Consider the following discrete nonlinear programming problem.Maximize Z = ,subject to x₁ + x₂ ≤ 0.75 and each variable is restricted to the values: .
(a)Reformulate this problem as a pure binary integer linear programming problem.
(b)Use the following outline in designing the main features of a branch-and-bound algorithm for solving this problem (and similar problems)directly without reformulation.(1)Specify the tightest possible nonlinear programming relaxation that has only continuous variables,and so can be solved efficiently by nonlinear programming techniques.(2)Specify the fathoming tests.(3)Specify a branching procedure that involves specifying two ranges of values for a single variable.

Question

Consider the following discrete nonlinear programming problem.Maximize Z =

,subject to x₁ + x₂ ≤ 0.75 and each variable is restricted to the values:

.
(a)Reformulate this problem as a pure binary integer linear programming problem.
(b)Use the following outline in designing the main features of a branch-and-bound algorithm for solving this problem (and similar problems)directly without reformulation.(1)Specify the tightest possible nonlinear programming relaxation that has only continuous variables,and so can be solved efficiently by nonlinear programming techniques.(2)Specify the fathoming tests.(3)Specify a branching procedure that involves specifying two ranges of values for a single variable.

Quizplus · Accepted Answer

The Answer is (a)We define the binary variable y_ij as follows.Let $(a)We define the binary variable yij as follows.Let for i = 1,2 and j = 1,2,3,4.For each i and j,calculate the contribution to the objective function associated with having yij = 1 because xi = 1/(j+1),as shown in the following table. These numbers in the second and third columns then become the coefficients of the respective yij in the objective function of the reformulated problem.The reformulated problem also requires some functional constraints on the yij.In particular,for each i = 1,2,exactly one of the yi1,yi2,yi3,and yi4 variables must be 1.Furthermore,because of the x1 + x2 ≤ 0.75 constraint,certain pairs of yij variables (y11 and y21,y11 and y22,y12 and y21)cannot both be 1.These observations lead to the following pure binary integer linear programming formulation.Maximize Z = (3/4)y11 + (5/9)y12 + (7/16)y12 + (9/25)y14 + (3/4)y11 + (2/3)y22 + (9/16)y23 + (12/25)y24,subject to y11 + y12 + y12 + y14 \le 1 y21 + y22 + y23 + y24 \le 1 y11 + y21 \le 1 y11 + y22 \le 1 y12 + y21 \le 1 and yij is binary,for i = 1,2 and j = 1,2,3,4. (b)(i)Use the following relaxation: Maximize Z = 2x1 - 3 + 3x2 - 3 ,subject to x1 + x2 \le 3/4 and 1/5 \le x1 \le 1/2 ,1/5 \le x2 \le 1/2. (ii)The fathoming tests are the same as those stated in Sec.12.6 of the textbook.A subproblem is fathomed if Test 1: The upper bound on Z for the subproblem is ≤ Z for the incumbent,or Test 2: Its relaxation has no feasible solutions,or Test 3: The optimal solution for its relaxation satisfies the original constraint that each variable is restricted to the values: . (iii)To branch from a value for xj* where 1/(m+1)< xj* < 1/m for some m = 2,3,4,we use the following two alternative constraints for the two new subproblems: xj \le 1/(m+1)or xj \ge 1/m.$ for i = 1,2 and j = 1,2,3,4.For each i and j,calculate the contribution to the objective function associated with having y_ij = 1 because x_i = 1/(j+1),as shown in the following table. $(a)We define the binary variable yij as follows.Let for i = 1,2 and j = 1,2,3,4.For each i and j,calculate the contribution to the objective function associated with having yij = 1 because xi = 1/(j+1),as shown in the following table. These numbers in the second and third columns then become the coefficients of the respective yij in the objective function of the reformulated problem.The reformulated problem also requires some functional constraints on the yij.In particular,for each i = 1,2,exactly one of the yi1,yi2,yi3,and yi4 variables must be 1.Furthermore,because of the x1 + x2 ≤ 0.75 constraint,certain pairs of yij variables (y11 and y21,y11 and y22,y12 and y21)cannot both be 1.These observations lead to the following pure binary integer linear programming formulation.Maximize Z = (3/4)y11 + (5/9)y12 + (7/16)y12 + (9/25)y14 + (3/4)y11 + (2/3)y22 + (9/16)y23 + (12/25)y24,subject to y11 + y12 + y12 + y14 \le 1 y21 + y22 + y23 + y24 \le 1 y11 + y21 \le 1 y11 + y22 \le 1 y12 + y21 \le 1 and yij is binary,for i = 1,2 and j = 1,2,3,4. (b)(i)Use the following relaxation: Maximize Z = 2x1 - 3 + 3x2 - 3 ,subject to x1 + x2 \le 3/4 and 1/5 \le x1 \le 1/2 ,1/5 \le x2 \le 1/2. (ii)The fathoming tests are the same as those stated in Sec.12.6 of the textbook.A subproblem is fathomed if Test 1: The upper bound on Z for the subproblem is ≤ Z for the incumbent,or Test 2: Its relaxation has no feasible solutions,or Test 3: The optimal solution for its relaxation satisfies the original constraint that each variable is restricted to the values: . (iii)To branch from a value for xj* where 1/(m+1)< xj* < 1/m for some m = 2,3,4,we use the following two alternative constraints for the two new subproblems: xj \le 1/(m+1)or xj \ge 1/m.$ These numbers in the second and third columns then become the coefficients of the respective y_ij in the objective function of the reformulated problem.The reformulated problem also requires some functional constraints on the y_ij.In particular,for each i = 1,2,exactly one of the y_i1,y_i2,y_i3,and y_i4 variables must be 1.Furthermore,because of the x₁ + x₂ ≤ 0.75 constraint,certain pairs of y_ij variables (y₁₁ and y₂₁,y₁₁ and y₂₂,y₁₂ and y₂₁)cannot both be 1.These observations lead to the following pure binary integer linear programming formulation.Maximize Z = (3/4)y₁₁ + (5/9)y₁₂ + (7/16)y₁₂ + (9/25)y₁₄ + (3/4)y₁₁ + (2/3)y₂₂ + (9/16)y₂₃ + (12/25)y₂₄,subject to y₁₁ + y₁₂ + y₁₂ + y₁₄

