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Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

A Parallel Bundle Method for Asynchronous Subspace Optimization in Lagrangian Relaxation

Frank Fischer, Christoph Helmberg

Chemnitz University of Technology

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Outline

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Problem Setting

Consider structured optimization Problems of the form (P) max cTx s.t. Ax = b x = (xT

1 , . . . , xT ω )T ∈ Ω := v∈V Ωv,

with V = {1, . . . , ω} and finite ground sets Ωv ⊂ Rnv , v ∈ V , where for all v ∈ V , y ∈ Rm the augmented subproblems (Pv(y)) max cT

v xv − (ATy)T v xv

s.t xv ∈ Ωv can be solved by an oracle that returns

• the optimal value P∗

v (y),

• an optimal solution x∗

v (y),

• a subgradient g(x∗

v ) := b − Ax∗ v .

Note: Inequalities constraints Dx ≤ d can be handled as well.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Problem Setting

Consider structured optimization Problems of the form (P) max cTx s.t. Ax = b x = (xT

1 , . . . , xT ω )T ∈ Ω := v∈V Ωv,

with V = {1, . . . , ω} and finite ground sets Ωv ⊂ Rnv , v ∈ V , where for all v ∈ V , y ∈ Rm the augmented subproblems (Pv(y)) max cT

v xv − (ATy)T v xv

s.t xv ∈ Ωv can be solved by an oracle that returns

• the optimal value P∗

v (y),

• an optimal solution x∗

v (y),

• a subgradient g(x∗

v ) := b − Ax∗ v .

Note: Inequalities constraints Dx ≤ d can be handled as well.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Problem Setting

Consider structured optimization Problems of the form (P) max cTx s.t. Ax = b x = (xT

1 , . . . , xT ω )T ∈ Ω := v∈V Ωv,

with V = {1, . . . , ω} and finite ground sets Ωv ⊂ Rnv , v ∈ V , where for all v ∈ V , y ∈ Rm the augmented subproblems (Pv(y)) max cT

v xv − (ATy)T v xv

s.t xv ∈ Ωv can be solved by an oracle that returns

• the optimal value P∗

v (y),

• an optimal solution x∗

v (y),

• a subgradient g(x∗

v ) := b − Ax∗ v .

Note: Inequalities constraints Dx ≤ d can be handled as well.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Problem Setting

Consider structured optimization Problems of the form (P) max cTx s.t. Ax = b x = (xT

1 , . . . , xT ω )T ∈ Ω := v∈V Ωv,

with V = {1, . . . , ω} and finite ground sets Ωv ⊂ Rnv , v ∈ V , where for all v ∈ V , y ∈ Rm the augmented subproblems (Pv(y)) max cT

v xv − (ATy)T v xv

s.t xv ∈ Ωv can be solved by an oracle that returns

• the optimal value P∗

v (y),

• an optimal solution x∗

v (y),

• a subgradient g(x∗

v ) := b − Ax∗ v .

Note: Inequalities constraints Dx ≤ d can be handled as well.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Example: Train Timetabling Problem

Problem: Find conflict free timetable for a set of trains in an infrastructure network. Model:

• time-discretized networks for trains ⇒ shortest-path

subproblems,

• coupling constraint for station capacities and headway times.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Example: Train Timetabling Problem

Problem: Find conflict free timetable for a set of trains in an infrastructure network. Model:

• time-discretized networks for trains ⇒ shortest-path

subproblems,

• coupling constraint for station capacities and headway times.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Example: Train Timetabling Problem

Problem: Find conflict free timetable for a set of trains in an infrastructure network. Model:

• time-discretized networks for trains ⇒ shortest-path

subproblems,

• coupling constraint for station capacities and headway times.
• t=1

t=3 t=4 t=6 t=7 t=2 t=5 Bhf 1 Bhf 3 Bhf 4 Bhf 2 Bhf 5

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Lagrangian Dual/Lagrangian Relaxation

Typical approach: get upper bounds from the Lagrangian Dual by relaxation of coupling constraints Ax = b. min

y∈Rm f (y)

where f (y) := sup

x∈Ω

L(x, y), L(x, y) :=cTx + (b − Ax)Ty =bTy +

• v∈V

(cv − ATy)Txv which can be solved, e. g., by a subgradient or bundle method.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Lagrangian Dual/Lagrangian Relaxation

Typical approach: get upper bounds from the Lagrangian Dual by relaxation of coupling constraints Ax = b. min

y∈Rm f (y)

where f (y) := sup

x∈Ω

L(x, y), L(x, y) :=cTx + (b − Ax)Ty =bTy +

• v∈V

(cv − ATy)Txv which can be solved, e. g., by a subgradient or bundle method.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Loosely Coupled Problems

In many problems the subproblems are loosely coupled in the following sense: Let

• j ∈ M = {1, . . . , m},
• Vj := {v ∈ V : Aj,v = 0},
• VJ :=

j∈J Vj.

For most rows Aj,•, j ∈ M, the sets Vj are small subsets of V ,

• i. e., j couples only few subproblems.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Loosely Coupled Problems

In many problems the subproblems are loosely coupled in the following sense: Let

• j ∈ M = {1, . . . , m},
• Vj := {v ∈ V : Aj,v = 0},
• VJ :=

j∈J Vj.

