lecture 10 - Math 482 (Lecture 9): Duality III: The dual...

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Math 482 (Lecture 9): Duality III: The dual simplex method This lecture concerns Sections 3.5, 3.6 and 3.7 of the textbook. What we have shown before Test 1 is the following: 1. Given any LP we can put it in standard form 2. Using phase 1 of the 2-phase simplex method we can find a basic feasible solution (column 0 is nonnegative) 3. Row 0 has negative entries, we attempt to drive towards nonnegativity by selecting columns that are negative and pivoting using the (row) ratio test. 4. When we terminate at a finite optimum both row 0 and column 0 are nonnegative. Complaint: Sometimes 2-phase is tedious. Response: For certain LP's it is better to use the dual simplex method: **** This is often the case when the cost vector c is nonnegative. Warning about possible confusion because of the name "dual simplex": The dual simplex method secretly uses duality theory to solve an LP, and is often helpful to solve LP's of the form min y'b, s.t., y'A≥ c and y≥ 0 which is the usual form of our duals. HOWEVER, it actually solves the LP it is given, not the dual of that LP. Example handed out in class. Summary of dual simplex method: The rules are "mirror images" of the rules for usual (primal) simplex: A. When to use dual simplex: The standard form of the problem has nonnegative row 0 The terminology for this is "primal optimal" or "dual feasible" (more later). However, column 0 is not nonnegative (primal infeasible)
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lecture 10 - Math 482 (Lecture 9): Duality III: The dual...

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