15.084J 非线性规划

MIT Course 15 · Operations Research/Statistics · 课号 15.084J

TL;DR

• 连续优化中的非线性理论，涵盖凸函数、凸集几何与广义一阶/二阶最优化极值条件。
• KKT（Karush-Kuhn-Tucker）条件系统刻画含不等式约束优化的一阶必要与充分条件。
• 牛顿法、拟牛顿法（BFGS）、共轭梯度法与增广拉格朗日乘子法支撑算法求解。

课程简介 / Description

This course introduces students to the fundamentals of nonlinear optimization theory and methods. Topics include unconstrained and constrained optimization, linear and quadratic programming, Lagrange and conic duality theory, interior-point algorithms and theory, Lagrangian relaxation, generalized programming, and semi-definite programming. Algorithmic methods used in the class include steepest descent, Newton's method, conditional gradient and subgradient optimization, interior-point methods and penalty and barrier methods.

讲义 / Lecture Notes

• 1
  • Unconstrained Optimization Optimality Conditions (PDF)
• 2
  • Convex Unconstrained Optimization Optimality Conditions
• 3
  • Newton's Method (PDF)
• 4
  • Quadratic Forms (PDF)
• 5
  • Steepest Descent Method (PDF - 2.2 MB)
• 6
  • Constrained Optimization Optimality Conditions I (PDF)
• 7
  • Constrained Optimization Optimality Conditions II
• 8
  • Constrained Optimization Optimality Conditions III
• 9
  • Projection Methods for Equality Constrained Problems (PDF)
• 10
  • Projection Methods/Penalty Methods (PDF)
• 11
  • Penalty Methods
• 12
  • Barrier Methods, Conditional Gradient Method (PDF)
• 13
  • Midterm Exam
• 14
  • Interior-Point Methods for Linear Optimization I (PDF)
• 15
  • Interior-Point Methods for Linear Optimization II
• 16
  • Analysis of Convex Sets (PDF)
• 17
  • Analysis of Convex Functions
• 18
  • Duality Theory I (PDF)
• 19
  • Duality Theory II
• 20
  • Duality Theory III
• 21
  • Duality Theory IV (PDF)
• 22
  • Generalized Programming and Subgradient Optimization (PDF)
• 23
  • Semidefinite Optimization I (PDF)
• 24
  • Semidefinite Optimization II
• 25
  • Semidefinite Optimization III
• 26
  • Extensions and Wrap-up

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