For most rows Aj,•, j ∈ M, the sets Vj are small subsets of V ,

• i. e., j couples only few subproblems.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Loosely Coupled Problems

In many problems the subproblems are loosely coupled in the following sense: Let

• j ∈ M = {1, . . . , m},
• Vj := {v ∈ V : Aj,v = 0},
• VJ :=

j∈J Vj.

For most rows Aj,•, j ∈ M, the sets Vj are small subsets of V ,

• i. e., j couples only few subproblems.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Example: Train Timetabling Problem

Trains are in conflict if

• they use same resources (tracks, stations), i. e., close in space,
• use they at the same time, i. e., close in time.

Typical situation on large networks:

• many local short distances trains in relatively loosely coupled subnetworks,
• few long-distance trains connecting those local areas.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Example: Train Timetabling Problem

Trains are in conflict if

• they use same resources (tracks, stations), i. e., close in space,
• use they at the same time, i. e., close in time.

Typical situation on large networks:

• many local short distances trains in relatively loosely coupled subnetworks,
• few long-distance trains connecting those local areas.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Example: Train Timetabling Problem

Trains are in conflict if

• they use same resources (tracks, stations), i. e., close in space,
• use they at the same time, i. e., close in time.

Typical situation on large networks:

• many local short distances trains in relatively loosely coupled subnetworks,
• few long-distance trains connecting those local areas.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Example: Train Timetabling Problem

Trains are in conflict if

• they use same resources (tracks, stations), i. e., close in space,
• use they at the same time, i. e., close in time.

Typical situation on large networks:

• many local short distances trains in relatively loosely coupled subnetworks,
• few long-distance trains connecting those local areas.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Example: Train Timetabling Problem

Trains are in conflict if

• they use same resources (tracks, stations), i. e., close in space,
• use they at the same time, i. e., close in time.

Typical situation on large networks:

• many local short distances trains in relatively loosely coupled subnetworks,
• few long-distance trains connecting those local areas.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Outline

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Classical Bundle Method

(see, e. g., Hiriart-Urruty, Lemar´ echal, 1993) Iterative method: generates sequences ˆ yk ∈ Rm, and ¯ xk ∈ conv Ω: At each iteration:

• current center of stability ˆ

y,

• ˆ

Ω ⊂ conv Ω, compact,

• 1. form the model fˆ

Ω(y) := sup{L(x, y): x ∈ ˆ

Ω}

• 2. compute the next candidate

¯ y := argmin{fˆ

Ω(y) + u 2 y − ˆ

y2 : y ∈ Rm}, and primal aggregate ¯ x ∈ Argmax

x∈conv ˆ Ω

[cT x + (b − Ax)T ˆ y −

1 2u g(x)2]

• 3. if the expected progress

∆ := f (ˆ y) − fˆ

Ω(¯

y) = f (ˆ y) − f{¯

x}(¯

y) = f (ˆ y) − L(¯ x, ˆ y) + 1

u g(¯

x)2 is small, then STOP.

• 4. If actual progress f (ˆ

y) − f (¯ y) is good in comparison with ∆, then

• do a descent step, ˆ

y ← ¯ y,

• otherwise do a null step by improving the

model ˆ Ω in ¯ y.

• 5. go to 1.

f (ˆ y) ˆ y x2 x1

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Classical Bundle Method

(see, e. g., Hiriart-Urruty, Lemar´ echal, 1993) Iterative method: generates sequences ˆ yk ∈ Rm, and ¯ xk ∈ conv Ω: At each iteration:

• current center of stability ˆ

y,

• ˆ

Ω ⊂ conv Ω, compact,

• 1. form the model fˆ

Ω(y) := sup{L(x, y): x ∈ ˆ

Ω}

• 2. compute the next candidate

¯ y := argmin{fˆ

Ω(y) + u 2 y − ˆ

y2 : y ∈ Rm}, and primal aggregate ¯ x ∈ Argmax

x∈conv ˆ Ω

[cT x + (b − Ax)T ˆ y −

1 2u g(x)2]

• 3. if the expected progress

∆ := f (ˆ y) − fˆ

Ω(¯

y) = f (ˆ y) − f{¯

x}(¯

y) = f (ˆ y) − L(¯ x, ˆ y) + 1

u g(¯

x)2 is small, then STOP.

• 4. If actual progress f (ˆ

y) − f (¯ y) is good in comparison with ∆, then

• do a descent step, ˆ

y ← ¯ y,

• otherwise do a null step by improving the

model ˆ Ω in ¯ y.

• 5. go to 1.

f (ˆ y) ˆ y x2 x1

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Classical Bundle Method

(see, e. g., Hiriart-Urruty, Lemar´ echal, 1993) Iterative method: generates sequences ˆ yk ∈ Rm, and ¯ xk ∈ conv Ω: At each iteration:

• current center of stability ˆ

y,

• ˆ

Ω ⊂ conv Ω, compact,

• 1. form the model fˆ

Ω(y) := sup{L(x, y): x ∈ ˆ

Ω}

• 2. compute the next candidate

¯ y := argmin{fˆ

Ω(y) + u 2 y − ˆ

y2 : y ∈ Rm}, and primal aggregate ¯ x ∈ Argmax

x∈conv ˆ Ω

[cT x + (b − Ax)T ˆ y −

1 2u g(x)2]

• 3. if the expected progress

∆ := f (ˆ y) − fˆ

Ω(¯

y) = f (ˆ y) − f{¯

x}(¯

y) = f (ˆ y) − L(¯ x, ˆ y) + 1

u g(¯

x)2 is small, then STOP.

