DECOMP: A Framework for Decomposition in Integer …DECOMP: A Framework for Decomposition in Integer...

Decomposition MethodsIntegrated Decomposition Methods

DECOMP Framework

DECOMP: A Framework for Decomposition in Integer Programming

Matthew Galati1 Ted Ralphs2

1SAS Institute – Analytical Solutions – Operations Research and Development

2Lehigh University – Department of Industrial and Systems Engineering

INFORMS 2006Computational Optimization and Software

Galati, Ralphs DECOMP: A Framework for Decomposition in IP

DECOMP Framework

Outline

1 Decomposition Methods

2 Integrated Decomposition Methods

3 DECOMP Framework

DECOMP Framework

Outline

3 DECOMP Framework

DECOMP Framework

Preliminaries

Consider the following integer linear program (ILP):

zIP = minx∈F

{c⊤x} = minx∈P

{c⊤x} = minx∈Zn

{c⊤x | Ax ≥ b}

F = {x ∈ Zn | A′x ≥ b′, A′′x ≥ b′′}

F ′ = {x ∈ Zn | A′x ≥ b′}

Q = {x ∈ Rn | A′x ≥ b′, A′′x ≥ b′′}

Q′ = {x ∈ Rn | A′x ≥ b′}

Q′′ = {x ∈ Rn | A′′x ≥ b′′}

Q′′

(2,1)P

Denote P = conv(F) and P ′ = conv(F ′).

OPT (c, X): Subroutine returns x ∈ X that minimizes c⊤x.

SEP (x, X): Subroutine returns (a, β) which separates x from X (if exists).

DECOMP Framework

Preliminaries

Assumption:OPT (c,P) and SEP (x,P) are “hard”.

OPT (c,P′) and SEP (x,P′) are “easy”.

Q′′ can be represented explicitly (description has polynomial size).

P′ must be represented implicitly (description has exponential size).

Example - Traveling Salesman Problem

x(δ({u})) = 2 ∀u ∈ Vx(E(S)) ≤ |S| − 1 ∀S ⊂ V, 3 ≤ |S| ≤ |V | − 1xe ∈ {0, 1} ∀e ∈ E

One classical decomposition of TSP is to look for a spanning subgraph with |V | edges (P ′

= 1-Tree) that satisfies the 2-degree constraints (Q′′).

DECOMP Framework

Preliminaries

Assumption:OPT (c,P) and SEP (x,P) are “hard”.

OPT (c,P′) and SEP (x,P′) are “easy”.

Q′′ can be represented explicitly (description has polynomial size).

P′ must be represented implicitly (description has exponential size).

Example - Traveling Salesman Problem

One classical decomposition of TSP is to look for a spanning subgraph with |V | edges (P ′

= 1-Tree) that satisfies the 2-degree constraints (Q′′).

DECOMP Framework

Bounding

Compute a lower bound on zIP by building an approximation to P.

The most common approach is to use the LP relaxation.

zLP = minx∈Q

{c⊤x} = minx∈Rn

{c⊤x | A′x ≥ b′, A′′x ≥ b′′}

Decomposition methods attempt to improve on this bound byutilizing the fact that OPT (c,P ′) or SEP (x,P ′) is easy.

zD = minx∈P′

{c⊤x | A′′x ≥ b′′} = minx∈P′∩Q′′

{c⊤x} ≥ zLP

P ′ is represented implicitly through solution of a subproblem.

Decomposition Methods

Cutting Plane Method (Outer Method)

Dantzig-Wolfe Method / Lagrangian Method (Inner Methods)

Q′′

Q′ ∩ Q′′

DECOMP Framework

Bounding

Compute a lower bound on zIP by building an approximation to P.

The most common approach is to use the LP relaxation.

zLP = minx∈Q

{c⊤x} = minx∈Rn

{c⊤x | A′x ≥ b′, A′′x ≥ b′′}

Decomposition methods attempt to improve on this bound byutilizing the fact that OPT (c,P ′) or SEP (x,P ′) is easy.

zD = minx∈P′

{c⊤x | A′′x ≥ b′′} = minx∈P′∩Q′′

{c⊤x} ≥ zLP

P ′ is represented implicitly through solution of a subproblem.

