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15.084

Nonlinear Optimization

4-0-8 units · Graduate

See description under subject 6.7220J.

Spring 2025

This course offers a unified analytical and computational approach to nonlinear optimization problems. Unconstrained optimization methods include gradient, conjugate direction, Newton, sub-gradient, and first-order methods. Constrained optimization methods include feasible directions, projection, interior point methods, and Lagrange multiplier methods. The curriculum covers convex analysis, Lagrangian relaxation, and nondifferentiable optimization, as well as applications in integer programming. It provides a comprehensive treatment of optimality conditions and Lagrange multipliers. The course also utilizes a geometric approach to duality theory. Finally, applications are drawn from control, communications, machine learning, and resource allocation problems.

Spring 2004 · Nonlinear Programming

Topics

  • Unconstrained Optimization Optimality Conditions
  • Convex Unconstrained Optimization Optimality Conditions
  • Newton’s Method
  • Quadratic Forms
  • Steepest Descent Method
  • Constrained Optimization Optimality Conditions I
  • Constrained Optimization Optimality Conditions II
  • Constrained Optimization Optimality Conditions III
  • Projection Methods for Equality Constrained Problems
  • Projection Methods/Penalty Methods
  • Penalty Methods
  • Barrier Methods, Conditional Gradient Method
  • Interior-Point Methods for Linear Optimization I
  • Interior-Point Methods for Linear Optimization II
  • Analysis of Convex Sets
  • Analysis of Convex Functions
  • Duality Theory I
  • Duality Theory II
  • Duality Theory III
  • Duality Theory IV
  • Generalized Programming and Subgradient Optimization
  • Semidefinite Optimization I
  • Semidefinite Optimization II
  • Semidefinite Optimization III
  • Extensions and Wrap-up

Readings

  • Bertsekas, Dimitri P. Nonlinear Programming . 2nd ed. Athena Scientific Press, 1999. ISBN: 1886529000.
    Required
  • Bazaraa, Mokhtar S., Hanif D. Sherali, and C. M. Shetty. Nonlinear Programming: Theory and Algorithms . New York: John Wiley & Sons, 1993. ISBN: 0471557935.
    Recommended
  • Linear Programming
    Listed reading
  • Luenberger, David G. Linear and Nonlinear Programming . 2nd ed. Reading, MA: Addison Wesley, 1984. ISBN: 0201157942.
    Listed reading
  • Chapter 2: Sections 2.1-2.5
    Listed reading
  • Chapter 3: Sections 3.1-3.5, 3.7
    Listed reading
  • Chapter 4: Sections 4.1-4.4
    Listed reading
  • Or: Bertsimas, Dimitris, and John Tsitsiklis. Introduction to Linear Optimization . Athena Scientific Press, 1997. ISBN: 1886529191.
    Listed reading
  • Chapter 1: Sections 1.1, 1.2, 1.4
    Listed reading
  • Chapter 2: Sections 2.1, 2.2
    Listed reading
  • Chapter 3: Sections 3.1, 3.2, 3.5
    Listed reading
  • Chapter 4: Sections 4.1, 4.2, 4.3
    Listed reading
  • Rudin, Walter. Principles of Mathematical Analysis . 3rd ed. McGraw-Hill, 1976. ISBN: 007054235X.
    Listed reading
  • Chapter 2: pp. 24-43 (the whole chapter)
    Listed reading
  • Chapter 3: pp. 47-61
    Listed reading
  • Chapter 4: pp. 83-95
    Listed reading
Spring 2003 · Nonlinear Programming

Topics

  • Introduction
  • Unconstrained Optimization - Optimality Conditions
  • Gradient Methods
  • Convergence Analysis of Gradient Methods
  • Rate of Convergence
  • Newton and Gauss - Newton Methods
  • Additional Methods
  • Optimization Over a Convex Set; Optimality Conditions
  • Feasible Direction Methods
  • Alternatives to Gradient Projection
  • Constrained Optimization; Lagrange Multipliers
  • Inequality Constraints
  • Introduction to Duality
  • Interior Point Methods
  • Penalty Methods
  • Augmented Lagrangian Methods
  • Duality Theory
  • Duality Theorems
  • Strong Duality
  • Dual Computational Methods
  • Additional Dual Methods

Readings

Prerequisites & course information

Same subject as 6.7220J Prereq: 18.06 and ( 18.100A , 18.100B , or 18.100Q ) G (Spring) 4-0-8 units

Sources