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fgoalattain

Solve multiobjective goal attainment problems

Equation

Finds the minimum of a problem specified by

weight, goal, 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.

Syntax

x = fgoalattain(fun,x0,goal,weight)
x = fgoalattain(fun,x0,goal,weight,A,b)
x = fgoalattain(fun,x0,goal,weight,A,b,Aeq,beq)
x = fgoalattain(fun,x0,goal,weight,A,b,Aeq,beq,lb,ub)
x = fgoalattain(fun,x0,goal,weight,A,b,Aeq,beq,lb,ub,nonlcon)
x = fgoalattain(fun,x0,goal,weight,A,b,Aeq,beq,lb,ub,nonlcon,... options)
x = fgoalattain(problem)
[x,fval] = fgoalattain(...)
[x,fval,attainfactor] = fgoalattain(...)
[x,fval,attainfactor,exitflag] = fgoalattain(...)
[x,fval,attainfactor,exitflag,output] = fgoalattain(...)
[x,fval,attainfactor,exitflag,output,lambda] = fgoalattain(...)

Description

fgoalattain solves the goal attainment problem, which is one formulation for minimizing a multiobjective optimization problem.

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

x = fgoalattain(fun,x0,goal,weight) tries to make the objective functions supplied by fun attain the goals specified by goal by varying x, starting at x0, with weight specified by weight.

x = fgoalattain(fun,x0,goal,weight,A,b) solves the goal attainment problem subject to the linear inequalities A*x ≤ b.

x = fgoalattain(fun,x0,goal,weight,A,b,Aeq,beq) solves the goal attainment problem subject to the linear equalities Aeq*x = beq as well. Set A = [] and b = [] if no inequalities exist.

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

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

x = fgoalattain(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] = fgoalattain(...) returns the values of the objective functions computed in fun at the solution x.

[x,fval,attainfactor] = fgoalattain(...) returns the attainment factor at the solution x.

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

[x,fval,attainfactor,exitflag,output] = fgoalattain(...) returns a structure output that contains information about the optimization.

[x,fval,attainfactor,exitflag,output,lambda] = fgoalattain(...) 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 fgoalattain. This section provides function-specific details for fun, goal, nonlcon, options, weight, 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 function file:

x = fgoalattain(@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 = fgoalattain(@(x)sin(x.*x),x0,goal,weight);

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

To make an objective function as near as possible to a goal value, (i.e., neither greater than nor less than) use optimoptions to set the GoalsExactAchieve option to the number of objectives required to be in the neighborhood of the goal values. Such objectives must be partitioned into the first elements of the vector F returned by fun.

If the gradient of the objective function can also be computed and the GradObj option is 'on', as set by

options = optimoptions('fgoalattain','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)).

    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.

goal

Vector of values that the objectives attempt to attain. The vector is the same length as the number of objectives F returned by fun. fgoalattain attempts to minimize the values in the vector F to attain the goal values given by goal.

nonlcon

The function that computes the nonlinear inequality constraints c(x) ≤ 0 and the 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 = fgoalattain(@myfun,x0,goal,weight,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('fgoalattain','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.

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) (i.e., 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) (i.e., the jth column of GCeq is the gradient of the jth equality constraint ceq(j)).

    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.

Passing Extra Parameters explains how to parameterize the nonlinear constraint function nonlcon, if necessary.

 

options

Options provides the function-specific details for the options values.

 

weight

A weighting vector to control the relative underattainment or overattainment of the objectives in fgoalattain. When the values of goal are all nonzero, to ensure the same percentage of under- or overattainment of the active objectives, set the weighting function to abs(goal). (The active objectives are the set of objectives that are barriers to further improvement of the goals at the solution.)

    Note   Setting a component of the weight vector to zero will cause the corresponding goal constraint to be treated as a hard constraint rather than as a goal constraint. An alternative method to set a hard constraint is to use the input argument nonlcon.