\le

1 y₂₁ + y₂₂ + y₂₃ + y₂₄

\le

1 y₁₁ + y₂₁

\le

1 y₁₁ + y₂₂

\le

1 y₁₂ + y₂₁

\le

1 and y_ijis binary,for i = 1,2 and j = 1,2,3,4.
(b)(i)Use the following relaxation: Maximize Z = 2x₁ - 3 $(a)We define the binary variable yij as follows.Let for i = 1,2 and j = 1,2,3,4.For each i and j,calculate the contribution to the objective function associated with having yij = 1 because xi = 1/(j+1),as shown in the following table. These numbers in the second and third columns then become the coefficients of the respective yij in the objective function of the reformulated problem.The reformulated problem also requires some functional constraints on the yij.In particular,for each i = 1,2,exactly one of the yi1,yi2,yi3,and yi4 variables must be 1.Furthermore,because of the x1 + x2 ≤ 0.75 constraint,certain pairs of yij variables (y11 and y21,y11 and y22,y12 and y21)cannot both be 1.These observations lead to the following pure binary integer linear programming formulation.Maximize Z = (3/4)y11 + (5/9)y12 + (7/16)y12 + (9/25)y14 + (3/4)y11 + (2/3)y22 + (9/16)y23 + (12/25)y24,subject to y11 + y12 + y12 + y14 \le 1 y21 + y22 + y23 + y24 \le 1 y11 + y21 \le 1 y11 + y22 \le 1 y12 + y21 \le 1 and yij is binary,for i = 1,2 and j = 1,2,3,4. (b)(i)Use the following relaxation: Maximize Z = 2x1 - 3 + 3x2 - 3 ,subject to x1 + x2 \le 3/4 and 1/5 \le x1 \le 1/2 ,1/5 \le x2 \le 1/2. (ii)The fathoming tests are the same as those stated in Sec.12.6 of the textbook.A subproblem is fathomed if Test 1: The upper bound on Z for the subproblem is ≤ Z for the incumbent,or Test 2: Its relaxation has no feasible solutions,or Test 3: The optimal solution for its relaxation satisfies the original constraint that each variable is restricted to the values: . (iii)To branch from a value for xj* where 1/(m+1)< xj* < 1/m for some m = 2,3,4,we use the following two alternative constraints for the two new subproblems: xj \le 1/(m+1)or xj \ge 1/m.$ + 3x₂ - 3 $(a)We define the binary variable yij as follows.Let for i = 1,2 and j = 1,2,3,4.For each i and j,calculate the contribution to the objective function associated with having yij = 1 because xi = 1/(j+1),as shown in the following table. These numbers in the second and third columns then become the coefficients of the respective yij in the objective function of the reformulated problem.The reformulated problem also requires some functional constraints on the yij.In particular,for each i = 1,2,exactly one of the yi1,yi2,yi3,and yi4 variables must be 1.Furthermore,because of the x1 + x2 ≤ 0.75 constraint,certain pairs of yij variables (y11 and y21,y11 and y22,y12 and y21)cannot both be 1.These observations lead to the following pure binary integer linear programming formulation.Maximize Z = (3/4)y11 + (5/9)y12 + (7/16)y12 + (9/25)y14 + (3/4)y11 + (2/3)y22 + (9/16)y23 + (12/25)y24,subject to y11 + y12 + y12 + y14 \le 1 y21 + y22 + y23 + y24 \le 1 y11 + y21 \le 1 y11 + y22 \le 1 y12 + y21 \le 1 and yij is binary,for i = 1,2 and j = 1,2,3,4. (b)(i)Use the following relaxation: Maximize Z = 2x1 - 3 + 3x2 - 3 ,subject to x1 + x2 \le 3/4 and 1/5 \le x1 \le 1/2 ,1/5 \le x2 \le 1/2. (ii)The fathoming tests are the same as those stated in Sec.12.6 of the textbook.A subproblem is fathomed if Test 1: The upper bound on Z for the subproblem is ≤ Z for the incumbent,or Test 2: Its relaxation has no feasible solutions,or Test 3: The optimal solution for its relaxation satisfies the original constraint that each variable is restricted to the values: . (iii)To branch from a value for xj* where 1/(m+1)< xj* < 1/m for some m = 2,3,4,we use the following two alternative constraints for the two new subproblems: xj \le 1/(m+1)or xj \ge 1/m.$ ,subject to x₁ + x₂