Decomposition Methods

Cutting Plane Method (Outer Method)

Dantzig-Wolfe Method / Lagrangian Method (Inner Methods)

Q′′

Q′ ∩ Q′′

P′ ∩ Q′′

DECOMP Framework

Traditional Decomposition Methods

Cutting Plane Method (CPM)

CPM builds an outer approximation of P ′ intersected with Q′′.

Master: zCP = minx∈Rn{c⊤x | Dx ≥ d}

Subproblem: SEP (xCP ,P ′)

Dantzig-Wolfe Method

DW builds an inner approximation of P ′ intersected with Q′′.

Master:z̄DW = min

λ∈RE+{c⊤(

s∈Esλs) | A′′(

s∈Esλs) ≥ b′′,

s∈Eλs = 1}

Subproblem: OPT (c⊤−u⊤DW

A′′,P ′)

Lagrangian Method

LD formulates a relaxation as finding the minimal extreme point of P ′ withrespect to a cost which is penalized if the point lies outside of Q′′.

Master: zLD = maxu∈R

m′′+

{mins∈E{c⊤s + u⊤(b′′ − A′′s)}}

Subproblem: OPT (c−u⊤LD

A′′,P ′)

DECOMP Framework

Master:z̄DW = min

λ∈RE+{c⊤(

s∈Eλs = 1}

A′′,P ′)

Lagrangian Method

m′′+

{mins∈E{c⊤s + u⊤(b′′ − A′′s)}}

A′′,P ′)

DECOMP Framework

Master:z̄DW = min

λ∈RE+{c⊤(

s∈Eλs = 1}

A′′,P ′)

Lagrangian Method

m′′+

{mins∈E{c⊤s + u⊤(b′′ − A′′s)}}

A′′,P ′)

DECOMP Framework

Common Framework

The continuous approximation of P is formed as the intersection of twoexplicitly defined polyhedra (both with a small description).

zLP = minx∈Rn

{c⊤x | x ∈ Q′ ∩ Q′′}

Decomposition Methods form an approximation as the intersection of oneexplicitly defined polyhedron (with a small description) and one implicitlydefined polyhedron (with a large description).

zCP = zDW = zLD = zD = minx∈Rn

{c⊤x | x ∈ P ′ ∩Q′′} ≥ zLP

Each of the traditional decomposition methods contain two primary steps

Master Problem: Update the primal or dual solution information.

Subproblem: Update the approximation of P: SEP (x,P′) or OPT (c,P′).

Integrated Decomposition Methods form an approximation as theintersection of two implicitly defined polyhedra (both with a largedescription).

So, we improve on the bound zD by building both an inner approximationPI of P ′ intersected with some outer approximation PO ⊂ Q′′.

Q′′

Q′ ∩ Q′′

DECOMP Framework

Common Framework

zLP = minx∈Rn

{c⊤x | x ∈ Q′ ∩ Q′′}

{c⊤x | x ∈ P ′ ∩Q′′} ≥ zLP

Q′′

Q′ ∩ Q′′

P′ ∩ Q′′

DECOMP Framework

Common Framework

zLP = minx∈Rn

{c⊤x | x ∈ Q′ ∩ Q′′}

{c⊤x | x ∈ P ′ ∩Q′′} ≥ zLP

Q′′

Q′ ∩ Q′′

P′ ∩ Q′′

DECOMP Framework

Outline

3 DECOMP Framework

DECOMP Framework

Integrated Decomposition Methods

Price and Cut (PC)

PC approximates P by building an inner approximation of P ′ (as in DW) inter-sected with an outer approximation of P (as in CPM).

Master:z̄PC = min

λ∈RE+{c⊤(

s∈Esλs) | D(

s∈Esλs) ≥ d,

s∈Eλs = 1}

Subproblem:

Pricing: OPT (c⊤−u⊤P CD,P′), or

Cutting: SEP (xP C ,P)

Relax and Cut (RC)

RC improves on the bound zD using LD and augmenting the multiplier space withvalid inequalities that are violated by the solution to the Lagrangian subproblem.

Master: zRC = maxu∈R

m′′+

{mins∈E{c⊤s + u⊤(d − Ds)}}

Subproblem:

Pricing: OPT (c−u⊤LDD,P′), or

Cutting: SEP (s,P)

DECOMP Framework

Integrated Decomposition Methods

Price and Cut (PC)

PC approximates P by building an inner approximation of P ′ (as in DW) inter-sected with an outer approximation of P (as in CPM).

