estimating a gradient.

Note

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

See Passing Extra Parameters for an explanation of how to parameterize the nonlinear constraint function nonlcon, if necessary.

Data Types: char | function_handle | string

Optimization options, specified as the output of optimoptions or a structure such as optimset returns.

Some options are absent from the optimoptions display. These options appear in italics in the following table. For details, see View Options.

For details about options that have different names for optimset, see Current and Legacy Option Names.

OptionDescription
AbsoluteMaxObjectiveCount

Number of elements of Fi(x) for which to minimize the absolute value of Fi. See Solve Minimax Problem Using Absolute Value of One Objective.

For optimset, the name is MinAbsMax.

ConstraintTolerance

Termination tolerance on the constraint violation (a positive scalar). The default is 1e-6. See Tolerances and Stopping Criteria.

For optimset, the name is TolCon.

Diagnostics

Display of diagnostic information about the function to be minimized or solved. The choices are 'on' or 'off' (the default).

DiffMaxChange

Maximum change in variables for finite-difference gradients (a positive scalar). The default is Inf.

DiffMinChange

Minimum change in variables for finite-difference gradients (a positive scalar). The default is 0.

Display

Level of display (see Iterative Display):

• 'off' or 'none' 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 only the final output, and gives the default exit message.

• 'final-detailed' displays only the final output, and gives the technical exit message.

FiniteDifferenceStepSize

Scalar or vector step size factor for finite differences. When you set FiniteDifferenceStepSize to a vector v, the forward finite differences delta are

delta = v.*sign′(x).*max(abs(x),TypicalX);

where sign′(x) = sign(x) except sign′(0) = 1. Central finite differences are

delta = v.*max(abs(x),TypicalX);

Scalar FiniteDifferenceStepSize expands to a vector. The default is sqrt(eps) for forward finite differences, and eps^(1/3) for central finite differences.

For optimset, the name is FinDiffRelStep.

FiniteDifferenceType

Type of finite differences used to estimate gradients, either 'forward' (default) or 'central' (centered). 'central' takes twice as many function evaluations, but is generally more accurate.

The algorithm is careful to obey bounds when estimating both types of finite differences. For example, it might take a backward difference, rather than a forward difference, to avoid evaluating at a point outside the bounds.

For optimset, the name is FinDiffType.

FunctionTolerance

Termination tolerance on the function value (a positive scalar). The default is 1e-6. See Tolerances and Stopping Criteria.

For optimset, the name is TolFun.

FunValCheck

Check that signifies whether the objective function and constraint 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.

MaxFunctionEvaluations

Maximum number of function evaluations allowed (a positive integer). The default is 100*numberOfVariables. See Tolerances and Stopping Criteria and Iterations and Function Counts.

For optimset, the name is MaxFunEvals.

MaxIterations

Maximum number of iterations allowed (a positive integer). The default is 400. See Tolerances and Stopping Criteria and Iterations and Function Counts.

For optimset, the name is MaxIter.

MaxSQPIter

Maximum number of SQP iterations allowed (a positive integer). The default is 10*max(numberOfVariables, numberOfInequalities + numberOfBounds).

MeritFunction

If this option is set to 'multiobj' (the default), use the goal attainment or minimax merit function. If this option is set to 'singleobj', use the fmincon merit function.

OptimalityTolerance

Termination tolerance on the first-order optimality (a positive scalar). The default is 1e-6. See First-Order Optimality Measure.

For optimset, the name is TolFun.

OutputFcn

One or more user-defined functions that an optimization function calls at each iteration. Pass a function handle or a cell array of function handles. The default is none ([]). See Output Function and Plot Function Syntax.

PlotFcn

Plots showing various measures of progress while the algorithm executes. Select from predefined plots or write your own. Pass a name, function handle, or cell array of names or function handles. For custom plot functions, pass function handles. The default is none ([]).

• 'optimplotx' plots the current point.

• 'optimplotfunccount' plots the function count.

• 'optimplotfval' plots the objective function values.

• 'optimplotconstrviolation' plots the maximum constraint violation.

