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# fminimax

Solve minimax constraint problem

## Equation

Finds 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 Notes.

## Syntax

x = fminimax(fun,x0)
x = fminimax(fun,x0,A,b)
x = fminimax(fun,x0,A,b,Aeq,beq)
x = fminimax(fun,x0,A,b,Aeq,beq,lb,ub)
x = fminimax(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)
x = fminimax(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)
x = fminimax(problem)
[x,fval] = fminimax(...)
[x,fval,maxfval] = fminimax(...)
[x,fval,maxfval,exitflag] = fminimax(...)
[x,fval,maxfval,exitflag,output] = fminimax(...)
[x,fval,maxfval,exitflag,output,lambda] = fminimax(...)

## Description

fminimax minimizes the worst-case (largest) value of a set of multivariable functions, starting at an initial estimate. This is generally referred to as the minimax problem.

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

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

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. Set A = [] and b = [] if no inequalities exist.

x = fminimax(fun,x0,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.

 Note:   See Iterations Can Violate Constraints.

x = fminimax(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon) subjects the minimax problem to the nonlinear inequalities c(x) or equality constraints ceq(x) defined in nonlcon. fminimax optimizes such that c(x) ≤ 0 and ceq(x) = 0. Set lb = [] and/or ub = [] if no bounds exist.

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

x = fminimax(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.

[x,fval] = fminimax(...) returns the value of the objective function fun at the solution x.

[x,fval,maxfval] = fminimax(...) returns the maximum of the objective functions in the input fun evaluated at the solution x.

[x,fval,maxfval,exitflag] = fminimax(...) returns a value exitflag that describes the exit condition of fminimax.

[x,fval,maxfval,exitflag,output] = fminimax(...) returns a structure output with information about the optimization.

[x,fval,maxfval,exitflag,output,lambda] = fminimax(...) 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 Input Arguments contains general descriptions of arguments passed into fminimax. This section provides function-specific details for fun, nonlcon, and 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 a file:

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 optimoptions to set the AbsoluteMaxObjectiveCount option to be the number of such objectives.

If the gradient of the objective function can also be computed and the SpecifyObjectiveGradient option is true, as set by

then the function fun must return, in the second output argument, the gradient value 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) (i.e., the jth column of G is the gradient of the jth objective function F(j)).

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
 Note:   Setting SpecifyObjectiveGradient to true is effective only when there is no nonlinear constraint, or when the nonlinear constraint has SpecifyConstraintGradient set to true as well. This is because 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.

nonlcon

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 SpecifyConstraintGradient option is true, as set by

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:   Setting SpecifyConstraintGradient to true is effective only when SpecifyObjectiveGradient is set to true as well. This is because 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 only accept inputs of type double, user-supplied objective and nonlinear constraint functions must return outputs of type double.

problem

objective

Objective function

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

lb

Vector of lower bounds

ub

Vector of upper bounds

nonlcon

Nonlinear constraint function

solver

'fminimax'

options

Options created with optimoptions

## Output Arguments

Function Input Arguments contains general descriptions of arguments returned by fminimax. This section provides function-specific details for exitflag, lambda, maxfval, 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 less than the specified tolerance and constraint violation less than options.ConstraintTolerance. 5 Magnitude of directional derivative less than the specified tolerance and constraint violation less than options.ConstraintTolerance. 0 Number of iterations exceeded options.MaxIterations or number of function evaluations exceeded options.MaxFunctionEvaluations. -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 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. firstorderopt Measure of first-order optimality constrviolation Maximum of constraint functions message Exit message

## Options

Optimization options used by fminimax. Use optimoptions to set or change options. See Optimization Options Reference for detailed information.

