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Home » Algebraic, Geometric, and Combinatorial Methods for Optimization

Summer Graduate School

Algebraic, Geometric, and Combinatorial Methods for Optimization August 02, 2010 - August 13, 2010
Organizers Matthias Köppe (University of California, Davis) and Jiawang Nie (University of California, San Diego)
Lecturer(s) No Lecturers Assigned Yet.
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Description This workshop is intended to introduce to graduate students the main ideas of algebraic, geometric and combinatorial methods in global optimization. We emphasize the major developments in the past few years from two viewpoints. The first one is that of the interaction of semidefinite programming and real algebraic geometry and includes topics such as linear matrix inequalities, positive polynomials, and sums of squares. The second viewpoint is that of primal methods and generating function methods in integer linear and nonlinear optimization.

The workshop consists of four parts of lectures and corresponding tutorials and computer experimentation with computer software in the area.


Week 1 -- Schedule Week 1 (PDF)

The first part (Jiawang Nie, University of California, San Diego) introduces the basic theory of semidefinite programming, which includes convex sets and linear matrix inequalities, duality theory, optimality conditions, and applications such as in control and optimization. We will also introduce how to use the existing software for solving semidefinite programming problems.
Lecture Notes Week 1 Nie (PDF), Homework Assigments Week 1 (PDF)

The second part (Greg Blekherman, Virginia Tech) introduces positive polynomials, sum of squares, basic real algebraic geometry, and its connections to semidefinite programming. The applications will also be covered. Existing software will be introduced.
Lecture Notes Week 1 Blekherman (PDF)


Week 2 -- Schedule Week 2 (PDF)

The third part (Shmuel Onn, Technion - Israel Institute of Technology) concentrates on primal methods of integer linear and nonlinear optimization, such as the theory of test sets, in particular the recent advances in Graver basis methods.

The fourth part (Matthias Köppe, University of California, Davis) is on tools from the geometry of numbers, with a focus on rational generating function techniques for integer programming. We introduce lattices, the LLL algorithm, Lenstra's algorithm for integer programming in fixed dimension, Barvinok's theory of short rational generating functions, and the summation method for polynomial integer programming.
Lecture Notes Week 2 Koeppe (PDF)

Teaching assistants: Cynthia Vinzant, University of California, Berkeley and Amitabh Basu, University of California, Davis

Bibliography (PDF)

Bibliography 2 (PDF)

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Schedule
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Aug 02, 2010
Monday
09:30 AM - 10:30 AM
  Nie Talk 1
Jiawang Nie (University of California, San Diego)
11:00 AM - 12:00 PM
  Blekherman Talk 1
Grigoriy Blekherman
02:00 PM - 03:00 PM
  Nie Talk 2
Jiawang Nie (University of California, San Diego)
Aug 03, 2010
Tuesday
09:30 AM - 10:30 AM
  Blekherman Talk 2
Grigoriy Blekherman
11:00 AM - 12:00 PM
  Nie Talk 3
Jiawang Nie (University of California, San Diego)
02:00 PM - 03:00 PM
  Blekherman Talk 3
Grigoriy Blekherman
Aug 04, 2010
Wednesday
09:30 AM - 10:30 AM
  Nie Talk 4
Jiawang Nie (University of California, San Diego)
11:00 AM - 12:00 PM
  Blekherman Talk 4
Grigoriy Blekherman
Aug 05, 2010
Thursday
09:30 AM - 10:30 AM
  Nie Talk 5
Jiawang Nie (University of California, San Diego)
11:00 AM - 12:00 PM
  Blekherman Talk 5
Grigoriy Blekherman
02:00 PM - 03:00 PM
  Nie Talk 6
Jiawang Nie (University of California, San Diego)
Aug 06, 2010
Friday
09:30 AM - 10:30 AM
  Blekherman Talk 6
Grigoriy Blekherman
11:00 AM - 12:00 PM
  Nie Talk 7
Jiawang Nie (University of California, San Diego)
02:00 PM - 03:00 PM
  Blekherman Talk 7
Grigoriy Blekherman
Aug 09, 2010
Monday
09:30 AM - 10:00 AM
  Onn Talk 1
Shmuel Onn
11:00 AM - 12:00 PM
  Koppe Talk 1
Matthias Köppe
02:00 PM - 03:00 PM
  Onn Talk 2
Shmuel Onn
Aug 10, 2010
Tuesday
09:30 AM - 10:00 AM
  Koppe Talk 2
Matthias Köppe
11:00 AM - 12:00 PM
  Onn Talk 3
Shmuel Onn
02:00 PM - 03:00 PM
  Koppe Talk 3
Matthias Köppe
Aug 11, 2010
Wednesday
09:00 AM - 10:00 AM
  Onn Talk 4
Shmuel Onn
11:00 AM - 12:00 PM
  Koppe Talk 4
Matthias Köppe
Aug 12, 2010
Thursday
09:30 AM - 10:00 AM
  Koppe Talk 5
Matthias Köppe
11:00 AM - 12:00 PM
  Onn Talk 5
Shmuel Onn
02:00 PM - 03:00 PM
  Onn Talk 6
Shmuel Onn
Aug 13, 2010
Friday
09:30 AM - 10:00 AM
  Koppe Talk 6
Matthias Köppe
11:00 AM - 12:00 PM
  Onn Talk 7
Shmuel Onn
02:00 PM - 03:00 PM
  Koppe Talk 7
Matthias Köppe