excluding a number before sending x vector to optimization?

Hi,

I am using a huristic algorithm for optimization. x-variable vector range is as given below.

xU = [16 16 16 16 1 1 1 1];%upper bound on decision variables

xL = [-16 -16 -16 -16 0 0 0 0];%lower bound on decision variables

If I want to exclude 0 from the first 4-variable how I can estaiblesh?

When I define sub groups from 1:16 and -16:-1 then combine as x vector for the first 4 variable, it increases the number of variables not just removing the 0.

thank you

[Askic V]

I'm not sure if I understand what you're asking, but if you want to exclude zero from a vector, then it is very easy:

x = -1:5;
x

x = 1×7

-1 0 1 2 3 4 5

x(x==0) = [];
x

x = 1×6

-1 1 2 3 4 5

[Gulcihan Ozdemir]

Thank you for the response but that was not the question.

I want to exclude zero just from some part of the vector not all zeros in the vector.

[Askic V]

Is your x a matrix with 8 columns where column k is bounded between xU(k) and xL(k)? You want to remove zeros from the first 4 columns of x?

[John D'Errico]

@Gulcihan Ozdemir - Please don't post an answer just to make a comment.

[Gulcihan Ozdemir]

Yes correct. that is the problem I asked. How to handle that

Answer 1

[John D'Errico]

You don't tell us WHICH heuristic algorithm you are using, so that means it is a homebrew algorithm of your own choosing or creation. It also tells me I cannot be more accurate about an answer.

And it is not even clear if your problem is an integer one. If the variables are continuous, then no, there is no tool where you can tell it that you have a continuous domain, that DOES not include some interior point of that domain. You cannot tell an optimizer like fmincon to not allow some specific value within the bounds for that variable.

The only optimizers that can solve an integer problem like this are tools like GA, so an algorithm that can handle integer variables. GA can handle it with no problem. And any algorithm that can handle integer variables in a nonlinear context should be ok too. So I'll show how to solve it using GA.

help ga

 GA    Constrained optimization using genetic algorithm.
    GA attempts to solve problems of the following forms:
        min F(X)  subject to:  A*X  <= B, Aeq*X  = Beq (linear constraints)
         X                     C(X) <= 0, Ceq(X) = 0 (nonlinear constraints)
                               LB <= X <= UB 
                               X(i) integer, where i is in the index
                               vector INTCON (integer constraints)
 
    Note: If INTCON is not empty, then no equality constraints are allowed.
    That is:-
    * Aeq and Beq must be empty
    * Ceq returned from NONLCON must be empty
 
    X = GA(FITNESSFCN,NVARS) finds a local unconstrained minimum X to the
    FITNESSFCN using GA. NVARS is the dimension (number of design
    variables) of the FITNESSFCN. FITNESSFCN accepts a vector X of size
    1-by-NVARS, and returns a scalar evaluated at X.
 
    X = GA(FITNESSFCN,NVARS,A,b) finds a local minimum X to the function
    FITNESSFCN, subject to the linear inequalities A*X <= B. Linear
    constraints are not satisfied when the PopulationType option is set to
    'bitString' or 'custom'. See the documentation for details.
 
    X = GA(FITNESSFCN,NVARS,A,b,Aeq,beq) finds a local minimum X to the
    function FITNESSFCN, subject to the linear equalities Aeq*X = beq as
    well as A*X <= B. (Set A=[] and B=[] if no inequalities exist.) Linear
    constraints are not satisfied when the PopulationType option is set to
    'bitString' or 'custom'. See the documentation for details.
 
    X = GA(FITNESSFCN,NVARS,A,b,Aeq,beq,lb,ub) defines a set of lower and
    upper bounds on the design variables, X, so that a solution is found in
    the range lb <= X <= ub. Use empty matrices for lb and ub if no bounds
    exist. Set lb(i) = -Inf if X(i) is unbounded below;  set ub(i) = Inf if
    X(i) is unbounded above. Linear constraints are not satisfied when the
    PopulationType option is set to 'bitString' or 'custom'. See the 
    documentation for details.
 
    X = GA(FITNESSFCN,NVARS,A,b,Aeq,beq,lb,ub,NONLCON) subjects the
    minimization to the constraints defined in NONLCON. The function
    NONLCON accepts X and returns the vectors C and Ceq, representing the
    nonlinear inequalities and equalities respectively. GA minimizes
    FITNESSFCN such that C(X)<=0 and Ceq(X)=0. (Set lb=[] and/or ub=[] if
    no bounds exist.) Nonlinear constraints are not satisfied when the
    PopulationType option is set to 'bitString' or 'custom'. See the 
    documentation for details.
 
