15.081J 数学规划导论

MIT Course 15 · Operations Research/Statistics · 课号 15.081J

TL;DR

• 研究生级数学规划理论基石，系统建立凸集、多面体几何与线性规划对偶理论。
• 严格证明强对偶性定理、互补松弛条件与 Farkas 引理，解构单纯形几何退化性。
• 内点法与多项式时间算法复杂度分析，奠定了现代凸优化的大规模计算理论基础。

课程简介 / Description

This course is an introduction to linear optimization and its extensions emphasizing the underlying mathematical structures, geometrical ideas, algorithms and solutions of practical problems. The topics covered include: formulations, the geometry of linear optimization, duality theory, the simplex method, sensitivity analysis, robust optimization, large scale optimization network flows, solving problems with an exponential number of constraints and the ellipsoid method, interior point methods, semidefinite optimization, solving real world problems problems with computer software, discrete optimization formulations and algorithms.

讲义 / Lecture Notes

• 1
  • Formulations
  • (PDF)
• 2
  • Geometry I
  • (PDF)
• 3
  • Geometry II
  • (PDF)
• 4
  • Geometry III
  • (PDF)
• 5
  • Simplex method I
  • (PDF)
• 6
  • Simplex method II
  • (PDF)
• 7
  • Simplex method III
  • (PDF)
• 8
  • Simplex method IV
  • (PDF)
• 9
  • Duality theory I
  • (PDF)
• 10
  • Duality theory II
  • (PDF)
• 11
  • Duality theory III
  • (PDF)
• 12
  • Sensitivity analysis
  • (PDF)
• 13
  • Robust optimization
  • (PDF)
• 14
  • Large scale optimization I
  • (PDF)
• 15
  • Large scale optimization II
  • (PDF)
• 16
  • Network flows I
  • (PDF)
• 17
  • Network flows II
  • Courtesy of Prof. Andreas Schulz. Used with permission.
  • (PDF)
• 18
  • The Ellipsoid method
  • (PDF)
• 19
  • Problems with exponentially many constraints
  • (PDF)
• 20
  • Interior point methods I
  • (PDF)
• 21
  • Interior point methods II
  • (PDF)
• 22
  • Interior point methods III
  • (PDF)
• 23
  • Semidefinite optimization
  • (PDF)
• 24
  • Discrete optimization I
  • (PDF)
• 25
  • Discrete optimization II
  • (PDF)

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