Master:z̄PC = min

λ∈RE+{c⊤(

s∈Esλs) | D(

s∈Esλs) ≥ d,

s∈Eλs = 1}

Subproblem:

Pricing: OPT (c⊤−u⊤P CD,P′), or

Cutting: SEP (xP C ,P)

Relax and Cut (RC)

RC improves on the bound zD using LD and augmenting the multiplier space withvalid inequalities that are violated by the solution to the Lagrangian subproblem.

Master: zRC = maxu∈R

m′′+

{mins∈E{c⊤s + u⊤(d − Ds)}}

Subproblem:

Pricing: OPT (c−u⊤LDD,P′), or

Cutting: SEP (s,P)

DECOMP Framework

Structured Separation

In general, the complexity of OPT (c, X) = SEP (x, X).

Observation: Restrictions on input or output can change their complexity.

Template Paradigm, restricts the output of SEP (x, X) to valid inequalities (a, β) thatconform to a certain structure. This class of inequalities forms a polyhedron C ⊃ X.

For example, let P be the convex hull of solutions to the TSP.

SEP (x,P) is NP -Complete.

SEP (x, C) is polynomially solvable, for C ⊃ P the Subtour Polytope (Min-Cut) or BlossomPolytope (Padberg-Rao).

Structured Separation, restricts the input of SEP (x, X), such that x conforms to somestructure. For example, if x is restricted to solutions to a combinatorial problem, thenseparation often becomes much easier.

DECOMP Framework

Example - TSP

Traveling Salesman Problem Formulation:

P ′ = 1-Tree Relaxation: OPT (c, 1 − Tree) in O(|E| log |V |)

x(δ({0})) = 2x(E(V \ {0})) = |V | − 2x(E(S)) ≤ |S| − 1 ∀S ⊂ V \ {0}, 3 ≤ |S| ≤ |V | − 1xe ∈ {0, 1} ∀e ∈ E

P ′ = 2-Matching Relaxation: OPT (c, 2 − Match) in polynomial time

x(δ(u)) = 2 ∀u ∈ Vxe ∈ {0, 1} ∀e ∈ E

DECOMP Framework

Example - TSP

Separation of Subtour Inequalities:

x(E(S)) ≤ |S| − 1

SEP (x, Subtour), for x ∈ Rn can be solved in O(|V |4) (Min-Cut)

SEP (s, Subtour), for s a 2-matching, can be solved in O(|V |)

Simply determine the connected components Ci, and set S = Ci for each component (each givesa violation of 1).

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DECOMP Framework

Example - TSP

x(E(S)) ≤ |S| − 1

DECOMP Framework

Example - TSP

x(E(S)) ≤ |S| − 1

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DECOMP Framework

Example - TSP

Separation of Comb Inequalities:

x(E(H)) +k

x(E(Ti)) ≤ |H| +k

(|Ti| − 1) − ⌈k/2⌉

SEP (x, Blossoms), for x ∈ Rn can be solved in O(|V |5) (Padberg-Rao)

SEP (s, Blossoms), for s a 1-Tree, can be solved in O(|V |2)Construct candidate handles H from BFS tree traversal and an odd (>= 3) set of edges with oneendpoint in H and one in V \ H as candidate teeth (each gives a violation of ⌈k/2⌉ − 1).

This can also be used as a quick heuristic to separate 1-Trees for more general comb structures, forwhich there is no known polynomial algorithm for separation of arbitrary vectors.

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DECOMP Framework

Example - TSP

x(E(H)) +k

x(E(Ti)) ≤ |H| +k

(|Ti| − 1) − ⌈k/2⌉

DECOMP Framework

Example - TSP

x(E(H)) +k

x(E(Ti)) ≤ |H| +k

(|Ti| − 1) − ⌈k/2⌉

0.30.7

DECOMP Framework

Structured Separation Motivation

Motivation: RC, solutions to LR, s ∈ E, have some nice combinatorial structure.

New Algorithms:

Price and Cut (Revisited) - see fig

Decomp and Cut - inverse process

Provides an alternative neccessary (but not sufficient) condition to find an improvinginequality which is very easy to implement and understand.

