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I try to find the local minimum of a function $$\Pi(\vec{d})$$ where $$\vec{d} = (d_1,d_2,...,d_n)$$ should satisfy $$d_i>0$$ for $i \in [1,n]$.

I use the FindMinimum function in Mathematica; i.e., use something like $$\mathtt{FindMinimum}[\{\Pi(\vec{d}),d_i>0\},\vec{d}].$$ But I don't know how to apply the constraints $d_i>0$. If it is indeed possible, it would very much help me to know how to formulate such a command. Thanks!

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  • $\begingroup$ FindMinimum[{pi[dvector], Apply[Min, dvector] > 0}, dvector] or you can learn to use the various abbreviations that MMA provides and then use FindMinimum[{pi[dvector], Min@@dvector > 0}, dvector] $\endgroup$ – Bill Aug 7 '17 at 6:30
  • $\begingroup$ Thank you so much. It works perfectly. Could you please show me where to find the various abbreviations that MMA provides as you said? $\endgroup$ – Wilhelm Aug 7 '17 at 7:11
  • $\begingroup$ This reference.wolfram.com/language/howto/UseShorthandNotations.html might get you started. $\endgroup$ – Bill Aug 8 '17 at 6:49
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ConicHullRegion[] is a useful constraining region to use for vectors with all positive components. In particular, one can express the first orthant for n-dimensional space in region form as ConicHullRegion[{ConstantArray[0, n]}, IdentityMatrix[n]].

Using the generalized Rosenbrock function as an example:

rosenbrock[v_?VectorQ] := With[{n = Length[v]}, 
  Sum[100 (Indexed[v, k + 1] - Indexed[v, k]^2)^2 + (1 - Indexed[v, k])^2, {k, 1, n - 1}]]

With[{n = 8}, 
     FindMinimum[{rosenbrock[v], 
                  v ∈ ConicHullRegion[{ConstantArray[0, n]}, IdentityMatrix[n]]},
                 {v, ConstantArray[0.5, n]}]]
   {2.46713*^-14, {v -> {1., 1., 1., 1., 1., 1., 1., 1.}}}
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