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TTK4135 Optimization and control B.Foss Spring semester 20051
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.
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TTK4135 Optimization and control B.Foss Spring semester 20052
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.
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TTK4135 Optimization and control B.Foss Spring semester 20053
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.
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TTK4135 Optimization and control B.Foss Spring semester 20054
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.
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TTK4135 Optimization and control B.Foss Spring semester 20055
Content of Lecture #1 - 2004-01-10
Optimization problems appear everywhereStock 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 termsConvexity and non-convexity
Global vs. local solution
Constrained vs. unconstrained problems
Feasible region
Reference: Chapter 1 in Nocedal and Wright (N&W)
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TTK4135 Optimization and control B.Foss Spring semester 20056
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
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TTK4135 Optimization and control B.Foss Spring semester 20057
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
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TTK4135 Optimization and control B.Foss Spring semester 20058
Content of Lecture #4 + #5 - 2004-01-21/28
Linear programming - LPMathematical 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
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TTK4135 Optimization and control B.Foss Spring semester 20059
Content of Lecture #6 - 2004-01-31
Quadratic programming - QPMathematical 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
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TTK4135 Optimization and control B.Foss Spring semester 200510
Content of Lecture #7 - 2004-02-04
Quadratic programming - QPThe 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
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TTK4135 Optimization and control B.Foss Spring semester 200511
Content of Lecture #8 - 2004-02-07
Quadratic programming - QPThe active-set method for non-convex problems
Reference: 16.5,16.8 in textbook
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Content of Lecture #9 - 2004-02-11
Repetition of LP, QP
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Optimality conditionsNecessary and sufficient conditions for optimality
Iterative solution methodsStarting point
Search direction
Step length
Termination criteria
Convergence
Reference: 2.1, 2.2 in textbook
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TTK4135 Optimization and control B.Foss Spring semester 200513
Content of Lecture #10 - 2004-02-14
Line search methodsChoice 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
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Content of Lecture #11 - 2004-02-18
Practical Newton-methodsApproximate Newton-step
Line search Newton
Modified Hessian
Reference: 6 - 6.3 in textbook
Computing gradients
Reference: 7 - 7.1 in textbook
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TTK4135 Optimization and control B.Foss Spring semester 200515
Content of Lecture #12 - 2004-02-21
Quasi Newton methodsDFP and BFGS methods
Rosenbrock example for illustration
Reference: 8 - 8.1 in textbook
Information on the mid.term examination
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Content of Lecture #13 - 2004-03-07
Mid-term examination
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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
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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
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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
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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
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Content of Lecture #18 - 2004-04-18
Nonlinear programming - SQPLine-search in nonlinear programming
l1 exact merit function
Exact merit function
Reference: 15.3,18.5,18.6 in textbook
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Content of Lecture #19 - 2004-04-25
Nonlinear programming - SQPComputing the search direction
Solving nonlinear equtions
Quasi-Newton method for computing the Hessian
Reference: 11.1,18.1-18.4,18.6 in textbook
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Content of Lecture #20 - 2004-05-02
Nonlinear programming - SQPReduced Hessian methods
Convergence rate
Maratos effect
Reference: 18.7,18.10,18.11 in textbook
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Content of Lecture #21 - 2004-05-09
SQP – final remarks including examples
RepetitionRepetition of main topics
Course evaluation----------------------