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Unformatted text preview: MS&E 211 Linear & Nonlinear Optimization Fall 2011 Prof Yinyu Ye Homework Assignment 6 Solutions 1. Log-Barrier Method: [25points] (a) [10points] Solve using the log-barrier method: gGGG: + 3 + : + = 1 , 0 Equalities cant be made into a barrier, and so we substitute: = 1 Thus the problem reduced to one variable: gGGG: 1 + : 0 The barrier function becomes: 1 + lo () Differentiating yields: 1 2 . Setting equal to zero and solving gives: 2 = 0 . Or = , which approaches 0 or Zero is the minimum, so using our substitution equation to get the other value, (0,1) is the optimal solution. (b) [10points] Solve using the log-barrier method: gGGG: 1 2 + 1 2 2 : + 4 The barrier function becomes: + 2 lo ( + 4) Differentiating and setting to zero yields: 1 = 0 2 = 0 Thus, in some optimality: = + 1 Substituting: 1 = 0 Simplifying: 2 5 + 3 = 0 Solving: g G = G Which converges to 1 and Using our substitution equation to get g , our two candidates are (1,2) and ( , ) Clearly one is the global minimum, and the other our answer. ( , ) (c) [5points] Explain why we can or cannot guarantee convergence to the optimal solution in...
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This note was uploaded on 01/16/2012 for the course MS&E 211 at Stanford.