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

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.


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.


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.


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.


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)



Karush Kuhn-Tucker (KKT) conditions

Sensitivities and Lagrange-multipliers

Reference: Chapter 12.1, 12.2 in N&W



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


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



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



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



Quadratic programming - QPThe active-set method for non-convex problems

Reference: 16.5,16.8 in textbook



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



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



Practical Newton-methodsApproximate Newton-step

Line search Newton

Modified Hessian

Reference: 6 - 6.3 in textbook

Computing gradients




Quasi Newton methodsDFP and BFGS methods

Rosenbrock example for illustration


Information on the mid.term examination



Mid-term examination



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



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



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



Model Predictive Control (MPC)

Feasibility and constraint handling

Target calculation

Robustness

Reference: Ch.4 – 6, 8, 9 – Note on MPC by M.Hovd



Nonlinear programming - SQPLine-search in nonlinear programming

l1 exact merit function

Exact merit function




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



Nonlinear programming - SQPReduced Hessian methods

Convergence rate

Maratos effect




SQP – final remarks including examples

RepetitionRepetition of main topics

Course evaluation----------------------

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

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