 

When the weighting function weight is positive, fgoalattain attempts to make the objectives less than the goal values. To make the objective functions greater than the goal values, set weight to be negative rather than positive. To make an objective function as near as possible to a goal value, use the GoalsExactAchieve option and put that objective as the first element of the vector returned by fun (see the preceding description of fun and options).

 
problem

objective

Vector of objective functions 

x0

Initial point for x 

goal

Goals to attain 

weight

Relative importance factors of goals 

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

'fgoalattain' 

options

Options created with optimoptions 

Output Arguments

Function Arguments contains general descriptions of arguments returned by fgoalattain. This section provides function-specific details for attainfactor, exitflag, lambda, and output:

attainfactor

The amount of over- or underachievement of the goals. attainfactor contains the value of γ at the solution. If attainfactor is negative, the goals have been overachieved; if attainfactor is positive, the goals have been underachieved.

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 solutions 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

Stopped by an output function or plot 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

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 final 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 fgoalattain. Use optimoptions to set or change options. See Optimization Options Reference 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 Inf.

DiffMinChange

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

Display

Level of 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 just the final output, and gives the default exit message.

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

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' (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.

GoalsExactAchieve

Specifies the number of objectives for which it is required for the objective fun to equal the goal goal (a nonnegative integer). Such objectives should be partitioned into the first few elements of F. The default is 0.

GradConstr

Gradient for nonlinear constraint functions defined by the user. When set to 'on', fgoalattain expects the constraint function to have four outputs, as described in nonlcon in the Input Arguments section. When set to the default, 'off', gradients of the nonlinear constraints are estimated by finite differences.

GradObj

Gradient for the objective function defined by the user. See the preceding description of fun to see how to define the gradient in fun. Set to 'on' to have fgoalattain use a user-defined gradient of the objective function. The default, 'off', causes fgoalattain to estimate gradients using finite differences.

MaxFunEvals

Maximum number of function evaluations allowed (a positive integer). The default is 100*numberOfVariables.

MaxIter

Maximum number of iterations allowed (a positive integer). The default is 400.

MaxSQPIter

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

MeritFunction

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

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

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 are considered too large. The default is none ([]).

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 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). fgoalattain 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.

Examples

Consider a linear system of differential equations.

An output feedback controller, K, is designed producing a closed loop system

The eigenvalues of the closed loop system are determined from the matrices A, B, C, and K using the command eig(A+B*K*C). Closed loop eigenvalues must lie on the real axis in the complex plane to the left of the points [-5,-3,-1]. In order not to saturate the inputs, no element in K can be greater than 4 or be less than -4.

The system is a two-input, two-output, open loop, unstable system, with state-space matrices.

The set of goal values for the closed loop eigenvalues is initialized as

goal = [-5,-3,-1];

To ensure the same percentage of under- or overattainment in the active objectives at the solution, the weighting matrix, weight, is set to abs(goal).

Starting with a controller, K = [-1,-1; -1,-1], first write a function file, eigfun.m.

function F = eigfun(K,A,B,C)
F = sort(eig(A+B*K*C)); % Evaluate objectives

Next, enter system matrices and invoke an optimization routine.

A = [-0.5 0 0; 0 -2 10; 0 1 -2];
B = [1 0; -2 2; 0 1];
C = [1 0 0; 0 0 1]; 
K0 = [-1 -1; -1 -1];       % Initialize controller matrix
goal = [-5 -3 -1];         % Set goal values for the eigenvalues
weight = abs(goal);        % Set weight for same percentage
lb = -4*ones(size(K0));    % Set lower bounds on the controller
ub = 4*ones(size(K0));     % Set upper bounds on the controller
options = optimoptions('fgoalattain','Display','iter');   % Set display parameter
[K,fval,attainfactor] = fgoalattain(@(K)eigfun(K,A,B,C),...
    K0,goal,weight,[],[],[],[],lb,ub,[],options)

You can run this example by using the script goaldemogoaldemo. (From the MATLAB Help browser or the MathWorks Web site documentation, you can click the goaldemo name to display the example.) After about 11 iterations, a solution is

