CO350Assignment 2Due: May 18 at 2pm.Question 1:Consider the following linear programs.(LPa)minx1+ 2x3Subject to2x1+x2-3x3≤43x1+ 2x2-4x3= 3-3x1-x2+x3≥3x1,x2≥0(LPb)minx1-x3Subject to2x1+x2+ 3x3≥4x1+ 2x2-2x3≤1-3x1+x3= 2x1≥0,x2≤0(a)Write (LPa) in standard inequality form.(b)Write (LPb) in standard equality form.Question 2:Consider the linear program max(cTx:Ax= 0, x≥0), whereA∈Rm×nandc∈Rn. Show that ifx= 0 is not an optimal solution, then the linear program is unbounded.Question 3:Consider the linear program max(cTx:Ax=b, x≥0), whereA∈Rm×n,b∈Rm, andc∈Rn. Show that, if the linear program has two distinct optimal solutions, thenit has infinitely many optimal solutions.Question 4:LetA∈Rm×nandb∈Rm.(a)Show that, if there existsy∈Rmsuch thatyTA≤0 andyTb >0, then the system(Ax=b, x≥0) is infeasible.(b)Show that the following linear program is infeasible by using the result proved in part (a)
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