Consider the simplex method applied to a standard form problem and assume that the rows of the matrix A are linearly independent. For each of the statements that follow, give either a proof or a counterexample. (a) (2 points) An iteration of the simplex method may move the feasible solution by a positive distance while leaving the cost unchanged. (b) (2 points) A variable that has just left the basis cannot reenter in the very next iteration. (c) (2 points) A variable that has just entered the basis cannot leave in the very next iteration. (d) (2 points) If there is a nondegenerate optimal basis, then there exists a unique optimal basis. (e) (2 points) If x is an optimal solution found by the simplex method, no more than m of its compo- nents can be positive, where m is the number of equality constraints.
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An iteration of the simplex method may move the feasible solution by positive distance while leaving the cost unchanged. Counterexample: Consider the following linear programming problem in standard form: Maximize $z = x_1 + x_2$ Subject to: $x_1 + x_2 \leq Show moreā¦
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Consider the following linear program (P) with two variables: maximize 2x1 + x2 such that Ax ā¤ b x1, x2 ā„ 0. (a) Give an example of a matrix A and a vector b so that (P) has a unique optimal solution, and prove that (P) has a unique optimal solution. (b) Give an example of a matrix A and a vector b so that (P) has infinitely many optimal solutions, and prove that (P) has infinitely many optimal solutions. (c) Is it possible to choose A and b so that (P) has precisely two optimal solutions? Prove or disprove. You may use that the feasible region of any linear program is a convex set.
The following question examines the case of a linear program in which two basic feasible solutions B1 and B2 correspond to the same extreme point. Give an example of a linear program in standard form in which two different non-optimal basic feasible solutions B1 and B2 correspond to the same extreme point w. Let q be the number of variables in this linear program, and let A1 and A2 be the set of columns corresponding to the non-zero variables in B1 and B2. Verify that the number of columns in A1 (A2) is less than q. Show that the columns in A1 (A2) are linearly independent. Add columns to A1 to form a basis corresponding to the extreme point w.
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