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Trying to define a variable as the index in another (vector) variable (MATLAB,YALMIP)

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Pira Perry
  • Gurobi-versary
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  • First Question

Hello everyone.

I want to define a decision variable that can indicate the first entry in another variable that is equal to  specified value. For example,

a = binvar(4,1); % 4x1 binary variable
b = intvar(1);

I want to add the constraint that let b change with a and indicate the index of first entry in a that is equal to 1, eg. a=(1,1,1,0) => b = 1; a=(0, 1, 1, 0) =>  b=2; a=(0,0,1,0) => b=3; etc.... The relation between a and b is just like function  b=find(a(:) == 1) 

Does anyone know how express this constraint in MATLAB? Or any similar constraint? Very much thanks!

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4 comments

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  • Official comment
    Simranjit Kaur
    • Gurobi Staff
    This post is more than two years old. Some information may not be up to date. For current information, please check the Gurobi Documentation or Knowledge Base. If you need more help, please create a new post in the community forum, or try Gurobot, our chatbot interface offering instant, expert-level support.
  • Riley Clement
    • Gurobi Staff
    Edited

    Hi Pira,

    Say you have binary variables \(a_1, \ldots, a_n\) and integer variable \(b\), then introduce binary variables \(x_1, \ldots, x_n\) and the following constraints.

    \begin{eqnarray}
    \sum_{i=1}^nx_i & \leq & 1\\
    x_i & \leq & a_i, \quad \forall i \in \{1,\ldots,n\}\\
    \sum_{j=1}^i x_j & \geq & a_i, \quad \forall i \in \{1,\ldots,n\}\\
    b & = & \sum_{i=1}^nix_i
    \end{eqnarray}

    The \(x\) variable corresponding to the leading 1, will be 1, all else will be 0.

    An example:

    \begin{eqnarray}
    && a = (0,0,1,0,1,1)\\
    && x = (0,0,1,0,0,0)\\
    && b = 1\cdot0 + 2\cdot0 + 1\cdot3 + 0\cdot4 + .... = 3\\
    \end{eqnarray}

    If \(a\) is all zero then \(b\) will be zero.

    - Riley

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  • Pira Perry
    • Gurobi-versary
    • First Comment
    • First Question

    Hi Riley,

    Thanks for your immediate reply.
    Seems like there's no existed function to express this constraint, but it's nice to use the index i to weight and sum. I will try it.

    Thanks again and best regards.
    Pira

    0
  • Riley Clement
    • Gurobi Staff

    Hi Pira,

    No problem, another formulation to try also:

    Introduce binary variables \(x_1, \ldots, x_n\) and the following constraints.

    \begin{eqnarray}
    a_i & \leq & x_i, \quad \forall i \in \{1,\ldots,n\}\\
    x_i & \leq & x_{i+1}, \quad \forall i \in \{1,\ldots,n-1\}\\
    \sum_{j=1}^ia_j & \geq & x_i \quad \forall i \in \{1,\ldots,n\}\\
    b & = & n+1 - \sum_{i=1}^nx_i
    \end{eqnarray}

    An example:

    \begin{eqnarray}
    && a = (0,0,1,0,1,1)\\
    && x = (0,0,1,1,1,1)\\
    && b = 7 - \sum_ix_i = 3\\
    \end{eqnarray}

    If \(a\) is all zero then \(b\) will be \(n+1\).

    I'm not sure which approach is best without running tests, but feel free to share if you find out.

    - Riley

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