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TTK4135 Optimization and control Spring semester 2005

TTK4135 Optimization and control Spring semester 2005. Scope - this you shall learn Optimization - important concepts and theory Formulating an engineering problem into an optimization problem Solving an optimization problem - algorithms, coding and testing. Course information

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TTK4135 Optimization and control Spring semester 2005

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  1. TTK4135 Optimization and controlSpring semester 2005 Scope - this you shall learn Optimization - important concepts and theory Formulating an engineering problem into an optimization problem Solving an optimization problem - algorithms, coding and testing Course information Lectures are given by professor Bjarne A. Foss The course assistant is Mr. K. Rambabu. TTK4135 Optimization and control B.Foss Spring semester 2005

  2. Course information All course information is provided on the web-pages for the course: www.itk.ntnu.no/fag/TTK4135. There will be no hand-out of material. Every student must access the course web-pages at least every week to keep updated course information (eg. changes in lecture times, information on mid-term exam) All students should subscribe to the email-list: 4135-optreg The deadlines for all assignments (“øvinger” and the helicopter lab. report) are absolute. There will be 1-2 “øvingstimer” with assistants present ahead of the deadline for every assignment. A minimum number of “øvinger” and the helicopter lab.report must be approved to enter the final examination. I will not cover the complete curriculum in my lectures; rather focus on the most important and difficult parts. TTK4135 Optimization and control B.Foss Spring semester 2005

  3. Grading • The final exam counts 70% on the final grade • The mid-term exam is graded. It counts 15% on the final grade. • Please note that only this semester’s mid-term exam counts. A mid-term grade from last year will not be acknowledged. • The project report (based on the helicopter laboratory) is graded. It counts 15% on the final grade. • Please note that only this semester’s report counts. A report grade from an earlier year will not be acknowledged. • To ensure participation from all students 4 groups will be selected for an oral presentation of their laboratory work. This presentation will influence the grade on the report. Finally I welcome constructive criticism on all aspects of the course, including my lectures. TTK4135 Optimization and control B.Foss Spring semester 2005

  4. Preliminary lecture plan The content of each lecture is specified in the following slides. All lectures are given in lecture halls EL 3 and EL 6. The mid-term examination is on 2004-03-11. The final examination is on 2004-05-23. TTK4135 Optimization and control B.Foss Spring semester 2005

  5. Content of Lecture #1 - 2004-01-10 • Optimization problems appear everywhere • Stock portfolio management • Resource allocation (airline companies, transport companies, oil well allocation problem) • Optimal adjustment of a PID-controller • Formulating an optimization problem: From an engineering problem to a mathematical description. • Case: a realistic production planning problem • Defining an optimization problem • Definition of important terms • Convexity and non-convexity • Global vs. local solution • Constrained vs. unconstrained problems • Feasible region • Reference: Chapter 1 in Nocedal and Wright (N&W) TTK4135 Optimization and control B.Foss Spring semester 2005

  6. Content of Lecture #2 - 2004-01-14 Karush Kuhn-Tucker (KKT) conditions Sensitivities and Lagrange-multipliers Reference: Chapter 12.1, 12.2 in N&W TTK4135 Optimization and control B.Foss Spring semester 2005

  7. Content of Lecture #3 - 2004-01-17 • Linear algebra (App. A.2 in N&W) • Norms of vectors and matrices • Positive definit and indefinite matrices • Condition number, well-conditioned and ill-conditioned linear equations • Subspaces; null space and range space of a matrix • Eigenvalue and singular-value decomposition • Matrix factorization: Cholesky factorization, LU factorization • Sequences (App.A.1, Ch.2.2 “Rates of …” in N&W) • Convergence to some points; convergence rate; order notation • Sets (App.A.1 in N&W) • Open, closed, bounded sets • Functions (App.A.1 in N&W) • Continuity, Lipschitz continuity • Directional derivatives TTK4135 Optimization and control B.Foss Spring semester 2005

  8. Content of Lecture #4 + #5 - 2004-01-21/28 • Linear programming - LP • Mathematical formulation • Condition for optimality - the Karush-Kuhn-Tucker (KKT) conditions • Basic solutions - basis for the Simplex method • The Simplex method • Understanding the solution - Lagrange variables • The dual problem • Obtaining an initial feasible solution • Efficiency of algorithms • LP example - production planning • Reference: Ch.12.2,13-13.5 in textbook TTK4135 Optimization and control B.Foss Spring semester 2005

