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Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Unbounded Knapsack Problemand the
Generalized Cordel Property
Lisa Schreiber
Friedrich-Schiller-Universitat Jena,Institut fur Angewandte Mathematik
November the 26th, 2010
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Knapsack Problem
Given n items and a knapsack with capacity C.
Every item has a value v(i) and a weight w(i).Question: Which items shall be packed into the knapsacksuch that the capacity C is not injured and the total value isas large as possible?
Definition: Knapsack Problem
maxx∈Gn
f (x) :=n
∑i=1
v(i)xi subject ton
∑i=1
w(i)xi ≤ C
G = {0,1}: 0-1 Knapsack ProblemG = N: Unbounded Knapsack Problem
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Knapsack Problem
Given n items and a knapsack with capacity C.Every item has a value v(i) and a weight w(i).
Question: Which items shall be packed into the knapsacksuch that the capacity C is not injured and the total value isas large as possible?
Definition: Knapsack Problem
maxx∈Gn
f (x) :=n
∑i=1
v(i)xi subject ton
∑i=1
w(i)xi ≤ C
G = {0,1}: 0-1 Knapsack ProblemG = N: Unbounded Knapsack Problem
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Knapsack Problem
Given n items and a knapsack with capacity C.Every item has a value v(i) and a weight w(i).Question: Which items shall be packed into the knapsacksuch that the capacity C is not injured and the total value isas large as possible?
Definition: Knapsack Problem
maxx∈Gn
f (x) :=n
∑i=1
v(i)xi subject ton
∑i=1
w(i)xi ≤ C
G = {0,1}: 0-1 Knapsack ProblemG = N: Unbounded Knapsack Problem
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Knapsack Problem
Given n items and a knapsack with capacity C.Every item has a value v(i) and a weight w(i).Question: Which items shall be packed into the knapsacksuch that the capacity C is not injured and the total value isas large as possible?
Definition: Knapsack Problem
maxx∈Gn
f (x) :=n
∑i=1
v(i)xi subject ton
∑i=1
w(i)xi ≤ C
G = {0,1}: 0-1 Knapsack ProblemG = N: Unbounded Knapsack Problem
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Knapsack Problem
Given n items and a knapsack with capacity C.Every item has a value v(i) and a weight w(i).Question: Which items shall be packed into the knapsacksuch that the capacity C is not injured and the total value isas large as possible?
Definition: Knapsack Problem
maxx∈Gn
f (x) :=n
∑i=1
v(i)xi subject ton
∑i=1
w(i)xi ≤ C
G = {0,1}: 0-1 Knapsack Problem
G = N: Unbounded Knapsack Problem
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Knapsack Problem
Given n items and a knapsack with capacity C.Every item has a value v(i) and a weight w(i).Question: Which items shall be packed into the knapsacksuch that the capacity C is not injured and the total value isas large as possible?
Definition: Knapsack Problem
maxx∈Gn
f (x) :=n
∑i=1
v(i)xi subject ton
∑i=1
w(i)xi ≤ C
G = {0,1}: 0-1 Knapsack ProblemG = N: Unbounded Knapsack Problem
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method
Aim: Compute not only an optimal solution but also one (ormore) alternative solutions.
two Criteria:1 An alternative should be good with respect to the objective
function.2 An alternative should not be too similar to the optimal
solution.
One good method: the Penalty Methodexamined by Schwarz (2003), Sameith (2005) andDornfelder (2009)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method
Aim: Compute not only an optimal solution but also one (ormore) alternative solutions.two Criteria:
1 An alternative should be good with respect to the objectivefunction.
2 An alternative should not be too similar to the optimalsolution.
One good method: the Penalty Methodexamined by Schwarz (2003), Sameith (2005) andDornfelder (2009)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method
Aim: Compute not only an optimal solution but also one (ormore) alternative solutions.two Criteria:
1 An alternative should be good with respect to the objectivefunction.
2 An alternative should not be too similar to the optimalsolution.
One good method: the Penalty Methodexamined by Schwarz (2003), Sameith (2005) andDornfelder (2009)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method
Aim: Compute not only an optimal solution but also one (ormore) alternative solutions.two Criteria:
