Optimisation of three function in two variables

Hi, i have three sperimantal eqation that describe: Temperature (T), Pressure (P) and refraction (R). Each function dependes from two variables x and y. So T=f(x,y), Pf(x,y) and R=f(x,y). I kwon the range of valure for x and y, so x1<x<x2 and y1<y<y2.

I want to maximise each of the functions, so i want to find the best couple of value for x and y that optimise the functions.

What is the best strategy?

Answer 1

[Star Strider]

To begin with, you need to parameterise the functions so that ‘x’ and ‘y’ are two elements in the same vector.

Your functions then become:

% b(1) = x, b(2) = y

T = f(b(1),b(2)) ...;

P = f(b(1),b(2)) ...;

R = f(b(1),b(2)) ...;

and the bounds become:

lb = [x1, y1];

ub = [x2, y2];

You can then use these with the fmincon functiion, for example. Optimise in ‘b’ and if you want to get the maxima, negate them in tthe fmincon call.

.

Answer 2

[John D'Errico]

There may be an issue here, if you think you can maximize them all simultaneously. The (x,y) coordinates that maximizes one function need not be (and almost certainly will not be) the same as that which maximizes the other responses.

Of course, if you are willing to find a different extremal point for each objective, then you just call a tool like fmincon three times, once for each objective.

Is there any recourse? I assume what you want to do is called multi-criteria optimization, or multi-objective optimization.

Here the common idea is to optimize a weighted sum of all three objectives. If, for example, the three objectives have completely different units or scaling, surely you want to weight them to counter that, possibly making them each roughly of the same importance in the result.

The tool in MATLAB to solve this general problem is fgoalattain.

help fgoalattain

 fgoalattain - Solve multiobjective goal attainment problems
    fgoalattain solves the goal attainment problem, a formulation for
    minimizing a multiobjective optimization problem.

    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,exitflag,output] = fgoalattain(___)
      [x,fval,attainfactor,exitflag,output,lambda] = fgoalattain(___)

    Input Arguments
      fun - Objective functions
        function handle | function name
      x0 - Initial point
        real vector | real array
      goal - Goal to attain
        real vector
      weight - Relative attainment factor
        real vector
      A - Linear inequality constraints
        real matrix
      b - Linear inequality constraints
        real vector
      Aeq - Linear equality constraints
        real matrix
      beq - Linear equality constraints
        real vector
      lb - Lower bounds
        real vector | real array
      ub - Upper bounds
        real vector | real array
      nonlcon - Nonlinear constraints
        function handle | function name
      options - Optimization options
        output of optimoptions | structure such as optimset returns
      problem - Problem structure
        structure

    Output Arguments
      x - Solution
        real vector | real array
      fval - Objective function values at solution
        real array
      attainfactor - Attainment factor
        real number
      exitflag - Reason fgoalattain stopped
        integer
      output - Information about optimization process
        structure
      lambda - Lagrange multipliers at solution
        structure

    Examples
      openExample('optim/BasicGoalAttainmentProblemExample')
      openExample('optim/GoalAttainmentWithLinearConstraintExample')
      openExample('optim/GoalAttainmentWithBoundsExample')
      openExample('optim/GoalAttainmentWithNonlinearConstraintExample')
      openExample('optim/GoalAttainmentUsingNondefaultOptionsExample')
      openExample('optim/ObtainObjectiveFunctionValuesInGoalAttainmentExample')
      openExample('optim/ObtainMoreOutputsInGoalAttainmentExample')
      openExample('optim/EffectsOfWeightsGoalsAndConstraintsInGoalAttainmentExample')

    See also fmincon, fminimax, optimoptions, Optimize

    Introduced in Optimization Toolbox before R2006a
    Documentation for fgoalattain
       doc fgoalattain

[Steven Lord]

Depending on what the poster wants the fminimax function may also be useful.