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I have an optimization problem which I would like to know whether it is linear or convex.

There are three variables $x_0$, $x_1$ and $x_2$ for which the optimization problem should be minimized.

Constraint: $\sum_{i=0}^{3}x_n = 1$

Boundaries: $0 \leq x_n \leq 1$

Objective: $Minimize\ \ \ || \pmb{v_3} - (x_0\pmb{v_0} + x_1\pmb{v_1} + x_2\pmb{v_2}) ||$

where $x_0$, $x_1$ and $x_2$ are scalars and $\pmb{v_0}$, $\pmb{v_1}$, $\pmb{v_2}$ and $\pmb{v_3}$ vectors of the same size.

Is my optimization problem linear or convex? And why?

MerklT
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  • What norm? Euclidean, sup, $p$... –  May 25 '20 at 13:43
  • @Renard the vectors have a size of ~10000 elements – MerklT May 26 '20 at 08:41
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    Your constraints are linear. Norms are convex and convexity is preserved under affine transformation so your problem is convex (since you minimize). If the norm is the 1 or infinity norm, then the problem can be converted into a linear problem. – Marc Dinh May 26 '20 at 09:54
  • @MarcDinh Thanks for the explanation. Could you help me converting it into a linear problem for norm 1? – MerklT May 27 '20 at 09:46
  • @MerklT You can find it here : https://math.stackexchange.com/questions/1639716/how-can-l-1-norm-minimization-with-linear-equality-constraints-basis-pu – Marc Dinh May 28 '20 at 10:38

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