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Follow-up exercise: try to model your decision process next time you are trying to converge with your friends or relatives, where to go out, or what to watch together. What is the topology of your movie-space?…

]]>Follow-up exercise: try to model your decision process next time you are trying to converge with your friends or relatives, where to go out, or what to watch together. What is the topology of your movie-space?

]]>Class notes (pretty incomplete, to be used just as a study guide), here and here.

**Homework**, due by Midnight May 1st:

- Consider the 4-dimensional space spanned by the polynomials

$$

p(x)=ax^6+bx^4+cx^2+d.

$$Find, using the Gram-Schmidt orthogonalization procedure,

the orthonormal basis, if the scalar product is given by

$$

(p_1,p_2)_Q=\int_{0}^\infty e^{-x}p_1(x)p_2(x) dx.

$$ - Consider quadratic form in $\Real^{100}$ given by

\[

Q=\left(

\begin{array}{ccccc}

2&1&1&\cdots&1\\

1&2&1&\cdots&1\\

1&1&2&\cdots&1\\

\vdots&\vdots&\vdots&&\vdots\\

1&1&1&\cdots&2\\

\end{array}

\right)

\]

(i.e.

Class notes (pretty incomplete, to be used just as a study guide), here and here.

**Homework**, due by Midnight May 1st:

- Consider the 4-dimensional space spanned by the polynomials

$$

p(x)=ax^6+bx^4+cx^2+d.

$$Find, using the Gram-Schmidt orthogonalization procedure,

the orthonormal basis, if the scalar product is given by

$$

(p_1,p_2)_Q=\int_{0}^\infty e^{-x}p_1(x)p_2(x) dx.

$$ - Consider quadratic form in $\Real^{100}$ given by

\[

Q=\left(

\begin{array}{ccccc}

2&1&1&\cdots&1\\

1&2&1&\cdots&1\\

1&1&2&\cdots&1\\

\vdots&\vdots&\vdots&&\vdots\\

1&1&1&\cdots&2\\

\end{array}

\right)

\]

(i.e. \(2\) on the diagonal,\(1\) elsewhere).- Find the spectrum of \(Q\).
- Solve

\[

Qx=b,

\]

where \(b=(1,0,\ldots,0)\). - Run two iterations of the conjugate gradient method for this system.

- Consider \(F(z)=z^3-9z+8\).
- Find roots of \(F\).
- Find (numerically) the for which starting points on the real line, the Newton method will converge to either of the roots.

Newton method, Conjugate gradients Guler, chapter 13.

Class notes (pretty incomplete, to be used just as a study guide), here and

here.…

Newton method, Conjugate gradients Guler, chapter 13.

Class notes (pretty incomplete, to be used just as a study guide), here and

here.

Gradient descent; Newton method: Guler, chapter 13.

Class notes (pretty incomplete, to be used just as a study guide), here and

here.…

Gradient descent; Newton method: Guler, chapter 13.

Class notes (pretty incomplete, to be used just as a study guide), here and

here.

Conic duality: Guler, Chapters 11, 13 and Barvinok’s “A Course in Convexity“, Chapter 4.

Class notes (pretty incomplete, to be used just as a study guide), here.

**Homework**, due Apr. 16.

- Warehouses numbered \(0,1,\ldots,N\) are located along the road. Initially, the warehouse \(k\) holds \(N-k\) units of goods. Management wants to redistribute the goods so that warehouse \(k\) holds \(k\) units. Moving one unit from \(k\) to \(l\) costs \(|k-l|\). What is the optimal movement plan?

Conic duality: Guler, Chapters 11, 13 and Barvinok’s “A Course in Convexity“, Chapter 4.

Class notes (pretty incomplete, to be used just as a study guide), here.

**Homework**, due Apr. 16.

- Warehouses numbered \(0,1,\ldots,N\) are located along the road. Initially, the warehouse \(k\) holds \(N-k\) units of goods. Management wants to redistribute the goods so that warehouse \(k\) holds \(k\) units. Moving one unit from \(k\) to \(l\) costs \(|k-l|\). What is the optimal movement plan?

Write this as a LP; find its dual and solve both. - Find Lovasz’ theta for the cyclic graph of order \(5\) (both computational and theoretical solutions admissible).
- Find Legendre dual for

\(f(x_1,x_2)=\min_{(s_1,s_2)}((x_1-s_1)^2+(x_2-s_2)^2)\), where the minimum is taken over all combinations of \(s_1,s_2=\pm 1\). - Find Legendre dual for

\(f(x_1,x_2)=\max_{(s_1,s_2)}((x_1-s_1)^2+(x_2-s_2)^2)\), where the minimum is taken over all combinations of \(s_1,s_2=\pm 1\). - Gradient descent with constant step size (choose one judiciously) is run for \(f=x_1x_2\). Estimate the size of the vector after \(100\)\th iteration for initial position \((1,1)\); \((.9999,1)\).

**Solutions** here.

Convex programming and duality: Guler, Chapter 11.

Class notes (pretty incomplete, to be used just as a study guide), here and here.

**Exercises**:

Find Legendre duals for the following functions:

- \(f(x)=\min(1-x,2,1+x)\);
- \(f(x)=\min((x-1)^2,(x+1)^2)\);
- \(f(x,y)= 0 \mathrm{\ if\ } x^2+y^2\leq 2; +\infty \mathrm{\ otherwise}\).

…

]]>Convex programming and duality: Guler, Chapter 11.

Class notes (pretty incomplete, to be used just as a study guide), here and here.

**Exercises**:

Find Legendre duals for the following functions:

- \(f(x)=\min(1-x,2,1+x)\);
- \(f(x)=\min((x-1)^2,(x+1)^2)\);
- \(f(x,y)= 0 \mathrm{\ if\ } x^2+y^2\leq 2; +\infty \mathrm{\ otherwise}\).

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