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week 10

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)

Simplex method

Tableau, pivot step, pivot rules. Exceptions. Phase one (finding a feasible point).


Sample problem

  1. For which \(h\) the following system is feasible?
    \[
    \begin{array}{ccc}
    &y&\leq 0\\
    x&&\geq 0\\
    x&-y&\leq
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mock midterm 2

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\)


Sample problems

  1. Find Fenchel dual (conjugate, or \(\check{f}(p):=\sup_x px-f(x)\)) for
    1. \(f(x)=|x-1|+x/2\);

      Answer:
      \[
      \check{f}(p)=p-1/2 \mathrm{\ if}\ -1/2\leq p\leq 3/2; +\infty \mathrm{\ otherwise}.
      \]
    2. f\((x)=x^2/2+1/2 \ \mathrm{for} \ |x|\leq 1; |x| \ \mathrm{otherwise}\).

      Answer:
      \[
      \check{f}(p)=p^2/2-1/2
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week 9

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)

Linear Proramming

Barvinok, 4.

Convex closed cones, their polars.

General LP problems. Dual problems for general LP.

Polyhedral cones, strong duality.


Sample problems

  1. Let \(K=\Real_+^3\) be the cone of vectors with
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week 8

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)

Convex optimization

Lange, ch.5.

Convex programs. Convexity of the set of optima.

Constraint qualification: Slater condition.

Duality

Boyd-Vandenberghe, 5.1-3,5.6.

Lagrange dual functions. Dual optimization problems. Fenchel duals (conjugate) functions.

Weak duality …

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week 7

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)

Convexity

Convex sets. Convex hulls. Supporting hyperplanes.

Convex functions; properties. Log-convex functions.


Sample problems

  1. Is the set of continuous functions on \(\Real^2\) positive at all points with rational coordinates numbers convex?
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weeks 5 and 6

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)

Quadratic forms

Positive definite quadratic forms. Signatures, and relations between the indices of inertia of the (that is the number \(n_+\) of positive, \(n_-\) negative and \(n_0\) ,zero coefficients in the …

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programming exercise 1.

\(
\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)

Use any language or library. The assignments are due by midnight of 3.14 (Monday).

3 credits students

Consider the piecewise linear function \(f\) on \([-4,4]\)
with nodes at the integer …

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mock midterm 1

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)

  1. Consider the following subsets of \(\Real^2\):
    \[
    A=\{(x,y):xy>0\}; B=\{(x,y):x=0\}; C=\{(x,y):y=0\}; D=\{(0,0)\}.
    \]
    Which of the following sets is open? closed?
    \[
    A; B; A\cup C; A\cup B\cup C; A\cup D
    \]
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week 4

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)

Differentials

Differentials. Slope functions.

Inverse function and implicit function theorems.

Matrix differentials


Sample problems:

  • Find the differential of
    \[
    (x,y)\mapsto (x^2-y^2,2xy)
    \]
  • The function \(z(x,y)\) is implicitly given by
    \[
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week 3

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)

Background in Linear Algebra and Analysis

Topology: closed and open sets. Convergence. Compact sets. Criterion of compactness in \(\Real^n\).

Continuity.

 


Sample problems:

  • Find an open and set in \(\Real\)
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