  9. Content of Lecture #6 - 2004-01-31 • Quadratic programming - QP • Mathematical formulation • Convex vs. non-convex problems • Condition for optimality - KKT conditions • Special case: No inequality conditions • Reduced space methods • The active-set method for convex problems • Understanding the solution - Lagrange variables • The dual problem • Obtaining an initial feasible solution • Efficiency of algorithms • QP example - production planning (varying sales price) • Reference: Ch.12.2,16.1-16.4,(16.5),16.8 in textbook TTK4135 Optimization and control B.Foss Spring semester 2005

  10. Content of Lecture #7 - 2004-02-04 • Quadratic programming - QP • The active-set method for convex problems • The active-set method for non-convex problems • QP example - production planning (varying sales price) • Reference: 16.4,16.5,16.8 in textbook TTK4135 Optimization and control B.Foss Spring semester 2005

  11. Content of Lecture #8 - 2004-02-07 • Quadratic programming - QP • The active-set method for non-convex problems • Reference: 16.5,16.8 in textbook TTK4135 Optimization and control B.Foss Spring semester 2005

  12. Content of Lecture #9 - 2004-02-11 • Repetition of LP, QP • --- • Optimality conditions • Necessary and sufficient conditions for optimality • Iterative solution methods • Starting point • Search direction • Step length • Termination criteria • Convergence • Reference: 2.1, 2.2 in textbook TTK4135 Optimization and control B.Foss Spring semester 2005

  13. Content of Lecture #10 - 2004-02-14 • Line search methods • Choice of • Wolfe-conditions • Back-tracking • Curve-fit and interpolation • Convergence of line-search methods - Theorem 3.2 • Convergence rate • Reference: 3.1-3.4 in textbook TTK4135 Optimization and control B.Foss Spring semester 2005

  14. Content of Lecture #11 - 2004-02-18 • Practical Newton-methods • Approximate Newton-step • Line search Newton • Modified Hessian • Reference: 6 - 6.3 in textbook • Computing gradients • Reference: 7 - 7.1 in textbook TTK4135 Optimization and control B.Foss Spring semester 2005

  15. Content of Lecture #12 - 2004-02-21 • Quasi Newton methods • DFP and BFGS methods • Rosenbrock example for illustration • Reference: 8 - 8.1 in textbook • Information on the mid.term examination TTK4135 Optimization and control B.Foss Spring semester 2005

  16. Content of Lecture #13 - 2004-03-07 Mid-term examination TTK4135 Optimization and control B.Foss Spring semester 2005

  17. Content of Lecture #14 - 2004-04-01 • Mid-term examination - once again • Model Predictive Control (MPC) • The MPC principle • Formulation of linear MPC • Formulating the optimisation problem which is a QP-problem • Reference: Ch.1 and 2 – Note on MPC by M.Hovd TTK4135 Optimization and control B.Foss Spring semester 2005

  18. Content of Lecture #15 - 2004-04-04 • Linear Quadratic Control (LQ-control) • Formulation of the LQ-problem • Finite horizon LQ-control • Reference: Ch.1-1.2 - Note on LQ-control by B.Foss TTK4135 Optimization and control B.Foss Spring semester 2005

  19. Content of Lecture #16 - 2004-04-08 • Linear Quadratic Control (LQ-control) • Infinite horizon LQ-control • State-estimation (repetition from TTK4115) • Reference: Ch.1.3-1.4 - Note on LQ-control by B.Foss TTK4135 Optimization and control B.Foss Spring semester 2005

  20. Content of Lecture #17 - 2004-04-11 Model Predictive Control (MPC) Feasibility and constraint handling Target calculation Robustness Reference: Ch.4 – 6, 8, 9 – Note on MPC by M.Hovd TTK4135 Optimization and control B.Foss Spring semester 2005

  21. Content of Lecture #18 - 2004-04-18 • Nonlinear programming - SQP • Line-search in nonlinear programming • l1 exact merit function • Exact merit function • Reference: 15.3,18.5,18.6 in textbook TTK4135 Optimization and control B.Foss Spring semester 2005

  22. Content of Lecture #19 - 2004-04-25 • Nonlinear programming - SQP • Computing the search direction • Solving nonlinear equtions • Quasi-Newton method for computing the Hessian • Reference: 11.1,18.1-18.4,18.6 in textbook TTK4135 Optimization and control B.Foss Spring semester 2005

  23. Content of Lecture #20 - 2004-05-02 • Nonlinear programming - SQP • Reduced Hessian methods • Convergence rate • Maratos effect • Reference: 18.7,18.10,18.11 in textbook TTK4135 Optimization and control B.Foss Spring semester 2005

  24. Content of Lecture #21 - 2004-05-09 SQP – final remarks including examples Repetition Repetition of main topics Course evaluation ---------------------- TTK4135 Optimization and control B.Foss Spring semester 2005

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