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linear algebra – How many inequalities do I necessity to make sure a exclusive resolution? Answer

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linear algebra – How many inequalities do I necessity to make sure a exclusive resolution?

Suppose there’s a ‘strictly constructive’ vector $ vec x in (0,1) ^ ok $that lies on the likelihood simplex $ sum_i x_i = 1 $. What is the smallest variety of inequalities of figure $ vec g_i ^ T vec x geq 0 $ required to make positive the answer clique is a singleton vector? Note that there is no such thing as a restriction on $ { vec g_i } $ so one can select the ‘greatest’ workable vectors (it ought to breathe constant within the sense that there has to breathe an answer on the simplex with constructive elements).

I cerebrate the respond would breathe $ 2 (k-1) $ – I do know that it’s undoubtedly sufficient, however apparently I can’t display that it’s too needful. Construction for sufficiency – take pairs of $ vec g_i, – vec g_i $ so primarily you’ll have $ k-1 $ Shape equations $ vec x ^ T vec g_i = 0 $ and $ sum_i x_i = 1 $. We have $ ok $ Equations in order that this has a exclusive resolution and it isn’t troublesome to do that by apt altenative of $ vec g_i $, we will add constructive elements to the answer.

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