Consider the following five constraints x1 + 2x2 ≤ 3, x1 − x2 ≥ 0, 2x1 + x2 ≤ 3, x1 + 5x2 ≤ 6, x1 − 2x2 ≥ −1. (a) Sketch the feasible region and find the degenerate vertex x0. (b) How many possible working sets are there at x0? (c) Suppose that we wish to minimize x1 + x2 subject to these constraints, starting at x0 and using the simplex method. Find a working set A0 for which the Lagrange multiplier vector λ (the solution of AT 0 λ = c) contains at least one negative component λs, but the simplex search direction satisfying A0p = es is not a feasible descent direction. Draw a picture showing p emanating from x0. What are the blocking constraints? (d) Under the same conditions as in part (c), find a working set A¯ 0 for which the Lagrange multiplier vector contains at least one negative component, but the associated search direction ¯p is a feasible descent direction. Draw a picture showing ¯p emanating from x0. (e) Can you find a feasible descent direction at x0 if we wish instead to minimize −x1 − x2? Explain your answer.

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Consider the following five constraints x1 + 2x2 ≤ 3, x1 − x2 ≥ 0, 2x1 + x2 ≤ 3, x1 + 5x2 ≤ 6, x1 − 2x2 ≥ −1. (a) Sketch the feasible region and find the degenerate vertex x0. (b) How many possible working sets are there at x0? (c) Suppose that we wish to minimize x1 + x2 subject to these constraints, starting at x0 and using the simplex method. Find a working set A0 for which the Lagrange multiplier vector λ (the solution of AT 0 λ = c) contains at least one negative component λs, but the simplex search direction satisfying A0p = es is not a feasible descent direction. Draw a picture showing p emanating from x0. What are the blocking constraints? (d) Under the same conditions as in part (c), find a working set A¯ 0 for which the Lagrange multiplier vector contains at least one negative component, but the associated search direction ¯p is a feasible descent direction. Draw a picture showing ¯p emanating from x0. (e) Can you find a feasible descent direction at x0 if we wish instead to minimize −x1 − x2? Explain your answer.

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