• 'optimplotstepsize' plots the step size.

Custom plot functions use the same syntax as output functions. See Output Functions for Optimization Toolbox and Output Function and Plot Function Syntax.

For optimset, the name is PlotFcns.

RelLineSrchBnd

Relative bound (a real nonnegative scalar value) on the line search step length such that the total displacement in x satisfies x(i)| ≤ relLineSrchBnd· max(|x(i)|,|typicalx(i)|). This option provides control over the magnitude of the displacements in x when the solver takes steps that are too large. The default is none ([]).

RelLineSrchBndDuration

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

Gradient for nonlinear constraint functions defined by the user. When this option is set to true, fminimax expects the constraint function to have four outputs, as described in nonlcon. When this option is set to false (the default), fminimax estimates gradients of the nonlinear constraints using finite differences.

For optimset, the name is GradConstr and the values are 'on' or 'off'.

Gradient for the objective function defined by the user. Refer to the description of fun to see how to define the gradient. Set this option to true to have fminimax use a user-defined gradient of the objective function. The default, false, causes fminimax to estimate gradients using finite differences.

For optimset, the name is GradObj and the values are 'on' or 'off'.

StepTolerance

Termination tolerance on x (a positive scalar). The default is 1e-6. See Tolerances and Stopping Criteria.

For optimset, the name is TolX.

TolConSQP

Termination tolerance on the inner iteration SQP constraint violation (a positive scalar). The default is 1e-6.

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). The fminimax function uses TypicalX for scaling finite differences for gradient estimation.

UseParallel

Option for using parallel computing. When this option is set to true, fminimax estimates gradients in parallel. The default is false. See Parallel Computing.

Example: optimoptions('fminimax','PlotFcn','optimplotfval')

Problem structure, specified as a structure with the fields in this table.

Field NameEntry

objective

Objective function fun

x0

Initial point for x

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

nonlcon

Nonlinear constraint function

solver

'fminimax'

options

Options created with optimoptions

You must supply at least the objective, x0, solver, and options fields in the problem structure.

Data Types: struct

## Output Arguments

collapse all

Solution, returned as a real vector or real array. The size of x is the same as the size of x0. Typically, x is a local solution to the problem when exitflag is positive. For information on the quality of the solution, see When the Solver Succeeds.

Objective function values at the solution, returned as a real array. Generally, fval = fun(x).

Maximum of the objective function values at the solution, returned as a real scalar. maxfval = max(fval(:)).

Reason fminimax stopped, returned as an integer.

 1 Function converged to a solution x 4 Magnitude of the search direction was less than the specified tolerance, and the constraint violation was less than options.ConstraintTolerance 5 Magnitude of the directional derivative was less than the specified tolerance, and the constraint violation was less than options.ConstraintTolerance 0 Number of iterations exceeded options.MaxIterations or the number of function evaluations exceeded options.MaxFunctionEvaluations -1 Stopped by an output function or plot function -2 No feasible point was found.

Information about the optimization process, returned as a structure with the fields in this table.

 iterations Number of iterations taken funcCount Number of function evaluations lssteplength Size of the line search step relative to the search direction constrviolation Maximum of the constraint functions stepsize Length of the last displacement in x algorithm Optimization algorithm used firstorderopt Measure of first-order optimality message Exit message

Lagrange multipliers at the solution, returned as a structure with the fields in this table.

 lower Lower bounds corresponding to lb upper Upper bounds corresponding to ub ineqlin Linear inequalities corresponding to A and b eqlin Linear equalities corresponding to Aeq and beq ineqnonlin Nonlinear inequalities corresponding to the c in nonlcon eqnonlin Nonlinear equalities corresponding to the ceq in nonlcon

## Algorithms

fminimax solves a minimax problem by converting it into a goal attainment problem, and then solving the converted goal attainment problem using fgoalattain. The conversion sets all goals to 0 and all weights to 1. See Equation 1 in Multiobjective Optimization Algorithms.

## Alternative Functionality

### App

The Optimize Live Editor task provides a visual interface for fminimax.

## Version History

Introduced before R2006a