This course gives an overview of numerical methods for mathematical optimization problems and introduces some applications of optimization to solve problems arising from engineering. The numerical methods covered in this course include primal-dual interior-point method for linear programming problems, and various methods for nonlinear optimization problems and constrained optimization problems. The topic also includes combinatorial optimization and its numerical methods. As the applications to engineering, the course introduces mechanical measurements and optimization methods that imitate nature.
The rapid developments of computers have brought significant changes to mathematical methodology. To use this progress, new mathematical computing technology has increased in importance. One of such mathematical technology is numerical methods for mathematical optimization problems. This course introduces various mathematical methods. The knowledge of multiple methods will be an advantage to tackle practical problems. In addition, the introduction of applications in engineering will give a chance to understand the rapid developments of computers and optimization methods more deeply.
At the end of this course, students will be able to:
(1) Formulate various problems (including applications in engineering) using mathematical optimization models.
(2) Explain the framework of primal-dual interior-point method for linear programming problems.
(3) Apply numerical methods to solve nonlinear optimization problems.
(4) Explain the relation in conic optimization problems.
(5) Explain numerical methods for combinatorial optimization problems.
Mathematical optimization, primal-dual interior-point method, numerical optimization method, optimality condition, second-order cone programming, semidefinite programming, optimization in engineering, combinatorial optimization, meta-heuristics, branch-and-bound method, semidefnite programming relaxation
Intercultural skills | Communication skills | ✔ Specialist skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
This course gives an overview of various types of optimization methods.
At the end of each class, exercise problems are given. Students must solve the exercise problems to review the class.
Course schedule | Required learning | |
---|---|---|
Class 1 | Overview of mathematical optimization | Explain main properties of mathematical optimization models |
Class 2 | Primal-dual initerior-point method for linear programming | Explain the framework of the primal-dual interior-point method and the estimation on computation complexity |
Class 3 | Numerical methods for nonlinear optimization problems | Explain the frameworks of the simplex method and line search. |
Class 4 | Steepest descent method, Newton method, quasi Newton method | Compare the merits and the demerits of the steepest decent method, Newton method. |
Class 5 | Numerical methods for constrained nonlinear optimization problems | Explain relations between optimal solutions and optimality conditions. |
Class 6 | Augmented Lagrangian method, trust-region method, primal-dual interior-point method | Explain the frameworks of the augmented Lagrangian method, the trust-region method, the primal-dual interior-point method |
Class 7 | Conic optimization problems | Show the relation of conic optimization problems. |
Class 8 | Second-order cone programming, semidefinite programming | Formulate optimization problems into semidefinite programming. |
Class 9 | Recent topics | Review the numerical methods covered for better understanding. |
Class 10 | Optimization in mechanical measurement | Explain what kind of problems in mechanical measurements are solved by optimization. |
Class 11 | Optimization methods that imitate nature | Find and explain optimization problems from daily life |
Class 12 | Efficiency improvements on optimization methods | Understand various improvements on optimization methods. |
Class 13 | Combinatorial optimization problems, meta-heuristics | Compare meta-heuristic methods. |
Class 14 | Brand-and-bound method | Explain the framework of the branch-and-bound method for integer programming. |
Class 15 | Semidefinite programming relaxation | Formulate semidefinite programming problems based on semidefinite programming relaxation |
None required. Parts of the course materials are based on the reference books below.
Reference books are follows.
D. P. Bertsekas, "Nonlinear Programming", Athena Scientific, 2003.
V. Chvatal, "Linear programming", Freedman, 1983
R. Horst, P. M. Pardalos, N. V. Thoai, "Introduction to Global optimization", Klewer Academic, 2000
D. Z. Zu, P. M. Pardalos, "Handbook of Combinatorial Optimization", Klewer Academic, 1998
Other reference books are listed in the course materials.
Students will be assessed on their understanding on mathematical formulation based on mathematical optimization problems and algorithmic framework of optimization methods.
Students' course scores are based on final exams (80%) and reports (20%).
Students require the following knowledge and skills.
* The simplex method for linear programming problems
* Linear algebra (in particular, positive semidefinite matrices)