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

## Examples

Find values of x that minimize the maximum value of

[f1(x), f2(x), f3(x), f4(x), f5(x)]

where

$\begin{array}{l}{f}_{1}\left(x\right)=2{x}_{1}^{2}+{x}_{2}^{2}-48{x}_{1}-40{x}_{2}+304,\\ {f}_{2}\left(x\right)=-{x}_{1}^{2}-3{x}_{2}^{2},\\ {f}_{3}\left(x\right)={x}_{1}+3{x}_{2}-18,\\ {f}_{4}\left(x\right)=-{x}_{1}-{x}_{2},\\ {f}_{5}\left(x\right)={x}_{1}+{x}_{2}-8.\end{array}$

First, write a file that computes the five functions at x.

function f = myfun(x)
f(1)= 2*x(1)^2+x(2)^2-48*x(1)-40*x(2)+304;     % Objectives
f(2)= -x(1)^2 - 3*x(2)^2;
f(3)= x(1) + 3*x(2) -18;
f(4)= -x(1)- x(2);
f(5)= x(1) + x(2) - 8;

Next, invoke an optimization routine.

x0 = [0.1; 0.1];       % Make a starting guess at solution
[x,fval] = fminimax(@myfun,x0);

After seven iterations, the solution is

x,fval

x =
4.0000
4.0000
fval =
0.0000  -64.0000  -2.0000  -8.0000  -0.0000

## Notes

You can solve problems of the form

$\underset{x}{\mathrm{min}}\underset{i}{\mathrm{max}}{G}_{i}\left(x\right),$

where

${G}_{i}\left(x\right)=\left\{\begin{array}{ll}|{F}_{i}\left(x\right)|\hfill & 1\le i\le m\hfill \\ {F}_{i}\left(x\right)\hfill & i>m.\hfill \end{array}$

Here m is the value of the AbsoluteMaxObjectiveCount option. The advantage of this formulation is you can minimize the absolute value of some components of F, even though the absolute value function is not smooth.

In order to use this option, reorder the elements of F, if necessary, so the first elements are those for which you want the minimum absolute value.

For example, consider the problem in Examples. Modify the problem to find the minimum of the maximum absolute values of all fi(x). Solve this problem by invoking fminimax with the commands

x0 = [0.1; 0.1]; % Make a starting guess at the solution
options = optimoptions('fminimax','AbsoluteMaxObjectiveCount',5); % Minimize abs. values
[x,fval] = fminimax(@myfun,x0,...
[],[],[],[],[],[],[],options);

After seven iterations, the solution is

x =
4.9256
2.0796
fval =
37.2356 -37.2356 -6.8357 -7.0052 -0.9948

## Limitations

The function to be minimized must be continuous. fminimax might only give local solutions.

collapse all

### Algorithms

fminimax internally reformulates the minimax problem into an equivalent Nonlinear Linear Programming problem by appending additional (reformulation) constraints of the form Fi(x) ≤ γ to the constraints given in Equation, and then minimizing γ over x. fminimax uses a sequential quadratic programming (SQP) method [1] to solve this problem.

Modifications are made to the line search and Hessian. In the line search an exact merit function (see [2] and [4]) is used together with the merit function proposed by [3] and [5]. The line search is terminated when either merit function shows improvement. The function uses a modified Hessian that takes advantage of the special structure of this problem. Using optimoptions to set the MeritFunction option to'singleobj' uses the merit function and Hessian used in fmincon.

See also SQP Implementation for more details on the algorithm used and the types of procedures printed under the Procedures heading when you set the Display option to'iter'.

## References

[1] Brayton, R. K., S. W. Director, G. D. Hachtel, and L. Vidigal, "A New Algorithm for Statistical Circuit Design Based on Quasi-Newton Methods and Function Splitting," IEEE Trans. Circuits and Systems, Vol. CAS-26, pp. 784-794, Sept. 1979.

[2] Grace, A. C. W., "Computer-Aided Control System Design Using Optimization Techniques," Ph.D. Thesis, University of Wales, Bangor, Gwynedd, UK, 1989.

[3] Han, S. P., "A Globally Convergent Method For Nonlinear Programming," Journal of Optimization Theory and Applications, Vol. 22, p. 297, 1977.

[4] Madsen, K. and H. Schjaer-Jacobsen, "Algorithms for Worst Case Tolerance Optimization," IEEE Trans. of Circuits and Systems, Vol. CAS-26, Sept. 1979.

[5] Powell, M. J. D., "A Fast Algorithm for Nonlinearly Constrained Optimization Calculations," Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics, Vol. 630, Springer Verlag, 1978.