1 An alternative should be good with respect to the objectivefunction.
2 An alternative should not be too similar to the optimalsolution.
One good method: the Penalty Method
examined by Schwarz (2003), Sameith (2005) andDornfelder (2009)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method
Aim: Compute not only an optimal solution but also one (ormore) alternative solutions.two Criteria:
1 An alternative should be good with respect to the objectivefunction.
2 An alternative should not be too similar to the optimalsolution.
One good method: the Penalty Methodexamined by Schwarz (2003), Sameith (2005) andDornfelder (2009)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
Important properties:
ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
Important properties:
ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
vε (i) =
{v (i) , if B0(i) = 0v (i) · [1− ε · B0(i)] , if B0(i) > 0
Important properties:
ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
vε (i) =
{v (i) , if B0(i) = 0v (i) · [1− ε · B0(i)] , if B0(i) > 0
Important properties:
ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
vε (i) =
{v (i) , if B0(i) = 0v (i) · [1− ε · B0(i)] , if B0(i) > 0
Important properties:ε increases→ punishment gets higher
B0(i) > B0(j)→ punishment of item i is higher
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
vε (i) =
{v (i) , if B0(i) = 0v (i) · [1− ε · B0(i)] , if B0(i) > 0
Important properties:ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
vε (i) = v (i) · [1− ε · B0(i)]
Important properties:ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
A first example
C = 13values v =[6,8,3,1]
weights w=[5,7,3,6]
optimal solution: B0 = (2,0,1,0) with f (B0) = 15
e.g. ε = 0.7. This leads to the following punished values:
vε(2) = v(2) = 8 andvε(4) = v(4) = 1
vε(1) = v (1) · [1− ε · B0(1)] = 6 · [1− 0.7 · 2] = −2.4
vε(3) = v (3) · [1− ε · B0(3)] = 3 · [1− 0.7 · 1] = 0.9
C, vε = [−2.4, 8, 0.9, 1] and w provide Bε = (0,1,2,0) asbest solution and penalty alternative
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
A first example
C = 13values v =[6,8,3,1]
weights w=[5,7,3,6]
optimal solution: B0 = (2,0,1,0) with f (B0) = 15
e.g. ε = 0.7. This leads to the following punished values:
vε(2) = v(2) = 8 andvε(4) = v(4) = 1
vε(1) = v (1) · [1− ε · B0(1)] = 6 · [1− 0.7 · 2] = −2.4
vε(3) = v (3) · [1− ε · B0(3)] = 3 · [1− 0.7 · 1] = 0.9
C, vε = [−2.4, 8, 0.9, 1] and w provide Bε = (0,1,2,0) asbest solution and penalty alternative
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
A first example
C = 13values v =[6,8,3,1]
weights w=[5,7,3,6]
optimal solution: B0 = (2,0,1,0) with f (B0) = 15
e.g. ε = 0.7. This leads to the following punished values:
vε(2) = v(2) = 8 andvε(4) = v(4) = 1
vε(1) = v (1) · [1− ε · B0(1)] = 6 · [1− 0.7 · 2] = −2.4
vε(3) = v (3) · [1− ε · B0(3)] = 3 · [1− 0.7 · 1] = 0.9
C, vε = [−2.4, 8, 0.9, 1] and w provide Bε = (0,1,2,0) asbest solution and penalty alternative
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
A first example
C = 13values v =[6,8,3,1]
weights w=[5,7,3,6]
optimal solution: B0 = (2,0,1,0) with f (B0) = 15
e.g. ε = 0.7. This leads to the following punished values:
vε(2) = v(2) = 8 andvε(4) = v(4) = 1
vε(1) = v (1) · [1− ε · B0(1)] = 6 · [1− 0.7 · 2] = −2.4
vε(3) = v (3) · [1− ε · B0(3)] = 3 · [1− 0.7 · 1] = 0.9
C, vε = [−2.4, 8, 0.9, 1] and w provide Bε = (0,1,2,0) asbest solution and penalty alternative
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
A first example
C = 13values v =[6,8,3,1]
weights w=[5,7,3,6]
optimal solution: B0 = (2,0,1,0) with f (B0) = 15
e.g. ε = 0.7. This leads to the following punished values:
vε(2) = v(2) = 8 andvε(4) = v(4) = 1
vε(1) = v (1) · [1− ε · B0(1)] = 6 · [1− 0.7 · 2] = −2.4