• 4. If actual progress f (ˆ

y) − f (¯ y) is good in comparison with ∆, then

• do a descent step, ˆ

y ← ¯ y,

• otherwise do a null step by improving the

model ˆ Ω in ¯ y.

• 5. go to 1.

x2 x1 f (ˆ y) ˆ y

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Classical Bundle Method

(see, e. g., Hiriart-Urruty, Lemar´ echal, 1993) Iterative method: generates sequences ˆ yk ∈ Rm, and ¯ xk ∈ conv Ω: At each iteration:

• current center of stability ˆ

y,

• ˆ

Ω ⊂ conv Ω, compact,

• 1. form the model fˆ

Ω(y) := sup{L(x, y): x ∈ ˆ

Ω}

• 2. compute the next candidate

¯ y := argmin{fˆ

Ω(y) + u 2 y − ˆ

y2 : y ∈ Rm}, and primal aggregate ¯ x ∈ Argmax

x∈conv ˆ Ω

[cT x + (b − Ax)T ˆ y −

1 2u g(x)2]

• 3. if the expected progress

∆ := f (ˆ y) − fˆ

Ω(¯

y) = f (ˆ y) − f{¯

x}(¯

y) = f (ˆ y) − L(¯ x, ˆ y) + 1

u g(¯

x)2 is small, then STOP.

• 4. If actual progress f (ˆ

y) − f (¯ y) is good in comparison with ∆, then

• do a descent step, ˆ

y ← ¯ y,

• otherwise do a null step by improving the

model ˆ Ω in ¯ y.

• 5. go to 1.

¯ x x2 x1 f (ˆ y) ¯ y ˆ y

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Classical Bundle Method

(see, e. g., Hiriart-Urruty, Lemar´ echal, 1993) Iterative method: generates sequences ˆ yk ∈ Rm, and ¯ xk ∈ conv Ω: At each iteration:

• current center of stability ˆ

y,

• ˆ

Ω ⊂ conv Ω, compact,

• 1. form the model fˆ

Ω(y) := sup{L(x, y): x ∈ ˆ

Ω}

• 2. compute the next candidate

¯ y := argmin{fˆ

Ω(y) + u 2 y − ˆ

y2 : y ∈ Rm}, and primal aggregate ¯ x ∈ Argmax

x∈conv ˆ Ω

[cT x + (b − Ax)T ˆ y −

1 2u g(x)2]

• 3. if the expected progress

∆ := f (ˆ y) − fˆ

Ω(¯

y) = f (ˆ y) − f{¯

x}(¯

y) = f (ˆ y) − L(¯ x, ˆ y) + 1

u g(¯

x)2 is small, then STOP.

• 4. If actual progress f (ˆ

y) − f (¯ y) is good in comparison with ∆, then

• do a descent step, ˆ

y ← ¯ y,

• otherwise do a null step by improving the

model ˆ Ω in ¯ y.

• 5. go to 1.

¯ x x2 x1 f (ˆ y) fˆ

Ω(¯

y) f (¯ y) ∆ ¯ y ˆ y

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Classical Bundle Method

(see, e. g., Hiriart-Urruty, Lemar´ echal, 1993) Iterative method: generates sequences ˆ yk ∈ Rm, and ¯ xk ∈ conv Ω: At each iteration:

• current center of stability ˆ

y,

• ˆ

Ω ⊂ conv Ω, compact,

• 1. form the model fˆ

Ω(y) := sup{L(x, y): x ∈ ˆ

Ω}

• 2. compute the next candidate

¯ y := argmin{fˆ

Ω(y) + u 2 y − ˆ

y2 : y ∈ Rm}, and primal aggregate ¯ x ∈ Argmax

x∈conv ˆ Ω

[cT x + (b − Ax)T ˆ y −

1 2u g(x)2]

• 3. if the expected progress

∆ := f (ˆ y) − fˆ

Ω(¯

y) = f (ˆ y) − f{¯

x}(¯

y) = f (ˆ y) − L(¯ x, ˆ y) + 1

u g(¯

x)2 is small, then STOP.

• 4. If actual progress f (ˆ

y) − f (¯ y) is good in comparison with ∆, then

• do a descent step, ˆ

y ← ¯ y,

• otherwise do a null step by improving the

model ˆ Ω in ¯ y.

• 5. go to 1.

¯ x x2 x1 f (ˆ y) fˆ

Ω(¯

y) f (¯ y) ∆ ¯ y ˆ y

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Classical Bundle Method

In short: In each iteration, the bundle method

• manages the current center of stability ˆ

y ∈ Rm,

• updates the global primal aggregate ¯

x ∈ conv Ω, And overall

• drives the expected progress

∆ = f (ˆ y) − fˆ

Ω(¯

y) = f (ˆ y) − f{¯

x}(¯

y) = f (ˆ y) − L(¯ x, ˆ y) + 1

ug(¯

x)2, to zero,

• f (ˆ

y) → infy∈Rm f (y), g(¯ x) → 0.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Classical Bundle Method

