\contentsline {part}{I\hspace {1em}Fundamentals}{1}{}%
\contentsline {chapter}{\numberline {1}Optimization problems}{3}{}%
\contentsline {chapter}{\numberline {2}Constraints}{7}{}%
\contentsline {part}{II\hspace {1em}Linear Programming}{9}{}%
\contentsline {chapter}{\numberline {3}Linear programs}{11}{}%
\contentsline {section}{\numberline {3.1}Graphical method}{12}{}%
\contentsline {section}{\numberline {3.2}Canonical form}{12}{}%
\contentsline {section}{\numberline {3.3}Application of convex analysis}{13}{}%
\contentsline {chapter}{\numberline {4}Simplex method}{15}{}%
\contentsline {section}{\numberline {4.1}Calculating arbitrary BFS}{16}{}%
\contentsline {section}{\numberline {4.2}Determining 'optimal' variable}{16}{}%
\contentsline {section}{\numberline {4.3}Ratio test}{17}{}%
\contentsline {section}{\numberline {4.4}Simplex method}{18}{}%
\contentsline {chapter}{\numberline {5}Simplex method revisions}{21}{}%
\contentsline {section}{\numberline {5.1}Artificial variables}{21}{}%
\contentsline {section}{\numberline {5.2}Big M simplex method}{21}{}%
\contentsline {section}{\numberline {5.3}Two phase simplex method}{21}{}%
\contentsline {section}{\numberline {5.4}Revised simplex method}{22}{}%
\contentsline {chapter}{\numberline {6}Duality theory}{23}{}%
\contentsline {section}{\numberline {6.1}Asymmetric dual problems}{23}{}%
\contentsline {section}{\numberline {6.2}Theorems of duality}{24}{}%
\contentsline {section}{\numberline {6.3}Dual simplex method}{25}{}%
\contentsline {chapter}{\numberline {7}Sensitivity analysis on LPs}{27}{}%
\contentsline {section}{\numberline {7.1}Cost vector entries $\mathbf {c}$}{28}{}%
\contentsline {subsection}{\numberline {7.1.1}Nonbasis}{28}{}%
\contentsline {subsection}{\numberline {7.1.2}Basis}{28}{}%
\contentsline {section}{\numberline {7.2}Constraint vector entries $\mathbf {b}$}{28}{}%
\contentsline {section}{\numberline {7.3}Constraint matrix entries $\mathbf {A}$}{29}{}%
\contentsline {subsection}{\numberline {7.3.1}Nonbasis}{29}{}%
\contentsline {subsection}{\numberline {7.3.2}Basis}{29}{}%
\contentsline {section}{\numberline {7.4}Additional constraint}{29}{}%
\contentsline {section}{\numberline {7.5}Additional variable}{29}{}%
\contentsline {part}{III\hspace {1em}Nonlinear programming}{31}{}%
\contentsline {chapter}{\numberline {8}Unconstrained}{33}{}%
\contentsline {section}{\numberline {8.1}Theoretical basis}{34}{}%
\contentsline {section}{\numberline {8.2}Derivative tests}{34}{}%
\contentsline {section}{\numberline {8.3}Gradient methods}{35}{}%
\contentsline {subsection}{\numberline {8.3.1}Steepest descent method}{35}{}%
\contentsline {subsection}{\numberline {8.3.2}Newton's method}{35}{}%
\contentsline {chapter}{\numberline {9}Constrained}{37}{}%
\contentsline {section}{\numberline {9.1}Reduced function}{37}{}%
\contentsline {subsection}{\numberline {9.1.1}Single linear condition}{37}{}%
\contentsline {subsection}{\numberline {9.1.2}Generalization to multiple linear conditions}{37}{}%
\contentsline {section}{\numberline {9.2}Lagrangian relaxation method}{38}{}%
\contentsline {part}{IV\hspace {1em}Integer programming}{39}{}%
\contentsline {chapter}{\numberline {10}Integer programs}{41}{}%
\contentsline {chapter}{\numberline {11}Branch-and-bound method}{43}{}%
\contentsline {section}{\numberline {11.1}Main idea}{43}{}%
\contentsline {section}{\numberline {11.2}Algorithm}{43}{}%
\contentsline {chapter}{\numberline {12}Hungarian method}{45}{}%
\contentsline {chapter}{\numberline {13}Cutting plane method}{47}{}%