vε(3) = v (3) · [1− ε · B0(3)] = 3 · [1− 0.7 · 1] = 0.9
C, vε = [−2.4, 8, 0.9, 1] and w provide Bε = (0,1,2,0) asbest solution and penalty alternative
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
A first example
C = 13values v =[6,8,3,1]
weights w=[5,7,3,6]
optimal solution: B0 = (2,0,1,0) with f (B0) = 15
e.g. ε = 0.7. This leads to the following punished values:
vε(2) = v(2) = 8 andvε(4) = v(4) = 1
vε(1) = v (1) · [1− ε · B0(1)] = 6 · [1− 0.7 · 2] = −2.4
vε(3) = v (3) · [1− ε · B0(3)] = 3 · [1− 0.7 · 1] = 0.9
C, vε = [−2.4, 8, 0.9, 1] and w provide Bε = (0,1,2,0) asbest solution and penalty alternative
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
A first example
C = 13values v =[6,8,3,1]
weights w=[5,7,3,6]
optimal solution: B0 = (2,0,1,0) with f (B0) = 15
e.g. ε = 0.7. This leads to the following punished values:
vε(2) = v(2) = 8 andvε(4) = v(4) = 1
vε(1) = v (1) · [1− ε · B0(1)] = 6 · [1− 0.7 · 2] = −2.4
vε(3) = v (3) · [1− ε · B0(3)] = 3 · [1− 0.7 · 1] = 0.9
C, vε = [−2.4, 8, 0.9, 1] and w provide Bε = (0,1,2,0) asbest solution and penalty alternative
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
Properties of Penalty Alternatives (Schwarz 2003)B penalty-alternative⇒ B is optimal for all parameters ε inan optimality interval IB = [ε1, ε2] and nowhere else
P1,P2 penalty alternatives with optimality intervals I1 and I2⇒ three possible cases
1 I1 = I22 I1 ∩ I2 = {ε}, intersection contains only a single epsilon3 I1 ∩ I2 = ∅, empty intersection
∞0ε0 ε1 ε2
P0 = B0 P1 P2 . . .
This provides an algorithm, how to compute all penaltyalternatives P0,P1, . . . ,Pk !
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
Properties of Penalty Alternatives (Schwarz 2003)B penalty-alternative⇒ B is optimal for all parameters ε inan optimality interval IB = [ε1, ε2] and nowhere elseP1,P2 penalty alternatives with optimality intervals I1 and I2⇒ three possible cases
1 I1 = I22 I1 ∩ I2 = {ε}, intersection contains only a single epsilon3 I1 ∩ I2 = ∅, empty intersection
∞0ε0 ε1 ε2
P0 = B0 P1 P2 . . .
This provides an algorithm, how to compute all penaltyalternatives P0,P1, . . . ,Pk !
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
Properties of Penalty Alternatives (Schwarz 2003)B penalty-alternative⇒ B is optimal for all parameters ε inan optimality interval IB = [ε1, ε2] and nowhere elseP1,P2 penalty alternatives with optimality intervals I1 and I2⇒ three possible cases
1 I1 = I2
2 I1 ∩ I2 = {ε}, intersection contains only a single epsilon3 I1 ∩ I2 = ∅, empty intersection
∞0ε0 ε1 ε2
P0 = B0 P1 P2 . . .
This provides an algorithm, how to compute all penaltyalternatives P0,P1, . . . ,Pk !
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
Properties of Penalty Alternatives (Schwarz 2003)B penalty-alternative⇒ B is optimal for all parameters ε inan optimality interval IB = [ε1, ε2] and nowhere elseP1,P2 penalty alternatives with optimality intervals I1 and I2⇒ three possible cases
1 I1 = I22 I1 ∩ I2 = {ε}, intersection contains only a single epsilon
3 I1 ∩ I2 = ∅, empty intersection
∞0ε0 ε1 ε2
P0 = B0 P1 P2 . . .
This provides an algorithm, how to compute all penaltyalternatives P0,P1, . . . ,Pk !
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
Properties of Penalty Alternatives (Schwarz 2003)B penalty-alternative⇒ B is optimal for all parameters ε inan optimality interval IB = [ε1, ε2] and nowhere elseP1,P2 penalty alternatives with optimality intervals I1 and I2⇒ three possible cases
1 I1 = I22 I1 ∩ I2 = {ε}, intersection contains only a single epsilon3 I1 ∩ I2 = ∅, empty intersection
∞0ε0 ε1 ε2
P0 = B0 P1 P2 . . .
This provides an algorithm, how to compute all penaltyalternatives P0,P1, . . . ,Pk !
