Worksheet 2-3

Worksheet 2-3#

Download: CMSE382-WS2_3.pdf

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Worksheet 2-3: Q1#

We are going to investigate the function \(f(x,y)=2x^2-8xy+y^2\).

  1. Write the function in the quadratic form \(f(x,y) = \mathbf{x}^\top A \mathbf{x} + 2\mathbf{b}^\top\mathbf{x} + c\) for some symmetric matrix \(A\), vector \(\mathbf{b}\), and scalar \(c\).

  2. What is the gradient \(\nabla f(x,y)\) and Hessian \(\nabla^2 f(x,y)\) of the function? (Hint: use your matrix \(A\) from above.)

  3. Using this, is \(f\) coercive? Why or why not?

  4. Is \(f\) convex? Why or why not?

  5. Find and classify the stationary points of \(f(x,y)=2x^2-8xy+y^2\). If there are local optima, are they also global?

Worksheet 2-3: Q2#

Now consider the function \(g(x,y)=2x^2 - 2xy + y^2 + 6x + 2y\).

  1. Write the function in the quadratic form \(g(x,y) = \mathbf{x}^\top A \mathbf{x} + 2\mathbf{b}^\top\mathbf{x} + c\).

  2. What is the gradient \(\nabla g(x,y)\) and Hessian \(\nabla^2 g(x,y)\)? (Hint: use your matrix \(A\).)

  3. Is \(g\) coercive? Why or why not?

  4. Is \(g\) convex? Why or why not?

  5. Find and classify the stationary points of \(g(x,y)=2x^2 - 2xy + y^2 + 6x + 2y\). Use the properties of quadratic functions to determine if any local optima are also global optima.

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