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?
    \[
    \begin{array}{ccc}
    &y&\leq 0\\
    x&&\geq 0\\
    x&-y&\leq 4-h\\
    x&+y&\leq 2-h\\
    x&+y&\leq h+2\\
    \end{array}
    \]
  2. Using simplex method, solve
    \[
    \begin{array}{ccl}
    2x&-3y&\to\min, \mathrm{subj\ \ to}\\
    -3x&+2y&\leq 6\\
    -x&+2y&\leq 10\\
    x&+y&\leq 8\\
    3x&+y&\leq 18\\
    x&-y&\leq 2\\
    x&&\geq 0\\
    &y&\geq 0\\
    \end{array}
    \]

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