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
Properties of Penalty Alternatives (Schwarz 2003)B penalty-alternative⇒ B is optimal for all parameters ε inan optimality interval IB = [ε1, ε2] and nowhere elseP1,P2 penalty alternatives with optimality intervals I1 and I2⇒ three possible cases
1 I1 = I22 I1 ∩ I2 = {ε}, intersection contains only a single epsilon3 I1 ∩ I2 = ∅, empty intersection
∞0ε0 ε1 ε2
P0 = B0 P1 P2 . . .
This provides an algorithm, how to compute all penaltyalternatives P0,P1, . . . ,Pk !
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
Properties of Penalty Alternatives (Schwarz 2003)B penalty-alternative⇒ B is optimal for all parameters ε inan optimality interval IB = [ε1, ε2] and nowhere elseP1,P2 penalty alternatives with optimality intervals I1 and I2⇒ three possible cases
1 I1 = I22 I1 ∩ I2 = {ε}, intersection contains only a single epsilon3 I1 ∩ I2 = ∅, empty intersection
∞0ε0 ε1 ε2
P0 = B0 P1 P2 . . .
This provides an algorithm, how to compute all penaltyalternatives P0,P1, . . . ,Pk !
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
vε (i) = v (i) · [1− ε · B0(i)]
⇒ fε(B) =n
∑i=1
vε(i) · B(i) =n
∑i=1
v(i) · [1− ε · B0(i)] · B(i)
=n
∑i=1
v(i) · B(i)︸ ︷︷ ︸=f (B)
−εn
∑i=1
v(i) · B0(i) · B(i)︸ ︷︷ ︸=:p(B)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
vε (i) = v (i) · [1− ε · B0(i)]
⇒ fε(B) =n
∑i=1
vε(i) · B(i) =n
∑i=1
v(i) · [1− ε · B0(i)] · B(i)
=n
∑i=1
v(i) · B(i)︸ ︷︷ ︸=f (B)
−εn
∑i=1
v(i) · B0(i) · B(i)︸ ︷︷ ︸=:p(B)
· · · ∞0ε0 ε1 ε2 εk
threshold parameters
f (P0) ≥ f (P1) > . . . > f (Pk)p (P0) ≥ p (P1) > . . . > p (Pk)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
vε (i) = v (i) · [1− ε · B0(i)]
⇒ fε(B) =n
∑i=1
vε(i) · B(i) =n
∑i=1
v(i) · [1− ε · B0(i)] · B(i)
=n
∑i=1
v(i) · B(i)︸ ︷︷ ︸=f (B)
−εn
∑i=1
v(i) · B0(i) · B(i)︸ ︷︷ ︸=:p(B)
· · · ∞0ε0 ε1 ε2 εk
threshold parameters
f (P0) ≥ f (P1) > . . . > f (Pk)p (P0) ≥ p (P1) > . . . > p (Pk)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Threshold Parameters
Let ε be the threshold parameter between the optimalityintervals of the two penalty alternatives Bl and Br .e.g.: Bl = Pi and Br = Pi+1Then we can compute ε the following way.
fε (Bl) = fε (Br )
⇔ f (Bl)− ε · p (Bl) = f (Br )− ε · p (Br )
⇔ ε =f (Bl)− f (Br )
p (Bl)− p (Br )
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
An Algorithm for Computing all Penalty Alternatives(Schwarz)
1 Initialization: Compute B0 (optimal solution) and B∞.Go to step 2 with [Bl = B0,Br = B∞].
2 Compute the possible threshold parameter ε between Bland Br and the penalty alternative Bε. Then we considerthe following two cases:fε (Bε) = fε (Bl) = fε (Br ):
bagagagsadgsagsadgdsaglabla
No further branching. ε is the real thresholdparameter between Bl and Br .
fε (Bε) 6= fε (Bl) = fε (Br ):
bagagagsadgsagsadgdsaglabla
With Bε we found a new penalty alternative,so we have to branch. Go to step 2 with[Bl = Bl ,Br = Bε] and [Bl = Bε,Br = Br ]
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
From Chess history
Siegbert Tarrasch(1862−1934)In every chess positionthere exists exactly onebest move!
Emanuel Lasker(1868−1941)In every chess positionthere exist as manyappropriate moves asthere are different players!
Oskar Cordel’s (1843−1913) “Three moves law”In every chess position there exists either
exactly one best moveor at least three equally best moves.