\le

3/4 and 1/5

\le

x₁

\le

1/2 ,1/5

\le

x₂

\le

1/2.
(ii)The fathoming tests are the same as those stated in Sec.12.6 of the textbook.A subproblem is fathomed if Test 1: The upper bound on Z for the subproblem is ≤ Z for the incumbent,or Test 2: Its relaxation has no feasible solutions,or Test 3: The optimal solution for its relaxation satisfies the original constraint that each variable is restricted to the values: $(a)We define the binary variable yij as follows.Let for i = 1,2 and j = 1,2,3,4.For each i and j,calculate the contribution to the objective function associated with having yij = 1 because xi = 1/(j+1),as shown in the following table. These numbers in the second and third columns then become the coefficients of the respective yij in the objective function of the reformulated problem.The reformulated problem also requires some functional constraints on the yij.In particular,for each i = 1,2,exactly one of the yi1,yi2,yi3,and yi4 variables must be 1.Furthermore,because of the x1 + x2 ≤ 0.75 constraint,certain pairs of yij variables (y11 and y21,y11 and y22,y12 and y21)cannot both be 1.These observations lead to the following pure binary integer linear programming formulation.Maximize Z = (3/4)y11 + (5/9)y12 + (7/16)y12 + (9/25)y14 + (3/4)y11 + (2/3)y22 + (9/16)y23 + (12/25)y24,subject to y11 + y12 + y12 + y14 \le 1 y21 + y22 + y23 + y24 \le 1 y11 + y21 \le 1 y11 + y22 \le 1 y12 + y21 \le 1 and yij is binary,for i = 1,2 and j = 1,2,3,4. (b)(i)Use the following relaxation: Maximize Z = 2x1 - 3 + 3x2 - 3 ,subject to x1 + x2 \le 3/4 and 1/5 \le x1 \le 1/2 ,1/5 \le x2 \le 1/2. (ii)The fathoming tests are the same as those stated in Sec.12.6 of the textbook.A subproblem is fathomed if Test 1: The upper bound on Z for the subproblem is ≤ Z for the incumbent,or Test 2: Its relaxation has no feasible solutions,or Test 3: The optimal solution for its relaxation satisfies the original constraint that each variable is restricted to the values: . (iii)To branch from a value for xj* where 1/(m+1)< xj* < 1/m for some m = 2,3,4,we use the following two alternative constraints for the two new subproblems: xj \le 1/(m+1)or xj \ge 1/m.$ .
(iii)To branch from a value for x_j^* where 1/(m+1)< x_j^* < 1/m for some m = 2,3,4,we use the following two alternative constraints for the two new subproblems: x_j

\le

1/(m+1)or x_j

\ge

1/m.

Consider the Following Discrete Nonlinear Programming Problem