    X = GA(FITNESSFCN,NVARS,A,b,Aeq,beq,lb,ub,NONLCON,options) minimizes
    with the default optimization parameters replaced by values in OPTIONS.
    OPTIONS can be created with the OPTIMOPTIONS function. See OPTIMOPTIONS
    for details. For a list of options accepted by GA refer to the
    documentation.
 
    X = GA(FITNESSFCN,NVARS,A,b,[],[],lb,ub,NONLCON,INTCON) requires that
    the variables listed in INTCON take integer values. Note that GA does
    not solve problems with integer and equality constraints. Pass empty
    matrices for the Aeq and beq inputs if INTCON is not empty.
 
    X = GA(FITNESSFCN,NVARS,A,b,[],[],lb,ub,NONLCON,INTCON,options)
    minimizes with integer constraints and the default optimization
    parameters replaced by values in OPTIONS. OPTIONS can be created with
    the OPTIMOPTIONS function. See OPTIMOPTIONS for details.
 
    X = GA(PROBLEM) finds the minimum for PROBLEM. PROBLEM is a structure
    that has the following fields:
        fitnessfcn: <Fitness function>
             nvars: <Number of design variables>
             Aineq: <A matrix for inequality constraints>
             bineq: <b vector for inequality constraints>
               Aeq: <Aeq matrix for equality constraints>
               beq: <beq vector for equality constraints>
                lb: <Lower bound on X>
                ub: <Upper bound on X>
           nonlcon: <Nonlinear constraint function>
            intcon: <Index vector for integer variables>
           options: <Options created with optimoptions('ga',...)>
          rngstate: <State of the random number generator>
 
    [X,FVAL] = GA(FITNESSFCN, ...) returns FVAL, the value of the fitness
    function FITNESSFCN at the solution X.
 
    [X,FVAL,EXITFLAG] = GA(FITNESSFCN, ...) returns EXITFLAG which
    describes the exit condition of GA. Possible values of EXITFLAG and the
    corresponding exit conditions are
 
      1 Average change in value of the fitness function over
         options.MaxStallGenerations generations less than 
         options.FunctionTolerance and constraint violation less than 
         options.ConstraintTolerance.
      3 The value of the fitness function did not change in
         options.MaxStallGenerations generations and constraint violation 
         less than options.ConstraintTolerance.
      4 Magnitude of step smaller than machine precision and constraint
         violation less than options.ConstraintTolerance. This exit 
         condition applies only to nonlinear constraints.
      5 Fitness limit reached and constraint violation less than
         options.ConstraintTolerance. 
      0 Maximum number of generations exceeded.
     -1 Optimization terminated by the output or plot function.
     -2 No feasible point found.
     -4 Stall time limit exceeded.
     -5 Time limit exceeded.
 
    [X,FVAL,EXITFLAG,OUTPUT] = GA(FITNESSFCN, ...) returns a
    structure OUTPUT with the following information:
              rngstate: <State of the random number generator before GA started>
           generations: <Total generations, excluding HybridFcn iterations>
             funccount: <Total function evaluations>
         maxconstraint: <Maximum constraint violation>, if any
               message: <GA termination message>
 
    [X,FVAL,EXITFLAG,OUTPUT,POPULATION] = GA(FITNESSFCN, ...) returns the
    final POPULATION at termination.
 
    [X,FVAL,EXITFLAG,OUTPUT,POPULATION,SCORES] = GA(FITNESSFCN, ...) returns
    the SCORES of the final POPULATION.
 
 
    Example:
      Unconstrained minimization of Rastrigins function:
        function scores = myRastriginsFcn(pop)
            scores = 10.0 * size(pop,2) + sum(pop.^2 - 10.0*cos(2*pi .* pop),2);
 
      numberOfVariables = 2
      x = ga(@myRastriginsFcn,numberOfVariables)
 
      Display plotting functions while GA minimizes
       options = optimoptions('ga','PlotFcn',...
         {@gaplotbestf,@gaplotbestindiv,@gaplotexpectation,@gaplotstopping});
       [x,fval,exitflag,output] = ga(fitfcn,2,[],[],[],[],[],[],[],options)
 