This assumption does not hold in every chess position!
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
From Chess history
Siegbert Tarrasch(1862−1934)In every chess positionthere exists exactly onebest move!
Emanuel Lasker(1868−1941)In every chess positionthere exist as manyappropriate moves asthere are different players!
Oskar Cordel’s (1843−1913) “Three moves law”In every chess position there exists either
exactly one best moveor at least three equally best moves.
This assumption does not hold in every chess position!
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
From Chess history
Siegbert Tarrasch(1862−1934)In every chess positionthere exists exactly onebest move!
Emanuel Lasker(1868−1941)In every chess positionthere exist as manyappropriate moves asthere are different players!
Oskar Cordel’s (1843−1913) “Three moves law”In every chess position there exists either
exactly one best moveor at least three equally best moves.
This assumption does not hold in every chess position!
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
From Chess history
Siegbert Tarrasch(1862−1934)In every chess positionthere exists exactly onebest move!
Emanuel Lasker(1868−1941)In every chess positionthere exist as manyappropriate moves asthere are different players!
Oskar Cordel’s (1843−1913) “Three moves law”In every chess position there exists either
exactly one best moveor at least three equally best moves.
This assumption does not hold in every chess position!
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Generalized Cordel Property (GeCoP)
Generalizing Cordel: examine differences betweencandidates
moves a1 and a2 are equally good, if |f (a1)− f (a2)| isvery small
Definition: Generalized Cordel Property (GeCoP)
a1,a2, and a3 solutions of a given maximization problemf (a1) ≥ f (a2) ≥ f (a3)
define differences di := f (ai)− f (ai+1)
We say that a1,a2, and a3 fulfill the Generalized CordelProperty (GeCoP), iff the following holds:
d1 ≥ d2 (GeCoP)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Generalized Cordel Property (GeCoP)
Generalizing Cordel: examine differences betweencandidatesmoves a1 and a2 are equally good, if |f (a1)− f (a2)| isvery small
Definition: Generalized Cordel Property (GeCoP)
a1,a2, and a3 solutions of a given maximization problemf (a1) ≥ f (a2) ≥ f (a3)
define differences di := f (ai)− f (ai+1)
We say that a1,a2, and a3 fulfill the Generalized CordelProperty (GeCoP), iff the following holds:
d1 ≥ d2 (GeCoP)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Generalized Cordel Property (GeCoP)
Generalizing Cordel: examine differences betweencandidatesmoves a1 and a2 are equally good, if |f (a1)− f (a2)| isvery small
Definition: Generalized Cordel Property (GeCoP)
a1,a2, and a3 solutions of a given maximization problemf (a1) ≥ f (a2) ≥ f (a3)
define differences di := f (ai)− f (ai+1)
We say that a1,a2, and a3 fulfill the Generalized CordelProperty (GeCoP), iff the following holds:
d1 ≥ d2 (GeCoP)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Generalized Cordel Property (GeCoP)
Generalizing Cordel: examine differences betweencandidatesmoves a1 and a2 are equally good, if |f (a1)− f (a2)| isvery small
Definition: Generalized Cordel Property (GeCoP)a1,a2, and a3 solutions of a given maximization problem
f (a1) ≥ f (a2) ≥ f (a3)
define differences di := f (ai)− f (ai+1)
We say that a1,a2, and a3 fulfill the Generalized CordelProperty (GeCoP), iff the following holds:
d1 ≥ d2 (GeCoP)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Generalized Cordel Property (GeCoP)
Generalizing Cordel: examine differences betweencandidatesmoves a1 and a2 are equally good, if |f (a1)− f (a2)| isvery small
Definition: Generalized Cordel Property (GeCoP)a1,a2, and a3 solutions of a given maximization problemf (a1) ≥ f (a2) ≥ f (a3)
define differences di := f (ai)− f (ai+1)
We say that a1,a2, and a3 fulfill the Generalized CordelProperty (GeCoP), iff the following holds:
d1 ≥ d2 (GeCoP)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Generalized Cordel Property (GeCoP)
Generalizing Cordel: examine differences betweencandidatesmoves a1 and a2 are equally good, if |f (a1)− f (a2)| isvery small
Definition: Generalized Cordel Property (GeCoP)a1,a2, and a3 solutions of a given maximization problemf (a1) ≥ f (a2) ≥ f (a3)
define differences di := f (ai)− f (ai+1)
We say that a1,a2, and a3 fulfill the Generalized CordelProperty (GeCoP), iff the following holds:
d1 ≥ d2 (GeCoP)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Generalized Cordel Property (GeCoP)
Generalizing Cordel: examine differences betweencandidatesmoves a1 and a2 are equally good, if |f (a1)− f (a2)| isvery small
Definition: Generalized Cordel Property (GeCoP)a1,a2, and a3 solutions of a given maximization problemf (a1) ≥ f (a2) ≥ f (a3)
define differences di := f (ai)− f (ai+1)
We say that a1,a2, and a3 fulfill the Generalized CordelProperty (GeCoP), iff the following holds:
d1 ≥ d2 (GeCoP)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Connection between Cordel and GeCoP
as a reminder: Cordel said there are never exactly twobest moves!