In short: In each iteration, the bundle method

• manages the current center of stability ˆ

y ∈ Rm,

• updates the global primal aggregate ¯

x ∈ conv Ω, And overall

• drives the expected progress

∆ = f (ˆ y) − fˆ

Ω(¯

y) = f (ˆ y) − f{¯

x}(¯

y) = f (ˆ y) − L(¯ x, ˆ y) + 1

ug(¯

x)2, to zero,

• f (ˆ

y) → infy∈Rm f (y), g(¯ x) → 0.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Classical Bundle Method

In short: In each iteration, the bundle method

• manages the current center of stability ˆ

y ∈ Rm,

• updates the global primal aggregate ¯

x ∈ conv Ω, And overall

• drives the expected progress

∆ = f (ˆ y) − fˆ

Ω(¯

y) = f (ˆ y) − f{¯

x}(¯

y) = f (ˆ y) − L(¯ x, ˆ y) + 1

ug(¯

x)2, to zero,

• f (ˆ

y) → infy∈Rm f (y), g(¯ x) → 0.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Classical Bundle Method

In short: In each iteration, the bundle method

• manages the current center of stability ˆ

y ∈ Rm,

• updates the global primal aggregate ¯

x ∈ conv Ω, And overall

• drives the expected progress

∆ = f (ˆ y) − fˆ

Ω(¯

y) = f (ˆ y) − f{¯

x}(¯

y) = f (ˆ y) − L(¯ x, ˆ y) + 1

ug(¯

x)2, to zero,

• f (ˆ

y) → infy∈Rm f (y), g(¯ x) → 0.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Outline

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Optimizing over Subspaces

The expected global progress is ∆ = f (ˆ y) − L(¯ x, ˆ y) + 1

u g(¯

x)2 =

• v∈V

[P∗

v (ˆ

y) − cT

v ¯

xv + ˆ y TA•,v ¯ xv] + 1

u g(¯

x)2. Let J ⊂ M be a subspace, ¯ J := {i ∈ M \ J : Vi ∩ VJ = ∅} (¯ J may couple VJ with V \ VJ). Then ∆J :=

• v∈VJ

[P∗

v (ˆ

y) − cT

v ¯

xv + ˆ y TA•,v ¯ xv] + 1

u g(¯

x)J2, ¯ ∆J :=

• v∈V \VJ

[P∗

v (ˆ

y) − cT

v ¯

xv + ˆ y TA•,v ¯ xv] + 1

u g(¯

x)M\(J∪¯

J)2,

δ¯

J := 1 u g(¯

x)¯

J2,

∆ = ∆J + ¯ ∆J + δ¯

J.

• ∆J is the expected progress on subspace J,
• ¯

∆J is not influenced by J or VJ,

• δ¯

J is depends on ¯

J, therefore on VJ and V \ VJ.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Optimizing over Subspaces

The expected global progress is ∆ = f (ˆ y) − L(¯ x, ˆ y) + 1

u g(¯

x)2 =

• v∈V

[P∗

v (ˆ

y) − cT

v ¯

xv + ˆ y TA•,v ¯ xv] + 1

u g(¯

x)2. Let J ⊂ M be a subspace, ¯ J := {i ∈ M \ J : Vi ∩ VJ = ∅} (¯ J may couple VJ with V \ VJ). Then ∆J :=

• v∈VJ

[P∗

v (ˆ

y) − cT

v ¯

xv + ˆ y TA•,v ¯ xv] + 1

u g(¯

x)J2, ¯ ∆J :=

• v∈V \VJ

[P∗

v (ˆ

y) − cT

v ¯

xv + ˆ y TA•,v ¯ xv] + 1

u g(¯

x)M\(J∪¯

J)2,

δ¯

J := 1 u g(¯

x)¯

J2,

∆ = ∆J + ¯ ∆J + δ¯

J.

• ∆J is the expected progress on subspace J,
• ¯

∆J is not influenced by J or VJ,

• δ¯

J is depends on ¯

J, therefore on VJ and V \ VJ.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Optimizing over Subspaces

The expected global progress is ∆ = f (ˆ y) − L(¯ x, ˆ y) + 1

u g(¯

x)2 =

• v∈V

[P∗

v (ˆ

y) − cT

v ¯

xv + ˆ y TA•,v ¯ xv] + 1

u g(¯

x)2. Let J ⊂ M be a subspace, ¯ J := {i ∈ M \ J : Vi ∩ VJ = ∅} (¯ J may couple VJ with V \ VJ). Then ∆J :=

• v∈VJ

[P∗

v (ˆ

y) − cT

v ¯

xv + ˆ y TA•,v ¯ xv] + 1

u g(¯

x)J2, ¯ ∆J :=

• v∈V \VJ

[P∗

v (ˆ

y) − cT

v ¯

xv + ˆ y TA•,v ¯ xv] + 1

u g(¯

x)M\(J∪¯

J)2,

δ¯

J := 1 u g(¯

x)¯

J2,

∆ = ∆J + ¯ ∆J + δ¯

J.