Active inequalities (to within options.TolCon = 1e-006):
  lower      upper     ineqlin   ineqnonlin
    1                                1
    2                                2
    4                                 


K = 
    -4.0000    -0.2564
    -4.0000    -4.0000

fval =
    -6.9313
    -4.1588
    -1.4099

attainfactor = 
    -0.3863

Discussion

The attainment factor indicates that each of the objectives has been overachieved by at least 38.63% over the original design goals. The active constraints, in this case constraints 1 and 2, are the objectives that are barriers to further improvement and for which the percentage of overattainment is met exactly. Three of the lower bound constraints are also active.

In the preceding design, the optimizer tries to make the objectives less than the goals. For a worst-case problem where the objectives must be as near to the goals as possible, use optimoptions to set the GoalsExactAchieve option to the number of objectives for which this is required.

Consider the preceding problem when you want all the eigenvalues to be equal to the goal values. A solution to this problem is found by invoking fgoalattain with the GoalsExactAchieve option set to 3.

options = optimoptions('fgoalattain','GoalsExactAchieve',3);
[K,fval,attainfactor] = fgoalattain(...
   @(K)eigfun(K,A,B,C),K0,goal,weight,[],[],[],[],lb,ub,[],...
options);

After about seven iterations, a solution is

K,fval,attainfactor

K = 
    -1.5954    1.2040
    -0.4201   -2.9046

fval =
    -5.0000
    -3.0000
    -1.0000

attainfactor = 
  1.1304e-022

In this case the optimizer has tried to match the objectives to the goals. The attainment factor (of 1.1304e-22 or so, depending on your system) indicates that the goals have been matched almost exactly.

Notes

This problem has discontinuities when the eigenvalues become complex; this explains why the convergence is slow. Although the underlying methods assume the functions are continuous, the method is able to make steps toward the solution because the discontinuities do not occur at the solution point. When the objectives and goals are complex, fgoalattain tries to achieve the goals in a least-squares sense.

Limitations

The objectives must be continuous. fgoalattain might give only local solutions.

More About

expand all

Algorithms

Multiobjective optimization concerns the minimization of a set of objectives simultaneously. One formulation for this problem, and implemented in fgoalattain, is the goal attainment problem of Gembicki [3]. This entails the construction of a set of goal values for the objective functions. Multiobjective optimization is discussed in Multiobjective Optimization Algorithms.

In this implementation, the slack variable γ is used as a dummy argument to minimize the vector of objectives F(x) simultaneously; goal is a set of values that the objectives attain. Generally, prior to the optimization, it is not known whether the objectives will reach the goals (under attainment) or be minimized less than the goals (overattainment). A weighting vector, weight, controls the relative underattainment or overattainment of the objectives.

fgoalattain uses a sequential quadratic programming (SQP) method, which is described in Sequential Quadratic Programming (SQP). Modifications are made to the line search and Hessian. In the line search an exact merit function (see [1] and [4]) is used together with the merit function proposed by [5] and [6]. The line search is terminated when either merit function shows improvement. A modified Hessian, which takes advantage of the special structure of the problem, is also used (see [1] and [4]). A full description of the modifications used is found in Goal Attainment Method in "Introduction to Algorithms." Setting the MeritFunction option to 'singleobj' with

options = optimoptions(options,'MeritFunction','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 displayed under the Procedures heading when the Display option is set 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 Transactions on Circuits and Systems, Vol. CAS-26, pp 784-794, Sept. 1979.

[2] Fleming, P.J. and A.P. Pashkevich, Computer Aided Control System Design Using a Multi-Objective Optimisation Approach, Control 1985 Conference, Cambridge, UK, pp. 174-179.

[3] Gembicki, F.W., "Vector Optimization for Control with Performance and Parameter Sensitivity Indices," Ph.D. Dissertation, Case Western Reserve Univ., Cleveland, OH, 1974.

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

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

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

See Also

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