best moves in chess position: a1,a2 and a3
assume a1 and a2 are equally good (two best moves)
⇒ d1 = f (a1)− f (a2) will be very small⇒ d2 = f (a2)− f (a3) has to be very small too,
because of GeCoP (d1 ≥ d2)⇒ at least three best moves!
My thesis: analyzing the validity of GeCoP for severalkinds of optimization problems andkinds of solutions (e.g. three best solutions,penalty-alternatives)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Connection between Cordel and GeCoP
as a reminder: Cordel said there are never exactly twobest moves!
best moves in chess position: a1,a2 and a3
assume a1 and a2 are equally good (two best moves)
⇒ d1 = f (a1)− f (a2) will be very small⇒ d2 = f (a2)− f (a3) has to be very small too,
because of GeCoP (d1 ≥ d2)⇒ at least three best moves!
My thesis: analyzing the validity of GeCoP for severalkinds of optimization problems andkinds of solutions (e.g. three best solutions,penalty-alternatives)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Connection between Cordel and GeCoP
as a reminder: Cordel said there are never exactly twobest moves!
best moves in chess position: a1,a2 and a3
assume a1 and a2 are equally good (two best moves)
⇒ d1 = f (a1)− f (a2) will be very small⇒ d2 = f (a2)− f (a3) has to be very small too,
because of GeCoP (d1 ≥ d2)⇒ at least three best moves!
My thesis: analyzing the validity of GeCoP for severalkinds of optimization problems andkinds of solutions (e.g. three best solutions,penalty-alternatives)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Connection between Cordel and GeCoP
as a reminder: Cordel said there are never exactly twobest moves!
best moves in chess position: a1,a2 and a3
assume a1 and a2 are equally good (two best moves)⇒ d1 = f (a1)− f (a2) will be very small
⇒ d2 = f (a2)− f (a3) has to be very small too,because of GeCoP (d1 ≥ d2)
⇒ at least three best moves!
My thesis: analyzing the validity of GeCoP for severalkinds of optimization problems andkinds of solutions (e.g. three best solutions,penalty-alternatives)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Connection between Cordel and GeCoP
as a reminder: Cordel said there are never exactly twobest moves!
best moves in chess position: a1,a2 and a3
assume a1 and a2 are equally good (two best moves)⇒ d1 = f (a1)− f (a2) will be very small⇒ d2 = f (a2)− f (a3) has to be very small too,
because of GeCoP (d1 ≥ d2)
⇒ at least three best moves!
My thesis: analyzing the validity of GeCoP for severalkinds of optimization problems andkinds of solutions (e.g. three best solutions,penalty-alternatives)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Connection between Cordel and GeCoP
as a reminder: Cordel said there are never exactly twobest moves!
best moves in chess position: a1,a2 and a3
assume a1 and a2 are equally good (two best moves)⇒ d1 = f (a1)− f (a2) will be very small⇒ d2 = f (a2)− f (a3) has to be very small too,
because of GeCoP (d1 ≥ d2)⇒ at least three best moves!
My thesis: analyzing the validity of GeCoP for severalkinds of optimization problems andkinds of solutions (e.g. three best solutions,penalty-alternatives)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Connection between Cordel and GeCoP
as a reminder: Cordel said there are never exactly twobest moves!
best moves in chess position: a1,a2 and a3
assume a1 and a2 are equally good (two best moves)⇒ d1 = f (a1)− f (a2) will be very small⇒ d2 = f (a2)− f (a3) has to be very small too,
because of GeCoP (d1 ≥ d2)⇒ at least three best moves!