• ∆J is the expected progress on subspace J,
• ¯

∆J is not influenced by J or VJ,

• δ¯

J is depends on ¯

J, therefore on VJ and V \ VJ.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Optimizing over Subspaces

The expected global progress is ∆ = f (ˆ y) − L(¯ x, ˆ y) + 1

u g(¯

x)2 =

• v∈V

[P∗

v (ˆ

y) − cT

v ¯

xv + ˆ y TA•,v ¯ xv] + 1

u g(¯

x)2. Let J ⊂ M be a subspace, ¯ J := {i ∈ M \ J : Vi ∩ VJ = ∅} (¯ J may couple VJ with V \ VJ). Then ∆J :=

• v∈VJ

[P∗

v (ˆ

y) − cT

v ¯

xv + ˆ y TA•,v ¯ xv] + 1

u g(¯

x)J2, ¯ ∆J :=

• v∈V \VJ

[P∗

v (ˆ

y) − cT

v ¯

xv + ˆ y TA•,v ¯ xv] + 1

u g(¯

x)M\(J∪¯

J)2,

δ¯

J := 1 u g(¯

x)¯

J2,

∆ = ∆J + ¯ ∆J + δ¯

J.

• ∆J is the expected progress on subspace J,
• ¯

∆J is not influenced by J or VJ,

• δ¯

J is depends on ¯

J, therefore on VJ and V \ VJ.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Optimizing over Subspaces

V \ VJ VJ J ¯ J M \ (J ∪ ¯ J) A =

• optimizing over J and VJ,
• ¯

J may not be changed,

• M \ (J ∪ ¯

J) and V \ VJ may be changed.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Optimizing over Subspaces

Idea: work only on subspace with large expected progress.

• Choose J ⊂ M so that ∆J ≥ τ1∆, τ1 ∈
• 0, 1

2 min

• 1

|M|, 1 |V |

• ,
• Optimize over J and VJ until ∆J is decreased enough, should

also decrease ∆. And

• during this process, ¯

xVJ and ˆ yJ must not be changed outside

• f this process,
• ˆ

yM\(J∪¯

J) and ¯

xV \VJ may be changed somewhere else.

Observation

When optimizing over J and VJ, we can simultaneously optimize

• ver other subspaces J′ ⊆ M \ (J ∪ ¯

J), too, without influencing ∆J.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Optimizing over Subspaces

Idea: work only on subspace with large expected progress.

• Choose J ⊂ M so that ∆J ≥ τ1∆, τ1 ∈
• 0, 1

2 min

• 1

|M|, 1 |V |

• ,
• Optimize over J and VJ until ∆J is decreased enough, should

also decrease ∆. And

• during this process, ¯

xVJ and ˆ yJ must not be changed outside

• f this process,
• ˆ

yM\(J∪¯

J) and ¯

xV \VJ may be changed somewhere else.

Observation

When optimizing over J and VJ, we can simultaneously optimize

• ver other subspaces J′ ⊆ M \ (J ∪ ¯

J), too, without influencing ∆J.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Optimizing over Subspaces

Idea: work only on subspace with large expected progress.

• Choose J ⊂ M so that ∆J ≥ τ1∆, τ1 ∈
• 0, 1

2 min

• 1

|M|, 1 |V |

• ,
• Optimize over J and VJ until ∆J is decreased enough, should

also decrease ∆. And

• during this process, ¯

xVJ and ˆ yJ must not be changed outside

• f this process,
• ˆ

yM\(J∪¯

J) and ¯

xV \VJ may be changed somewhere else.

Observation

When optimizing over J and VJ, we can simultaneously optimize

• ver other subspaces J′ ⊆ M \ (J ∪ ¯

J), too, without influencing ∆J.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Optimizing over Subspaces

Idea: work only on subspace with large expected progress.

• Choose J ⊂ M so that ∆J ≥ τ1∆, τ1 ∈
• 0, 1

2 min

• 1

|M|, 1 |V |

• ,
• Optimize over J and VJ until ∆J is decreased enough, should

also decrease ∆. And

• during this process, ¯

xVJ and ˆ yJ must not be changed outside

• f this process,
• ˆ

yM\(J∪¯

J) and ¯

xV \VJ may be changed somewhere else.

Observation

When optimizing over J and VJ, we can simultaneously optimize

• ver other subspaces J′ ⊆ M \ (J ∪ ¯

J), too, without influencing ∆J.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Optimizing over Subspaces

Idea: work only on subspace with large expected progress.

• Choose J ⊂ M so that ∆J ≥ τ1∆, τ1 ∈
• 0, 1

2 min

• 1

|M|, 1 |V |

• ,
• Optimize over J and VJ until ∆J is decreased enough, should

also decrease ∆. And

• during this process, ¯

xVJ and ˆ yJ must not be changed outside

• f this process,
• ˆ

yM\(J∪¯

J) and ¯

xV \VJ may be changed somewhere else.

Observation

When optimizing over J and VJ, we can simultaneously optimize

• ver other subspaces J′ ⊆ M \ (J ∪ ¯

J), too, without influencing ∆J.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Conceptual Algorithm

Global data:

• center of stability ˆ

y ∈ Rm,

• primal aggregate ¯

x ∈ Ω,

• set of blocked subproblems B ⊆ V .
• 1. (Initialization): ˆ

y ← 0, ¯ x ∈ Ω, B ← ∅.

