Single-Course English 5 ECTS

Constrained Optimization

Overall Course Objectives

When modelling a technical or economical problem it often happens that free parameters are determined by the solution to an optimization problem subject to some constraints on the solution. The simplest example is that the parameters must be positive or lie in certain intervals due to physical constraints in the underlying physical problem.
In this course the participant learns about efficient algorithms for constrained optimization. The student will be able both to develop algorithms and to use existing software for numerical solution of optimization problems with constraints. The course concerns numerical algorithms for linear programming (LP), convex quadratic programming (QP), convex optimization, and non-linear programming (NLP).

Learning Objectives

  • derive and explain the KKT optimality conditions for constrained optimization
  • use the KKT conditions to construct active set and interior point algorithms
  • derive, implement and use interior-point algorithms for LP, QP and NLP problems
  • derive, implement and use interior-point algorithms for convex optimization
  • derive, implement and use active set and interior point algorithms for convex quadratic programming (QP) and linear programming (LP)
  • explain the principles of the SQP algorithm for nonlinear, constrained optimization problems
  • combine the LP and QP algorithms into an SQP algorithm
  • develop, test and compare alternative optimization algorithms in Matlab for the solution of a given problem
  • use existing Matlab software libraries for constrained optimization
  • use optimization algorithms for the solution of problems from engineering and finance
  • apply convex optimization
  • apply optimization algorithms to numerically solve dynamic optimization problems and optimal control problems

Course Content

First and second order optimalitty conditions (KKT conditions. Active set and interior point algorithms for linear programming (LP) and convex quadratic programming (QP). Methods for nonlinear programming (NLP): Sequential quadratic programming (SQP) algorithms and augmented Lagrange algorithms. Development of simple numerical algorithms and use of existing software libraries for constrained optimization (LP, QP, NLP). Apply convex optimization (SOCP, SDP). Optimization of dynamic systems and optimal control. Application examples from Engineering and Finance.

Recommended prerequisites

02610/02002/02631/02633/02601, An introductory optimization course, experience with Matlab programming, and an introduction to numerical algorithms,

Teaching Method

Lectures and project work.



The course may be followed by 02619 “Model Predictive Control”, and a master thesis project.

See course in the course database.





13 weeks




DTU Lyngby Campus

Course code 02612
Course type Candidate
Semester start Week 5
Semester end Week 19
Days Tues 13-17

7.500,00 DKK