My thesis: analyzing the validity of GeCoP for severalkinds of optimization problems andkinds of solutions (e.g. three best solutions,penalty-alternatives)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
GeCoP for Penalty Alternatives
0ε0 ε1 ε2P0 P1 P2 . . .
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
GeCoP for Penalty Alternatives
0ε0 ε1 ε2P0 P1 P2 . . .
d1 := f(P0)− f(P1) d2 := f(P1)− f(P2)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
GeCoP for Penalty Alternatives
0ε0 ε1 ε2P0 P1 P2 . . .
d1 := f(P0)− f(P1) d2 := f(P1)− f(P2)≥?
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Random instances
according to Martello, Toth: “Knapsack Problems” (1990)two parameters v , r , e.g. v = 1000, r = 100w(i) uniformly randomly generated in the int range [1, v ]
C = 0.5 ·∑ni=1 w(i)
Three different types of instances:
Uncorrelated: v(i) uniformly randomly generatedin the range [1, v ].
Weakly correlated: v(i) uniformly randomly generated inthe range [w(i)− r ,w(i) + r ]with respect to v(i) > 0.
Strongly correlated: v(i) = w(i) + r .
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Random instances
according to Martello, Toth: “Knapsack Problems” (1990)two parameters v , r , e.g. v = 1000, r = 100w(i) uniformly randomly generated in the int range [1, v ]C = 0.5 ·∑n
i=1 w(i)
Three different types of instances:
Uncorrelated: v(i) uniformly randomly generatedin the range [1, v ].
Weakly correlated: v(i) uniformly randomly generated inthe range [w(i)− r ,w(i) + r ]with respect to v(i) > 0.
Strongly correlated: v(i) = w(i) + r .
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Random instances
according to Martello, Toth: “Knapsack Problems” (1990)two parameters v , r , e.g. v = 1000, r = 100w(i) uniformly randomly generated in the int range [1, v ]C = 0.5 ·∑n
i=1 w(i)Three different types of instances:
Uncorrelated: v(i) uniformly randomly generatedin the range [1, v ].
Weakly correlated: v(i) uniformly randomly generated inthe range [w(i)− r ,w(i) + r ]with respect to v(i) > 0.
Strongly correlated: v(i) = w(i) + r .
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Random instances
according to Martello, Toth: “Knapsack Problems” (1990)two parameters v , r , e.g. v = 1000, r = 100w(i) uniformly randomly generated in the int range [1, v ]C = 0.5 ·∑n
i=1 w(i)Three different types of instances:
Uncorrelated: v(i) uniformly randomly generatedin the range [1, v ].
Weakly correlated: v(i) uniformly randomly generated inthe range [w(i)− r ,w(i) + r ]with respect to v(i) > 0.
Strongly correlated: v(i) = w(i) + r .
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Random instances
according to Martello, Toth: “Knapsack Problems” (1990)two parameters v , r , e.g. v = 1000, r = 100w(i) uniformly randomly generated in the int range [1, v ]C = 0.5 ·∑n
i=1 w(i)Three different types of instances:
Uncorrelated: v(i) uniformly randomly generatedin the range [1, v ].
Weakly correlated: v(i) uniformly randomly generated inthe range [w(i)− r ,w(i) + r ]with respect to v(i) > 0.
Strongly correlated: v(i) = w(i) + r .
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Random instances
according to Martello, Toth: “Knapsack Problems” (1990)two parameters v , r , e.g. v = 1000, r = 100w(i) uniformly randomly generated in the int range [1, v ]C = 0.5 ·∑n
i=1 w(i)Three different types of instances:
Uncorrelated: v(i) uniformly randomly generatedin the range [1, v ].
Weakly correlated: v(i) uniformly randomly generated inthe range [w(i)− r ,w(i) + r ]with respect to v(i) > 0.
Strongly correlated: v(i) = w(i) + r .
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Comparison of P (d1 ≥ d2) and P (d2 ≥ d3)
w(i), v(i) ∈ [1,100], v(i) ∈ [w(i)− 10,w(i) + 10], v(i) = w(i) + 10
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Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Comparison of P (d1 ≥ d2) and P (d2 ≥ d3)
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Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Comparison of P (d1 ≥ d2) and P (d2 ≥ d3)
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Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Results
1 Prob (d1 ≥ d2) �Prob (d2 ≥ d3)
2 The behavior changes heavily with the rangesize v .
The first phenomenon is unique according to our wider studieson other optimization problems!