    An example with inequality constraints and lower bounds
     A = [1 1; -1 2; 2 1];  b = [2; 2; 3];  lb = zeros(2,1);
     fitfcn = @(x)0.5*x(1)^2 + x(2)^2 -x(1)*x(2) -2*x(1) - 6.0*x(2);
 
     % Use mutation function which can handle constraints
     options = optimoptions('ga','MutationFcn',@mutationadaptfeasible);
     [x,fval,exitflag] = ga(fitfcn,2,A,b,[],[],lb,[],[],options);
 
    If FITNESSFCN or NONLCON are parameterized, you can use anonymous
    functions to capture the problem-dependent parameters. Suppose you want
    to minimize the fitness given in the function myfit, subject to the
    nonlinear constraint myconstr, where these two functions are
    parameterized by their second argument a1 and a2, respectively. Here
    myfit and myconstr are MATLAB file functions such as
 
         function f = myfit(x,a1)
         f = exp(x(1))*(4*x(1)^2 + 2*x(2)^2 + 4*x(1)*x(2) + 2*x(2) + a1);
 
    and
 
         function [c,ceq] = myconstr(x,a2)
         c = [1.5 + x(1)*x(2) - x(1) - x(2);
               -x(1)*x(2) - a2];
         % No nonlinear equality constraints:
          ceq = [];
 
    To optimize for specific values of a1 and a2, first assign the values
    to these two parameters. Then create two one-argument anonymous
    functions that capture the values of a1 and a2, and call myfit and
    myconstr with two arguments. Finally, pass these anonymous functions to
    GA:
 
      a1 = 1; a2 = 10; % define parameters first
      % Mutation function for constrained minimization
      options = optimoptions('ga','MutationFcn',@mutationadaptfeasible);
      x = ga(@(x)myfit(x,a1),2,[],[],[],[],[],[],@(x)myconstr(x,a2),options)
 
    Example: Solving a mixed-integer optimization problem
    An example of optimizing a function where a subset of the variables are
    required to be integers:
    
    % Define the objective and call GA. Here variables x(2) and x(3) will
    % be integer. 
    fun = @(x) (x(1) - 0.2)^2 + (x(2) - 1.7)^2 + (x(3) -5.1)^2; 
    x = ga(fun,3,[],[],[],[],[],[],[],[2 3])
           
    See also OPTIMOPTIONS, FITNESSFUNCTION, GAOUTPUTFCNTEMPLATE, PATTERNSEARCH, @.

    Documentation for ga
       doc ga

For example, suppose I want to find the minimum of a VERY simple, two variable function. So minimize X^2+Y^2, subject to the constraint that X lies in the set [-5 -4 -3 2 4 9 12], and Y lies in the set [-14 -11 -5 3 6 10 16]? These are totally arbitrary sets of integers.

Yesh, I know, utterly trivial. And I can surely find many ways to solve the problem. I might use a table lookup, for example.

xvalid = [-6 -5 -4 -3 2 4 9 12];
yvalid = [-14 -11 -5 3 6 10 16];
fun = @(xyind) xvalid(xyind(1)).^2 + yvalid(xyind(2)).^2;

intcon = [1 2];
LB = [1 1];
UB = [8 7];

[xyind,fval,exitflag] = ga(fun,2,[],[],[],[],LB,UB,[],intcon)

Optimization terminated: average change in the penalty fitness value less than options.FunctionTolerance
and constraint violation is less than options.ConstraintTolerance.

xyind = 1×2

5 4

fval = 13

exitflag = 1

[xvalid(xyind(1)),yvalid(xyind(2))]

ans = 1×2

2 3

As you can see, there is no problem at all, since the algorithm itself thinks the set of choices are the integers 1:8 for X and 1:7 for Y.

Anything you will do will probably be in some way equivalent to that.

[Gulcihan Ozdemir]

Thanks for your effort.

Yes integer variables I am having.

"As you can see, there is no problem at all, since the algorithm itself thinks the set of choices are the integers 1:8 for X and 1:7 for Y."

What if you want to remove the third element in the search space of x above given (1:8 range and you do not want to see in the solution set the value 3, you do not want to include it) , how do you handle?

[Torsten]

The value "3" does not stem from 1:8 and 1:7. The "2" comes from xvalid and the "3" comes from yvalid. If you don't want to see certain numbers in the solution, you just have to remove these numbers from xvalid resp. yvalid. 1:8 resp. 1:7 are just the number of elements of the two valid number sets.