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Solve minimax constraint problem
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 maxmin problems with fminimax, using the identity
You can solve problems of the form
by using the MinAbsMax option; see Notes.
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(...)
fminimax minimizes the worstcase (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.
Function Arguments contains general descriptions of arguments passed into fminimax. This section provides functionspecific details for fun, nonlcon, and problem:
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 userdefined 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 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 = optimoptions('fminimax','GradObj','on') 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 nbym 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

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 = optimoptions('fminimax','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.  
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 
Function Arguments contains general descriptions of arguments returned by fminimax. This section provides functionspecific 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.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.MaxFunEvals.  
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 firstorder optimality  
constrviolation  Maximum of constraint functions  
message  Exit message 
Optimization options used by fminimax. Use optimoptions to set or change options. See Optimization Options Reference for detailed information.
DerivativeCheck  Compare usersupplied derivatives (gradients of objective or constraints) to finitedifferencing 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 finitedifference gradients (a positive scalar). The default is Inf. 
DiffMinChange  Minimum change in variables for finitedifference gradients (a positive scalar). The default is 0. 
Display  Level of display:

FinDiffRelStep  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. 
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. 
GradConstr  Gradient for the userdefined constraints. When set to 'on', fminimax expects the constraint function to have four outputs, as described in nonlcon in Input Arguments. When set to the default 'off', fminimax estimates gradients of the nonlinear constraints by finite differences. 
GradObj  Gradient for the userdefined objective function. See the preceding description of fun to see how to define the gradient in fun. Set to 'on' to have fminimax use a userdefined gradient of the objective function. The default 'off' causes fminimax to estimate gradients using finite differences. 
MaxFunEvals  Maximum number of function evaluations allowed, a positive integer. The default value is 100*numberOfVariables. 
MaxIter  Maximum number of iterations allowed, a positive integer. The default value is 400. 
MaxSQPIter  Maximum number of SQP iterations allowed, a positive integer. The default is 10*max(numberOfVariables, numberOfInequalities + numberOfBounds). 
MeritFunction  Use the goal attainment/minimax merit function if set to 'multiobj' (default). Use the fmincon merit function if set to 'singleobj'. 
MinAbsMax  Number of elements of F_{i}(x) to minimize the maximum absolute value of F_{i}. See Notes. The default is 0. 
OutputFcn  Specify one or more userdefined 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 ([]).
For information on writing a custom plot function, see Plot Functions. 
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 it 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 1e6. 
TolConSQP  Termination tolerance on inner iteration SQP constraint violation, a positive scalar. The default is 1e6. 
TolFun  Termination tolerance on the function value, a positive scalar. The default is 1e6. 
TolX  Termination tolerance on x, a positive scalar. The default value is 1e6. 
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). fminimax uses TypicalX for scaling finite differences for gradient estimation. 
UseParallel  When true, estimate gradients in parallel. Disable by setting to the default false. See Parallel Computing. 
Find values of x that minimize the maximum value of
[f_{1}(x), f_{2}(x), f_{3}(x), f_{4}(x), f_{5}(x)]
where
First, write a file that computes the five functions at x.
function f = myfun(x) f(1)= 2*x(1)^2+x(2)^248*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
You can solve problems of the form
where
Here m is the value of the MinAbsMax 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 f_{i}(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','MinAbsMax',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
The function to be minimized must be continuous. fminimax might only give local solutions.
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[2] Grace, A.C.W., "ComputerAided 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. SchjaerJacobsen, "Algorithms for Worst Case Tolerance Optimization," IEEE Trans. of Circuits and Systems, Vol. CAS26, 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.