For some theoretical details see:

T. Ralphs and M.G., Decomposition and Dynamic Cut Generation in Integer Programming,Mathematical Programming 106 (2006), 261

T. Ralphs and M.G., Decomposition in Integer Programming, in Integer Programming: Theory andPractice, John Karlof, ed. (2005), 57

DECOMP Framework

Structured Separation Motivation

Motivation: RC, solutions to LR, s ∈ E, have some nice combinatorial structure.

New Algorithms:

Price and Cut (Revisited) - see fig

Decomp and Cut - inverse process

Provides an alternative neccessary (but not sufficient) condition to find an improvinginequality which is very easy to implement and understand.

For some theoretical details see:

T. Ralphs and M.G., Decomposition and Dynamic Cut Generation in Integer Programming,Mathematical Programming 106 (2006), 261

T. Ralphs and M.G., Decomposition in Integer Programming, in Integer Programming: Theory andPractice, John Karlof, ed. (2005), 57

DECOMP Framework

Price and Cut (Revisited)

The violated subtour found by separating the 2-Matching also violates the fractional point,but was found at little cost.

0.60.8

x̂ λ̂0 = 0.2 λ̂1 = 0.2 λ̂2 = 0.6

Similarly, the violated blossom found by separating the 1-Tree also violates the fractionalpoint, but was found at little cost.

DECOMP Framework

Price and Cut (Revisited)

The violated subtour found by separating the 2-Matching also violates the fractional point,but was found at little cost.

0.60.8

x̂ λ̂0 = 0.2 λ̂1 = 0.2 λ̂2 = 0.6

Similarly, the violated blossom found by separating the 1-Tree also violates the fractionalpoint, but was found at little cost.

0.30.7

λ̂4 = 0.1 λ̂5 = 0.1

λ̂0 = 0.3 λ̂1 = 0.2 λ̂2 = 0.2

λ̂3 = 0.1

DECOMP Framework

Decomp and Cut

In the context of the traditional CPM, we can construct (inverse DW) the decomposition λfrom the current fractional solution xCP by solving the following LP

maxλ∈R

{0⊤λ|X

sλs = xCP ,X

λs = 1},

If we find a decomposition D, then we separate each s ∈ D, as in revised PC.

If we fail, then the LP proof of infeasibility (Farkas Cut) gives us a separating hyperplanewhich can be used to cut off the current fractional point.

Farkas ineqaulity

xCP xCP

s ∈ E : λs > 0

(a) xCP ∈ P′ (b) xCP /∈ P′

valid ineqaulity for P

DECOMP Framework

Decomp and Cut

In the context of the traditional CPM, we can construct (inverse DW) the decomposition λfrom the current fractional solution xCP by solving the following LP

maxλ∈R

{0⊤λ|X

sλs = xCP ,X

λs = 1},

If we find a decomposition D, then we separate each s ∈ D, as in revised PC.

If we fail, then the LP proof of infeasibility (Farkas Cut) gives us a separating hyperplanewhich can be used to cut off the current fractional point.

Farkas ineqaulity

xCP xCP

s ∈ E : λs > 0

(a) xCP ∈ P′ (b) xCP /∈ P′

valid ineqaulity for P

DECOMP Framework

Structured Separation - Useful?

Case 1: There exists a polynomial algorithm for SEP (x, C).TSP/1-Tree: SEP (x, Blossoms) Padberg-Rao, SEP (s, Blossoms) BFS.Unlikely that SS will ever outperform a well written exact method, but very easy to implement.

Case 2: There exists no polynomial algorithm for SEP (x, C).VRP,k-Tree: SEP (x, GSECs) heuristics, SEP (s, GSECs) in O(|E|).SS provides an alternative and typically simple heuristic for separation.

Case 3: There exists no algorithm for SEP (x, C), but ineq’s in C are facet-defining.KCCP,CP: SEP (x, MaxSet) heuristics, SEP (s, MaxSet) in O(|E|).SS provides a starting point for constructing separation heuristics.

Case 4: We have a new problem class for which we are searching for new valid inequalities.Trying to analyze integral points (as in SS) seems much more promising.