• 2. Each process does:

(a) (Subspace selection):

• compute ∆ = f (ˆ

y) − L(¯ x, ˆ y) + 1

u g(¯

x)2,

• select J ⊂ M with VJ ∩ B = ∅, ∆J ≥ τ1∆,
• B ← B ∪ VJ,

(b) (Solve subspace problem): ∆0 ← ∆, solve minyJ ∈R|J| fJ(yJ) where fJ(y) = sup{LJ(xVJ , yJ): xVJ ∈ ΩVJ } LJ(xVJ , yJ) = bT

J yJ +

• v∈VJ

[ˆ cT

v xv − y T J AJ,vxv],

ˆ cv = cv − AT

J,vyJ.

until ∆J ≤ τ2∆0

J → ˆ

y σ

J , ¯

xσ

VJ ,

(c) (Update global data): ˆ yJ ← ˆ y σ

J , ¯

xσ ← ¯ xσ

VJ , B ← B \ VJ.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Conceptual Algorithm

• any number of parallel processes may run at the same time,
• at most one process is allowed to interact with global data

at the same time,

• the order in which processes start and stop is undefined, i. e.,

the subspace problems are solve asynchronously,

• each process drives ∆J on subspace J to zero → should also

drive ∆ to zero.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Conceptual Algorithm

• any number of parallel processes may run at the same time,
• at most one process is allowed to interact with global data

at the same time,

• the order in which processes start and stop is undefined, i. e.,

the subspace problems are solve asynchronously,

• each process drives ∆J on subspace J to zero → should also

drive ∆ to zero.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Conceptual Algorithm

• any number of parallel processes may run at the same time,
• at most one process is allowed to interact with global data

at the same time,

• the order in which processes start and stop is undefined, i. e.,

the subspace problems are solve asynchronously,

• each process drives ∆J on subspace J to zero → should also

drive ∆ to zero.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Conceptual Algorithm

• any number of parallel processes may run at the same time,
• at most one process is allowed to interact with global data

at the same time,

• the order in which processes start and stop is undefined, i. e.,

the subspace problems are solve asynchronously,

• each process drives ∆J on subspace J to zero → should also

drive ∆ to zero.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Subspace Dependencies

Parallel processes may influence each other: assume the following situation

• process A starts with expected global progress

∆0 = ∆0

J + ¯

∆0

J + δ0 ¯ J,

• in the meantime, process B changes the global data,

∆1 = ∆1

J + ¯

∆1

J + δ1 ¯ J,

• A writes its solution back with

∆2 = ∆2

J + ¯

∆2

J + δ2 ¯ J.

We need ∆2 ≤ α∆1 for some constant α ∈ [0, 1) to get global progress. We know ∆0

J = ∆1 J and ¯

∆1

J = ¯

∆2

J, but in general δ0 ¯ J = δ1 ¯ J = δ2 ¯ J:

∆1 − ∆2 = (∆1

J − ∆2 J) + ( ¯

∆1

J − ¯

∆2

J) + (δ1 ¯ J − δ2 ¯ J)

= (∆0

J − ∆2 J) + (δ1 ¯ J − δ2 ¯ J)

≥ (1 − τ2)∆0

J + (δ1 ¯ J − δ2 ¯ J)

• ???

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Subspace Dependencies

Parallel processes may influence each other: assume the following situation

• process A starts with expected global progress

∆0 = ∆0

J + ¯

∆0

J + δ0 ¯ J,

• in the meantime, process B changes the global data,

∆1 = ∆1

J + ¯

∆1

J + δ1 ¯ J,

• A writes its solution back with

∆2 = ∆2

J + ¯

∆2

J + δ2 ¯ J.

We need ∆2 ≤ α∆1 for some constant α ∈ [0, 1) to get global progress. We know ∆0

J = ∆1 J and ¯

∆1

J = ¯

∆2

J, but in general δ0 ¯ J = δ1 ¯ J = δ2 ¯ J:

∆1 − ∆2 = (∆1

J − ∆2 J) + ( ¯

∆1

J − ¯

∆2

J) + (δ1 ¯ J − δ2 ¯ J)

= (∆0

J − ∆2 J) + (δ1 ¯ J − δ2 ¯ J)

≥ (1 − τ2)∆0

J + (δ1 ¯ J − δ2 ¯ J)

• ???

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Subspace Dependencies

Parallel processes may influence each other: assume the following situation

• process A starts with expected global progress

∆0 = ∆0

J + ¯

∆0

J + δ0 ¯ J,

• in the meantime, process B changes the global data,

∆1 = ∆1

J + ¯

∆1

J + δ1 ¯ J,

• A writes its solution back with

∆2 = ∆2

J + ¯

∆2

J + δ2 ¯ J.

We need ∆2 ≤ α∆1 for some constant α ∈ [0, 1) to get global progress. We know ∆0

J = ∆1 J and ¯

∆1

J = ¯

∆2

J, but in general δ0 ¯ J = δ1 ¯ J = δ2 ¯ J:

∆1 − ∆2 = (∆1

J − ∆2 J) + ( ¯

∆1

J − ¯

∆2

J) + (δ1 ¯ J − δ2 ¯ J)

= (∆0

J − ∆2 J) + (δ1 ¯ J − δ2 ¯ J)

≥ (1 − τ2)∆0

J + (δ1 ¯ J − δ2 ¯ J)

• ???

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Subspace Dependencies

Parallel processes may influence each other: assume the following situation

• process A starts with expected global progress

∆0 = ∆0

J + ¯

∆0

J + δ0 ¯ J,

• in the meantime, process B changes the global data,

∆1 = ∆1

J + ¯

∆1

J + δ1 ¯ J,

• A writes its solution back with

∆2 = ∆2

J + ¯

∆2

J + δ2 ¯ J.

