Let a and B E R. Consider the linear optimization model:(Pa,8)mins.t.X1 + x₂αX1 + βα2 >1x1 ≥ 0, x₂ € R.Write the linear programming dual of (Pa,ß). Call it (Da‚ß).α,β determine all values of a and ß for which(Da‚ß) is infeasible. What does that mean about (Pa‚ß)?(Da‚ß) is feasible, but not unbounded. What does that mean about (Pa,ß)?(Da‚ß) is unbounded. What does that mean about (Pa,p)?(Pa,ß) has multiple optimal solutions. (You reasoning should use complementarity slackness relation-ships between primal and dual optimal solutions.)

Let .Consider the linear optimization model (Pα,β)..Subject to:We need to write the linear programming dual of (Pα,β).Also, we need to find the values of such that the linear programming dual is feasible or infeasible.Observe that the variable is unrestricted in sign.Let , where .Since the linear programming is in minimization form, therefore we will convert all the constraints in form.Since is unrestricted in sign, therefore we will replace by , where .Therefore the linear programming model (Pα,β) is written as follows..Subject to:Since there is only one constraint, therefore we will be choosing as corresponding dual variable.Therefore the linear programming dual (Dα,β) is calculated as follows..Subject to:(a)We know that a linear programming problem is feasible, if there exists at least one solution.Therefore to have infeasible solution, we don't have a feasible region.Since , therefore either (i) and or (ii) and is true.Hence for infeasible solution, either (i) and or (ii) and .Hence the primal problem have no feasible solution or unbounded region.(b)To have a feasible solution, all of the following constraints must have a feasible region.Since , therefore .Hence for feasible region, .Therefore either the primal problem has an optimal solution or have a bounded region.(c)Consider the given constraints.Therefore the dual problem is unbounded if .Therefore the primal problem is infeasible.(d)Suppose the primal problem has multiple optimal solutions.If the primal problem has multiple solutions, then complementary slackness conditions indicates that, some primal variables are positive, this implies that dual problem has multiple optimal solutions.Hence the linear programming dual (Dα,β) is written as follows..Subject to:(a)For all infeasible dual solution, either (i) and or (ii) and .Hence primal problem have no feasible solution or unbounded region.(b)Hence for either , or the dual problem is feasible.Hence there exists a optimal primal solution or unbounded region.(c)Hence the dual problem is unbounded.Therefore the primal problem is infeasible.(d)If the primal problem has multiple solutions, then complementary slackness conditions indicate that, some primal variables are positive, this implies that dual problem has multiple optimal solutions.