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Results
1 Prob (d1 ≥ d2) �Prob (d2 ≥ d3)
2 The behavior changes heavily with the rangesize v .
The first phenomenon is unique according to our wider studieson other optimization problems!
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Results
1 Prob (d1 ≥ d2) �Prob (d2 ≥ d3)
2 The behavior changes heavily with the rangesize v .
The first phenomenon is unique according to our wider studieson other optimization problems!
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Average number of penalty alternatives
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Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Average number of penalty alternatives
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Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Average number of penalty alternatives
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Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Open Questions
1 Do these phenomenona also occur at other optimizationproblems?
2 Is there a better method to generate alternatives for theunbounded knapsack problem?Is there a good method, that generates more than only 6alternatives in average?
3 How can the results presented be used in practice?Is Cordel more than a Gedanken-Experiment?
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Open Questions
1 Do these phenomenona also occur at other optimizationproblems?
2 Is there a better method to generate alternatives for theunbounded knapsack problem?Is there a good method, that generates more than only 6alternatives in average?
3 How can the results presented be used in practice?Is Cordel more than a Gedanken-Experiment?
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Open Questions
1 Do these phenomenona also occur at other optimizationproblems?
2 Is there a better method to generate alternatives for theunbounded knapsack problem?Is there a good method, that generates more than only 6alternatives in average?
3 How can the results presented be used in practice?Is Cordel more than a Gedanken-Experiment?
additional slides
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
vε (i) = v (i) · [1− ε · B0(i)]
Important properties:ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
additional slides
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from a given reference solution Br .have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
vε (i) = v (i) · [1− ε · Br (i)]
Important properties:ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
additional slides
Reference Solutions
Possible reference solutions are:1 An optimal solution.
2 A Greedy solution
sort items in decreasing order of value per unit of weightput every item in sequence with maximal frequency in theknapsack such that the capacity is not exceeded
Example: v = [6,8,3,1], w = [5,7,3,6], C = 13
optimal solution: B0 = (2,0,1,0)
additional slides
Reference Solutions
Possible reference solutions are:1 An optimal solution.2 A Greedy solution
sort items in decreasing order of value per unit of weightput every item in sequence with maximal frequency in theknapsack such that the capacity is not exceeded
Example: v = [6,8,3,1], w = [5,7,3,6], C = 13vw ≈ [1.2, 1.14, 1, 0.17]→ order: 1,2,3,4
(2,0,0,0)→ (2,0,0,0)→ (2,0,1,0)
additional slides
Reference Solutions
Possible reference solutions are:1 An optimal solution.2 A Greedy solution
sort items in decreasing order of value per unit of weightput every item in sequence with maximal frequency in theknapsack such that the capacity is not exceeded
Example: v = [6,8,3,1], w = [5,7,3,6], C = 13vw ≈ [1.2, 1.14, 1, 0.17]→ order: 1,2,3,4(2,0,0,0),C ′ = 3
additional slides
Reference Solutions
Possible reference solutions are:1 An optimal solution.2 A Greedy solution
sort items in decreasing order of value per unit of weightput every item in sequence with maximal frequency in theknapsack such that the capacity is not exceeded
Example: v = [6,8,3,1], w = [5,7,3,6], C = 13vw ≈ [1.2, 1.14, 1, 0.17]→ order: 1,2,3,4(2,0,0,0),C ′ = 3→ (2,0,0,0),C ′ = 3
additional slides
Reference Solutions
Possible reference solutions are:1 An optimal solution.2 A Greedy solution
sort items in decreasing order of value per unit of weightput every item in sequence with maximal frequency in theknapsack such that the capacity is not exceeded
Example: v = [6,8,3,1], w = [5,7,3,6], C = 13vw ≈ [1.2, 1.14, 1, 0.17]→ order: 1,2,3,4(2,0,0,0),C ′ = 3→ (2,0,0,0),C ′ = 3→ (2,0,1,0),C ′ = 0
additional slides
How often is the Greedy Solution Optimal?
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additional slides
Optimal Solution as Reference Solution
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additional slides
Greedy Solution as Reference Solution
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additional slides
Other Studied Optimization-Problems
Shortest Path ProblemBinary Knapsack ProblemMinimal Spanning TreesAssignment Problemsome theoretical problem types