We need ∆2 ≤ α∆1 for some constant α ∈ [0, 1) to get global progress. We know ∆0

J = ∆1 J and ¯

∆1

J = ¯

∆2

J, but in general δ0 ¯ J = δ1 ¯ J = δ2 ¯ J:

∆1 − ∆2 = (∆1

J − ∆2 J) + ( ¯

∆1

J − ¯

∆2

J) + (δ1 ¯ J − δ2 ¯ J)

= (∆0

J − ∆2 J) + (δ1 ¯ J − δ2 ¯ J)

≥ (1 − τ2)∆0

J + (δ1 ¯ J − δ2 ¯ J)

• ???

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Subspace Dependencies

Parallel processes may influence each other: assume the following situation

• process A starts with expected global progress

∆0 = ∆0

J + ¯

∆0

J + δ0 ¯ J,

• in the meantime, process B changes the global data,

∆1 = ∆1

J + ¯

∆1

J + δ1 ¯ J,

• A writes its solution back with

∆2 = ∆2

J + ¯

∆2

J + δ2 ¯ J.

We need ∆2 ≤ α∆1 for some constant α ∈ [0, 1) to get global progress. We know ∆0

J = ∆1 J and ¯

∆1

J = ¯

∆2

J, but in general δ0 ¯ J = δ1 ¯ J = δ2 ¯ J:

∆1 − ∆2 = (∆1

J − ∆2 J) + ( ¯

∆1

J − ¯

∆2

J) + (δ1 ¯ J − δ2 ¯ J)

= (∆0

J − ∆2 J) + (δ1 ¯ J − δ2 ¯ J)

≥ (1 − τ2)∆0

J + (δ1 ¯ J − δ2 ¯ J)

• ???

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Subspace Dependencies

Parallel processes may influence each other: assume the following situation

• process A starts with expected global progress

∆0 = ∆0

J + ¯

∆0

J + δ0 ¯ J,

• in the meantime, process B changes the global data,

∆1 = ∆1

J + ¯

∆1

J + δ1 ¯ J,

• A writes its solution back with

∆2 = ∆2

J + ¯

∆2

J + δ2 ¯ J.

We need ∆2 ≤ α∆1 for some constant α ∈ [0, 1) to get global progress. We know ∆0

J = ∆1 J and ¯

∆1

J = ¯

∆2

J, but in general δ0 ¯ J = δ1 ¯ J = δ2 ¯ J:

∆1 − ∆2 = (∆1

J − ∆2 J) + ( ¯

∆1

J − ¯

∆2

J) + (δ1 ¯ J − δ2 ¯ J)

= (∆0

J − ∆2 J) + (δ1 ¯ J − δ2 ¯ J)

≥ (1 − τ2)∆0

J + (δ1 ¯ J − δ2 ¯ J)

• ???

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Subspace Dependencies

Let term d := δ2

¯ J − δ1 ¯ J represents interdependencies between

subspace J and M \ J.

• if d is large compared with ∆0

J, the gain in progress on J may

be defeated by the loss caused by d,

• have to ensure that this does not happen too often.

Idea: Introduce directed dependency graph D = (M, E) with (i, j) ∈ E if and only if a subspace including i should also include j.

• start with empty graph D = (M, ∅),
• in subspace selection ensure ∀ j ∈ J, ∀ (j, k) ∈ E : k ∈ J,
• when a process updates the global data,
• check if δ2

¯ J − δ1 ¯ J > τ3∆0 J,

• if so, add some (j, j∗) to E with j ∈ J,

j∗ ∈ Argmax{δ2

j − δ1 j : j ∈ ¯

J}.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Subspace Dependencies

Let term d := δ2

¯ J − δ1 ¯ J represents interdependencies between

subspace J and M \ J.

• if d is large compared with ∆0

J, the gain in progress on J may

be defeated by the loss caused by d,

• have to ensure that this does not happen too often.

Idea: Introduce directed dependency graph D = (M, E) with (i, j) ∈ E if and only if a subspace including i should also include j.

• start with empty graph D = (M, ∅),
• in subspace selection ensure ∀ j ∈ J, ∀ (j, k) ∈ E : k ∈ J,
• when a process updates the global data,
• check if δ2

¯ J − δ1 ¯ J > τ3∆0 J,

• if so, add some (j, j∗) to E with j ∈ J,

j∗ ∈ Argmax{δ2

j − δ1 j : j ∈ ¯

J}.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Subspace Dependencies

Let term d := δ2

¯ J − δ1 ¯ J represents interdependencies between

subspace J and M \ J.

• if d is large compared with ∆0

J, the gain in progress on J may

be defeated by the loss caused by d,

• have to ensure that this does not happen too often.

Idea: Introduce directed dependency graph D = (M, E) with (i, j) ∈ E if and only if a subspace including i should also include j.

• start with empty graph D = (M, ∅),
• in subspace selection ensure ∀ j ∈ J, ∀ (j, k) ∈ E : k ∈ J,
• when a process updates the global data,
• check if δ2

¯ J − δ1 ¯ J > τ3∆0 J,

• if so, add some (j, j∗) to E with j ∈ J,

j∗ ∈ Argmax{δ2

j − δ1 j : j ∈ ¯

J}.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Subspace Dependencies

Let term d := δ2

¯ J − δ1 ¯ J represents interdependencies between

subspace J and M \ J.

