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CO 769. Paper presentation.
A New Method for Quick Solving Quadratic
Assignment Problem
by Zhongzhen Zhang
2
0. Before intro

• Focus. A New Method for Quickly Solving
• Quadratic Assignment Problems
• Key concept. Pivoting Algorithms
• Linear Programming
• well established in Quadratic Programming
• Follows from. An efficient method for solving
  the local minimum of indefinite quadratic
  programming
• More work same person same topic.
• Book (in Chinese,2004) Convex Programming
  Pivoting Algorithms for Portfolio Selection and
  Network Optimization

CO 769
An algorithm for solving Quadratic Assignment
Problems
3
1. Intro Pivoting

• Start with a set of vectors
• Definition. The subset is
  called a basis if it is linearly independent and
  maximal i.e.
• Assume is a basis. Represent this
  situation with the table

Basic vectors
Nonbasic vectors
CO 769
An algorithm for solving Quadratic Assignment
Problems
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1. Pivoting

• Pivoting. Swap one basic vector with a nonbasic
  vector
• When it can be done?

swapped
non zero

• How to do it? What comes out?

CO 769
An algorithm for solving Quadratic Assignment
Problems
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1. Pivoting

• Pivoting. Swap one basic vector with a nonbasic
  vector
• Update the table !

non zero



CO 769
An algorithm for solving Quadratic Assignment
Problems
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2.Solving systems of linear equalitiesinequalitie
s
B can be empty l can be 0

• Solve

System defined by nm vectors

• Take large Mgt0

• Set up the pivoting table

CO 769
An algorithm for solving Quadratic Assignment
Problems
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2.Solving systems of linear equalitiesinequalitie
s
Let x1 be the solution of



measure the offset
x1 is a solution
CO 769
An algorithm for solving Quadratic Assignment
Problems
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3. Preprocessing move the equalities in the
basis.



Assume is lt0

• Find a positive element
• no such thing infeasible problem
• Do a pivoting on a positive element
• for example a11

Theorem. gt0


CO 769
An algorithm for solving Quadratic Assignment
Problems
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3. Preprocessing move the equalities in the
basis.
Assume is gt0

• Find a negative element
• no such thing infeasible problem
• Do a pivoting on a negative element
• for example

Theorem.

Once an equality goes in the basis it stays there
! ! !

CO 769
An algorithm for solving Quadratic Assignment
Problems
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3. Preprocessing move the equalities in the
basis.
Assume is 0

• If everything 0 the equality is redundant. Just
  throw it away.
• Else do a pivoting on a nonzero element

CO 769
An algorithm for solving Quadratic Assignment
Problems
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3. Main iterations





assume lt0

• Find a positive element
• no such thing infeasible
• Do a pivoting on it

• Pivoting rules the smallest deviation rule, the
  largest distance rule, the smallest index rule
  etc.
• Theorem. When cycling doesnt occur.

CO 769
An algorithm for solving Quadratic Assignment
Problems
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4. Quadratic programming

• What Im about to do can be done for general
  quadratic problems
• For simplicity, Ill work directly with the
  quadratic problem that will come from QAP
• Same ideas

CO 769
An algorithm for solving Quadratic Assignment
Problems
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4. Quadratic programming

• Consider

where

• As before

• Set up KKT for this

CO 769
An algorithm for solving Quadratic Assignment
Problems
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4. Quadratic programming

• Consider

where

• KKT

0

• Step 1 get rid of ?s

CO 769
An algorithm for solving Quadratic Assignment
Problems
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4. Quadratic programming. Preprocessing.

• New KKT

The unknowns
Ignore this!







Step 1. Preprocessing. ! ! Must become basic ! !

CO 769
An algorithm for solving Quadratic Assignment
Problems
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4. QP. Preprocessing.

• New KKT

The unknowns
Ignore this!







Step 1. Preprocessing. ! ! Must become basic ! !

CO 769
An algorithm for solving Quadratic Assignment
Problems
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4. QP. Preprocessing.

• KKT

The unknowns
!Dont destroy other complementarity!
Ignore this!
Choose pivot!







Step 1. Preprocessing. ! ! Must become basic ! !

CO 769
An algorithm for solving Quadratic Assignment
Problems
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4. QP. Preprocessing.

• KKT

The unknowns
Ignore this!







Step 1. Preprocessing. ! ! Must become basic ! !

CO 769
An algorithm for solving Quadratic Assignment
Problems
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4. QP. Preprocessing.
0 or 1
0 or 1 or -1

• Preprocessing leads to a feasible solution
• Choosing pivots in preprocessing can be done in
  more than one way. Pivot selection plays a role
  in the implementation. If a choice fails, another
  is pursued.

Remark 1. The method described in the paper is
heuristic. 2. Attempts to gain insight are
made. 3. No insights for pivot selection in
preprocessing.
CO 769
An algorithm for solving Quadratic Assignment
Problems
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4. QP. Main Iterations Local search.

• Not any local search is good.
• Goal
• Decrease in the objective!
• Nothing else.
• No other insight.

Local search
What happens The offset column must be
updated. Feasibility can be lost!!
CO 769
An algorithm for solving Quadratic Assignment
Problems
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4. QP. Main Iterations Pivoting.

• In general for QP pivoting is done such that
  complementarity is preserved !
• Pivoting can destroy the feasibility ! Keep
  feasibility all the time
• Pivoting modifies the objective ! Heuristic goal
  pivot to decrease the objective

Feasibility Objective Decrease
Forward pivoting. Assume the pivot is

• produces a feasible solution

CO 769
An algorithm for solving Quadratic Assignment
Problems
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4. QP. Main Iterations Pivoting.

• In general for QP pivoting is done such that
  complementarity is preserved !
• Pivoting can destroy the feasibility ! Keep
  feasibility all the time
• Pivoting modifies the objective ! Heuristic goal
  pivot to decrease the objective

Feasibility Objective Decrease
Backward pivoting. Assume the pivot is

• produces a feasible solution

CO 769
An algorithm for solving Quadratic Assignment
Problems
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4. QP. Main Iterations Pivoting.

• In general for QP pivoting is done such that
  complementarity is preserved !
• Pivoting can destroy the feasibility ! Keep
  feasibility all the time
• Pivoting modifies the objective ! Heuristic goal
  pivot to decrease the objective

Par pivoting. Assume the pivot is
Feasibility No Decrease in Objective
CO 769
An algorithm for solving Quadratic Assignment
Problems
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4. QP. Main Iterations Pivoting.

• This pivoting against heuristic.
• Experiments show
  equivalent with

Par pivoting. Assume the pivot is
Feasibility No Decrease in Objective
CO 769
An algorithm for solving Quadratic Assignment
Problems
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4. QP. Main Iterations Pivoting.
The contraction effect of pivoting operations
occurs to us the formation of substance that is
under certain conditions or control while the
formation of a good solution of QAP is depending
on the appropriate control of pivoting
operations. It is hopeful to discover the deep
mechanism of QAP and develop more efficient
computing method.
Z.Zhang
Thanks!
CO 769
An algorithm for solving Quadratic Assignment
Problems