Quiz1_sample - Quiz 1 Sample Questions IE406 Introduction...

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Quiz 1 Sample Questions IE406 – Introduction to Mathematical Programming Dr. Ralphs These questions are from previous years and should you give you some idea of what to expect on Quiz 1. 1. Consider the linear program pictured here, where the feasible region is the shaded area, the objective function vector is the vector pictured with its tail at the point labeled A, and the goal is to MINIMIZE. C D E B 2 3 4 5 6 1 A obj fcn The numbers indicate the indices of the variables corresponding to each (nonnega- tivity) constraint when the program is expressed in standard form, while the letters indicate the extremal elements of the feasible region. The following questions refer to solution of this linear program by the primal simplex method. (a) (10 points) What is the optimal solution to this linear program? Argue geomet- rically why the solution must be optimal. Is the solution unique? Why or why not? (b) (10 points) List all sequences of BFS’s that can occur when solving this linear program with the simplex method. For this part, you SHOULD NOT assume that the rule for selecting the variable to enter the basis is to select the one with the most negative reduced cost. In the case of a degenerate pivot, a single BFS may be listed more than once in the list. Indicate the sequences that involve the smallest and largest number of simplex iterations. Justify your answer. (c) (10 points) Of the above sequences, which ones can occur if the rule for selecting the variable to enter the basis is to select the variable with the most negative reduced cost (assuming the basic direction vectors all have the same norm)? Justify your answer. (d) (10 points) Which path occurs if the tiebreaking rule for selecting the leaving variable in each iteration is to select the variable with the smallest subscript? With the largest subscript? Justify your answer. 2. For the questions below, provide either a short argument supporting the validity of the statement or a specific counterexample. All questions refer to the primal simplex 1
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algorithm applied to a linear program in standard form with feasible region P and a constraint matrix of full rank. (a) (10 points) If the only allowable pivots are degenerate, then the current basis is optimal. (b) (10 points) If the current solution is degenerate, then the objective function value will remain unchanged after the next pivot. (c) (10 points) If there is a tie in the ratio test, then the next BFS will be degenerate. (d) (10 points) The existence of a redundant constraint implies the existence of a degenerate BFS. 3. (a) (10 points) Consider a linear program in standard form. An alternative to the method of finding an initial BFS discussed in class is to introduce a single artificial variable with index n + 1 such that A n +1 = b . In the modified LP, the solution ˆ x R n +1 such that ˆ x n +1 = 1 and ˆ x i = 0 for all i 1 ..n is then feasible. Describe a method of finding an initial BFS by introducing such an artificial variable in the case, assuming all of the original constraints are inequalities. Be specific.
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