A Parallel Bundle Method for Asynchronous Subspace Optimization in - PowerPoint PPT Presentation

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 c T x max ( P ) s.t. Ax = b ω ) T ∈ Ω := � x = ( x T 1 , . . . , x T v ∈ V Ω v , with V = { 1 , . . . , ω } and finite ground sets Ω v ⊂ R n v , v ∈ V , where for all v ∈ V , y ∈ R m the augmented subproblems c T v x v − ( A T y ) T max v x v ( P v ( y )) s.t x v ∈ Ω v 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 c T x max ( P ) s.t. Ax = b ω ) T ∈ Ω := � x = ( x T 1 , . . . , x T v ∈ V Ω v , with V = { 1 , . . . , ω } and finite ground sets Ω v ⊂ R n v , v ∈ V , where for all v ∈ V , y ∈ R m the augmented subproblems c T v x v − ( A T y ) T max v x v ( P v ( y )) s.t x v ∈ Ω v 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 c T x max ( P ) s.t. Ax = b ω ) T ∈ Ω := � x = ( x T 1 , . . . , x T v ∈ V Ω v , with V = { 1 , . . . , ω } and finite ground sets Ω v ⊂ R n v , v ∈ V , where for all v ∈ V , y ∈ R m the augmented subproblems c T v x v − ( A T y ) T max v x v ( P v ( y )) s.t x v ∈ Ω v 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 c T x max ( P ) s.t. Ax = b ω ) T ∈ Ω := � x = ( x T 1 , . . . , x T v ∈ V Ω v , with V = { 1 , . . . , ω } and finite ground sets Ω v ⊂ R n v , v ∈ V , where for all v ∈ V , y ∈ R m the augmented subproblems c T v x v − ( A T y ) T max v x v ( P v ( y )) s.t x v ∈ Ω v 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. Bhf 1 Bhf 2 Bhf 3 Bhf 4 Bhf 5 � � t=1 � � � � t=2 � � � � � � � � t=3 � � � � �� �� �� �� � � � � �� �� �� �� t=4 � � � � �� �� �� �� � � � � �� �� �� �� t=5 � � � � � � �� �� � � � � � � � � �� �� � � t=6 � � � � �� �� � � � � � � �� �� � � t=7 � � � � �� �� � � � � � � �� �� � �

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 . y ∈ R m f ( y ) min where f ( y ) := sup L ( x , y ) , x ∈ Ω L ( x , y ) := c T x + ( b − Ax ) T y = b T y + � ( c v − A T y ) T x v v ∈ V 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 . y ∈ R m f ( y ) min where f ( y ) := sup L ( x , y ) , x ∈ Ω L ( x , y ) := c T x + ( b − Ax ) T y = b T y + � ( c v − A T y ) T x v v ∈ V 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 } , • V j := { v ∈ V : A j , v � = 0 } , • V J := � j ∈ J V j . For most rows A j , • , j ∈ M , the sets V j 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 } , • V j := { v ∈ V : A j , v � = 0 } , • V J := � j ∈ J V j . For most rows A j , • , j ∈ M , the sets V j 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 } , • V j := { v ∈ V : A j , v � = 0 } , • V J := � j ∈ J V j . For most rows A j , • , j ∈ M , the sets V j 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