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ECE Ph.D. Qualifying Exam

Automatic Control (AC)

Question 3: Optimization

August 2017



Student answers and discussions for Part 1,2,3,4,5

1.(20 pts) Considern the following linear program,

minimize $ 2x_{1} + x_{2} $,

subject to $ x_{1} + 3x_{2} \geq 6 $

$ 2x_{1} + x_{2} \geq 4 $

$ x_{1} + x_{2} \leq 3 $

$ x_{1} \geq 0 $, $ x_{2} \geq 0 $.

Convert the above linear program into standard form and find an initial basix feasible solution for the program in standard form.


2.(20 pts)

  • (15 pts) FInd the largest range of the step-size, $ \alpha $, for which the fixed step gradient descent algorithm is guaranteed to convege to the minimizer of the quadratic function
$ f = \frac{1}{2} x^{T}Qx - b^{T}x $

starting from an arbitary initial condition $ x^{(0)} \in \mathbb{R}^{n} $


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