• if d is large compared with ∆0

J, the gain in progress on J may

be defeated by the loss caused by d,

• have to ensure that this does not happen too often.

Idea: Introduce directed dependency graph D = (M, E) with (i, j) ∈ E if and only if a subspace including i should also include j.

• start with empty graph D = (M, ∅),
• in subspace selection ensure ∀ j ∈ J, ∀ (j, k) ∈ E : k ∈ J,
• when a process updates the global data,
• check if δ2

¯ J − δ1 ¯ J > τ3∆0 J,

• if so, add some (j, j∗) to E with j ∈ J,

j∗ ∈ Argmax{δ2

j − δ1 j : j ∈ ¯

J}.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

The Parallel Bundle Method

Global data:

• center of stability ˆ

y ∈ Rm,

• primal aggregate ¯

x ∈ Ω,

• set of blocked subproblems B ⊆ V .
• dependency graph D = (M, E).
• 1. (Initialization): ˆ

y ← 0, ¯ x ∈ Ω, M ← ∅, E ← ∅.

• 2. Each process does:

(a) (Subspace selection):

• compute ∆ = f (ˆ

y) − L(¯ x, ˆ y) + 1

u g(¯

x)2,

• select J ⊂ M with VJ ∩ B = ∅ w. r. t. D, ∆J ≥ τ1∆,
• B ← B ∪ VJ,

(b) (Solve subspace problem): ∆0 ← ∆, solve minyJ ∈R|J| fJ(yJ) until ∆J ≤ τ2∆0

J → ˆ

y σ, ¯ xσ

VJ ,

(c) (Update global data): ˆ yJ ← ˆ y σ

J , ¯

xσ ←ˆ ¯ xσ

VJ , B ← B \ VJ,

update D if required.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

The Parallel Bundle Method: Convergence

Convergence guaranteed, because:

• M is finite,
• D = (M, E) is only extended ⇒ will only change finitely often,
• if D does not change, the subspace progress always implies

global progress. Note:

• it may happen that E = M × M if the algorithm runs long

enough,

• in this case the parallel bundle method behaves like a classical

bundle method.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

The Parallel Bundle Method: Convergence

Convergence guaranteed, because:

• M is finite,
• D = (M, E) is only extended ⇒ will only change finitely often,
• if D does not change, the subspace progress always implies

global progress. Note:

• it may happen that E = M × M if the algorithm runs long

enough,

• in this case the parallel bundle method behaves like a classical

bundle method.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Extension: Stronger Coupled Problems

Can be extended to stronger coupled problems. Till now:

• Vj should be small for most j ∈ M.

Now:

• ¯

Vj := {v ∈ V : Aj,v ¯ xv = 0 for all former global aggregates ¯ x } (v ∈ V has not interfere with j ∈ J before),

• when selecting J ⊂ M, optimize only over ¯

Vj,

• needs dependency detection between j and Vj.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Extension: Stronger Coupled Problems

Can be extended to stronger coupled problems. Till now:

• Vj should be small for most j ∈ M.

Now:

• ¯

Vj := {v ∈ V : Aj,v ¯ xv = 0 for all former global aggregates ¯ x } (v ∈ V has not interfere with j ∈ J before),

• when selecting J ⊂ M, optimize only over ¯

Vj,

• needs dependency detection between j and Vj.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Outline

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Test-instances

Generated instances of TTP:

• n × n lattice,
• k × k sublattices,
• both with random routes,
• nl trains per sublattice (local trains),
• ng trains in full lattice (global trains),
• simple cost function penalizing waiting.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Test-instances

Generated instances of TTP:

• n × n lattice,
• k × k sublattices,
• both with random routes,
• nl trains per sublattice (local trains),
• ng trains in full lattice (global trains),
• simple cost function penalizing waiting.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Test-instances

Generated instances of TTP:

• n × n lattice,
• k × k sublattices,
• both with random routes,
• nl trains per sublattice (local trains),
• ng trains in full lattice (global trains),
• simple cost function penalizing waiting.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Test-instances

Generated instances of TTP:

• n × n lattice,
• k × k sublattices,
• both with random routes,
• nl trains per sublattice (local trains),
• ng trains in full lattice (global trains),
• simple cost function penalizing waiting.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Tests

• solve with classical bundle method and parallel bundle

method,

• use number of subproblem evaluations as performance

measure (current parallel implementation not very efficient),

• stop algorithm when subgradient norm is smaller than 10−3,
• use final accuracy level
• f (ˆ

y) − L(¯ x, ¯ y) f (ˆ y)

• as quality measure of approximate primal solution.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Numerical Results

20 40 60 80 100 120 140 50000 100000 150000 200000 250000 300000 # instances # of evaluations single parallel

Number of instances solved within the given number of evaluations.

20 40 60 80 100 120 140 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 # instances relative accuracy single parallel

Number of instances solved within the given relative accuracy.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Outline

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Next steps

• efficient implementation (parallel hardware),
• test on real world instances,
• allow deletion of dependencies,
• weaker assumptions on problem structure.

Problem Setting Classical Bundle Method Parallel Bundle Method Numerical Tests Outlook

Thank you for your attention.