Automatic Control (AC)

Question 3: Optimization

August 2016

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

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

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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 converge to the minimizer of the quadratic function

starting from an arbitary initial condition $ x^{(0)} \in \mathbb{R}^{n} $, where $ x \in \mathbb{R}^{n} $, and $ Q = Q^{T} > 0 $.

- (5 pts) Find the largest range of the step size, $ \alpha $, for which the fixed step gradient descent algorithm is guaranteed to converge to the minimizer of the quadratic function

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

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3. (20 pts) Is the function

locally convex, concave, or neither in the neighborhood of the point $ [2 -1]^{T} $? Justify your answer by giving all the details of your argument.

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4. (20 pts) Solve the following optimization problem:

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5. (20 pts) Solve the following optimization problem:

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