Algorithmic Methods Fall Semester, 2010/11 Exercise 1: November 8, 2010 Summary: Algorithmic Methods Fall Semester, 2010/11 Exercise 1: November 8, 2010 Lecturer: Prof. Yossi Azar Write short but full and accurate answers. Each question should start on a new separate page and each of its parts should not exceed a page. 1. (a) Show that there can exist a degenerate vertex whose corresponding basis is unique. (b) Prove or disprove: every vertex of an LP is non-degenerate implies that the optimal solution is unique. 2. Complete the proof of the strong part of the duality theorem for the general form. Note that in class we assumed the primal is LPS (standard form LP). 3. (a) Prove that if an LPS (standard form LP) has a non-degenerate vertex which is an optimal solution then the dual problem has a unique optimal solution. (b) Does the above remain true if the LPS has both a non-degenerate vertex and a degenerate vertex which are optimal ? (c) Does the above remain true if the LPS has an optimal solution (not necessarily a vertex) with m variables of non-zero values (the LPS has m equations ) ? 4. Prove that a variable which has just left the basis in the simplex algorithm cannot reenter on the very next pivot step. 5. We are given a set of n points in R2, (x1, y1), . . . , (xn, yn). Our goal to find a function f(x) = ax+b such that Collections: Computer Technologies and Information Sciences