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I have a question like this:

Show $$x^* = [1,1,1,1]$$ is optimal for the following problem: $$ min\ 6x_1 + .... - 10x_4$$ such that $$ Ax \preccurlyeq b$$

and I am given the matrix A which is 5x4 and vector b length 4 entries. Generally speaking what is the process behind "solving"/showing this? I know this is an inequality form LP, analogous to inequality form SDP per my text.

EDIT $$g(\lambda) = -b^TA$$ for $$ A^T\lambda + c = 0$$ -inf otherwise

bikeactuary
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  • The number of columns in $A$ should be equal to the dimension of $x$ in order to multiply $A$ and $x$ – Kendall Nov 30 '23 at 02:53
  • There is a procedure called Certificate of optimality using the dual of this problem. If you can find a dual feasible solution whose objective value is equal to the objective value of this problem (primal) at $x^$, then $x^$ is optimal for the primal. – Kendall Nov 30 '23 at 02:58
  • sorry, yes i edit to say A is 5 row by 4 columns – bikeactuary Nov 30 '23 at 03:23
  • Also, this should be something solvable by hand on a single sheet of paper.. – bikeactuary Nov 30 '23 at 03:25
  • It will be. You might need to work out some cases but you should be able to write the full answer in a single sheet – Kendall Nov 30 '23 at 03:38
  • i have the dual, I will edit it into the question. I do not see anything useful in text on certificate of optimality – bikeactuary Nov 30 '23 at 04:05

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