fminimax - Solve minimax constraint problem

Equation

Finds the minimum of a problem specified by

where x, b, beq, lb, and ub 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.

Syntax

x = fminimax(fun,x0) x = fminimax(fun,x0,A,b) x = fminimax(fun,x,A,b,Aeq,beq) x = fminimax(fun,x,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,x,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,x,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.

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 the structure options. Use optimset to set these options.

x = fminimax(problem) finds the minimum for problem, where problem is a structure described in Input Arguments.

Create the structure problem by exporting a problem from Optimization Tool, as described in Exporting to the MATLAB Workspace.

[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 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 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. 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 GradObj to 'on' is effective only when there is no nonlinear constraint, or when the nonlinear constraint has GradConstr set to 'on' 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 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 Setting GradConstr to 'on' is effective only when GradObj is set to 'on' 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 structure created with optimset

Output Arguments

Function 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.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
	`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. You can use optimset to set or change the values of these fields in the options structure options. See Optimization Options for detailed information.

`DerivativeCheck`	Compare user-supplied derivatives (gradients of objective or constraints) to finite-differencing 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 finite-difference gradients (a positive scalar). The default is `0.1`.
`DiffMinChange`	Minimum change in variables for finite-difference gradients (a positive scalar). The default is `1e-8`.
`Display`	Level of display: `'off'` 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 just the final output, and gives the default exit message. `'final-detailed'` displays just the final output, and gives the technical exit message.
`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 user-defined 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 user-defined 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 user-defined 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 F(x) to minimize the worst case absolute values. The default is `0`.
`OutputFcn`	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.
`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 (`[]`). `@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.
`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 `1e-6`.
`TolConSQP`	Termination tolerance on inner iteration SQP constraint violation, a positive scalar. The default is `1e-6`.
`TolFun`	Termination tolerance on the function value, a positive scalar. The default is `1e-6`.
`TolX`	Termination tolerance on `x`, a positive scalar. The default value 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)`. `fminimax` uses `TypicalX` for scaling finite differences for gradient estimation.
`UseParallel`	When `'always'`, estimate gradients in parallel. Disable by setting to the default `'never'`.

Examples

Find values of x that minimize the maximum value of

[f₁(x), f₂(x), f₃(x), f₄(x), f₅(x)]

where

First, write an M-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 = 
     4.0000
     4.0000
fval =
     0.0000  -64.0000  -2.0000  -8.0000  -0.0000

Notes

You can set the number of objectives for which the worst case absolute values of F are minimized in the MinAbsMax option using optimset. You should partition these objectives into the first elements of F.

For example, consider the preceding problem, which requires finding values of x that minimize the maximum absolute value of

[f₁(x), f₂(x), f₃(x), f₄(x), f₅(x)]

Solve this problem by invoking fminimax with the commands

x0 = [0.1; 0.1]; % Make a starting guess at the solution
options = optimset('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

If equality constraints are present, and dependent equalities are detected and removed in the quadratic subproblem, 'dependent' is displayed under the Procedures heading (when the Display option is set to 'iter'). The dependent equalities are only removed when the equalities are consistent. If the system of equalities is not consistent, the subproblem is infeasible and 'infeasible' is displayed under the Procedures heading.

Algorithm

fminimax internally reformulates the minimax problem into an equivalent Nonlinear Linear Programming problem by appending additional (reformulation) constraints of the form F_i(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 optimset 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'.

Limitations

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

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.