Given the optimal solution to the following LO problem x = (0, 3, 2). Write down the dual problem and use the strong duality theorem to find the optimal solution to the dual problem.

min z=3x1 +2x2 +1x3

s.t.

2x1 +x2 4;

3x1 + 2x2 6;

x1 + x2 + x3 = 5;

x1 0,x2 0.

Determine which of the following statements are correct for the given problem setting. Note that there may exist multiple correct answers to some subproblems.

1. Given a primal LO maxAxb,x0 cT x, and its dual problem minAT yc,y0 bT y. (a) For any feasible solutions x and y to the primal and dual problem, respectively, it holds cTx bTy; (b) For any feasible solutions x and y to the primal and dual problem, respectively, both x and y are optimal solutions if and only if cT x = bT y; (c) At the optimal solution pair (x,y), it holds xiyi = 0,i = 1,,n;

2. Given a primal LO maxAxb,x0 cT x, and its dual problem minAT yc,y0 bT y. (a) If the primal problem is infeasible, then the dual problem is unbounded; (b) If the dual problem is unbounded, then the primal problem is infeasible; (c) If the primal problem has an unbounded feasible set, then so is the dual problem;

3. Given a primal LO maxA1xb1,A2x=b2,x0 cT x. (a) If the shadow prices regarding the constraint A1x b1 are always non-negative; (b) If the shadow prices regarding the constraint A1x b1 are always non-positive; (c) If the shadow prices regarding the constraint A2x = b2 are always non-negative; (d) The reduced cost for a non-basic variable are always positive.

4. LetS1 ={x:f(x)0},S2 ={x:g(x)0}wherebothf(x)andg(x)areconvex functions.

   (a) The function f (x) + g(x) is convex; (b) The function f(x)g(x) is convex; (c) The set S1 S2 (the intersection of S1 and S2 ) is convex; (d) The set S1 S2 (the union of S1 and S2 ) is convex.

5. the multiplicity of the optimal solutions to an linear optimization problem is caused by (a) There is a tie in the z-row for the minimum element; (b) There is a zero reduced cost for some nonbasic variable in the final tableau; (c) Some basic variable has a value 0; (d) All of the above;