1 TTK4135 Optimization and control B.Foss 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 Lectures are given by professor Bjarne A. Foss The course assistant is Mr. K. Rambabu.

2 TTK4135 Optimization and control B.Foss Spring semester 2005 Course information All course information is provided on the web-pages for the course: 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 -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.

3 TTK4135 Optimization and control B.Foss Spring semester 2005 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.

4 TTK4135 Optimization and control B.Foss Spring semester 2005 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 The final examination is on

5 TTK4135 Optimization and control B.Foss Spring semester 2005 Content of Lecture # 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)

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

7 TTK4135 Optimization and control B.Foss Spring semester 2005 Content of Lecture # 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

8 TTK4135 Optimization and control B.Foss Spring semester 2005 Content of Lecture #4 + # /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, in textbook

9 TTK4135 Optimization and control B.Foss Spring semester 2005 Content of Lecture # 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, ,(16.5),16.8 in textbook

10 TTK4135 Optimization and control B.Foss Spring semester 2005 Content of Lecture # 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

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

12 TTK4135 Optimization and control B.Foss Spring semester 2005 Content of Lecture # 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

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

14 TTK4135 Optimization and control B.Foss Spring semester 2005 Content of Lecture # Practical Newton-methods Approximate Newton-step Line search Newton Modified Hessian Reference: in textbook Computing gradients Reference: in textbook

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

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

17 TTK4135 Optimization and control B.Foss Spring semester 2005 Content of Lecture # 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

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

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

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

21 TTK4135 Optimization and control B.Foss Spring semester 2005 Content of Lecture # Nonlinear programming - SQP Line-search in nonlinear programming l 1 exact merit function Exact merit function Reference: 15.3,18.5,18.6 in textbook

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

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

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