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DECOMP Framework

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DECOMP Framework

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DECOMP Framework

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DECOMP Framework

Structured Separation - Applications

Steiner Tree ProblemMinimum Spanning Tree : Lifted SECs, Partition - RC* [Lucena 92]

Traveling Salesman ProblemOne-Tree: Blossoms, CombsMatching: SECs

Vehicle Routing Problemk-Traveling Salesman Problem : GSECs - DC [Ralphs, et al. 03]k-Tree : GSECs, Combs, Multistars - RC* [Marthinhon, et al. 01]

Axial Assignment ProblemAssignment Problem : Clique-Facets - RC [Balas, Saltzman 91]

Knapsack Constrained Circuit ProblemKnapsack Problem : Cycle Cover, Maximal-Set InequalitiesCircuit Problem: Cycle Cover, Maximal-Set Inequalities

Edge-Weighted Clique ProblemTree Relaxation : Trees, Cliques - RC [Hunting, et al. 01]

Subtour Elimination Problem [G. Benoit / S. Boyd]

Fractional 2-Factor Problem : SECs - DC / LP Context [Benoit, Boyd 03]

DECOMP Framework

Multiple Polytopes

Common paradigm - tighten bounds by solving SEP (x, X), for various X.

X is a polyhedron - the closure of the set of all cuts in a particular template.

Successive outer approximation from the intersection of various polyhedra.

TSP: X = subtour closure, blossom closure, comb closure, etc...

MILP: X = knapsack closure, flow cover closure, clique closure, etc...

Common paradigm - column generation based on one oracle Why only one?

New paradigm - generate vars from multiple polytopes, solving OPT (c, X), for various X.

Successive inner approximation from the intersection of various polyhedra.

TSP: X = one-tree, 2-matching, TSP, etc...

DECOMP Framework

Multiple Polytopes

DECOMP Framework

Multiple Polytopes

DECOMP Framework

Outline

3 DECOMP Framework

DECOMP Framework

DECOMP provides a flexible software framework for testing and extending the theoreticalframework presented thus far, with the primary goal of minimal user responsibility.

DECOMP was built around data structures and interfaces provided by COIN-OR:

COmputational INfrastructure for Operations Research

BCP provides a framework for parallel PC with LP-Based Bounding.

A generalization of BCP currently under development:

ALPs: Abstract Library for Parallel Search (INFORMS’06 - MD02)

BiCePS: Branch, Constrain and Price [Generic Bounding]

BLIS: BiCePS Linear Integer Solver = BCP

DECOMP will provide an implementation of the BiCePS layer.

The DECOMP framework, written in C++, is accessed through two user interfaces:

Applications Interface: DecompApp

Algorithms Interface: DecompAlgo

DECOMP Framework

DECOMP - Applications

The base class DecompApp provides all default algorithms: CPM, DW, LD, PC, RC, DC.

In order to develop an application, the user must derive the following methods/objects. Allother methods have appropriate defaults but are virtual and may be overridden.

DecompApp::createModel(). Define [A′′, b′′] and [A′, b′] (optional).TSP: [A′′, b′′] define the degree constraints. [A′, b′] is empty.

DecompApp::isFeasible(). Does x∗ define a feasible solution?TSP: do we have a feasible tour?

DecompApp::solveRelaxed(). Provide a subroutine for OPT (c,P ′).This is optional as well, if [A′, b′] is defined (it will call the built in IP solver, currently CBC).TSP 1-Tree: provide a solver for 1-tree.TSP 2-Match: provide a solver for 2-matching.

DecompApp::generateCuts(s). Provide a subroutine for SEP (s, C)TSP 1-Tree: provide separation for comb/blossoms which violate a 1-tree (BFS).TSP 2-Match: provide separation for subtours which violate a 2-matching (connect-comp).

To perform traditional CPM, if known, the user can also derive a subroutine to solveSEP (x, C), for separation of arbitrary real vectors. Note: the user also has the option toturn on CGL cuts.

DECOMP Framework

A key feature of DECOMP is that the user only needs to provide methods for theirapplication in the original space (x-space), rather than in the space of a particularreformulation.

Automatic reformulation allows for users to consider cuts and variables in their mostintuitive form and greatly simplifies the process of expansion into rows and columns.

This is a major differentiator for DECOMP, compared to others BCP, ABACUS, MINTO, etc.

Structured separation allows for fast and easy prototyping without the need forimplementation of difficult separation routines.

Features:

One interface to all default algorithms: CPM/DC, DW, LD, PC, RC.

Built on top of the COIN/OSI interface, so easily interchange LP solvers.

Column generation based on multiple polytopes can be easily defined and employed.

Active LP compression, variable and cut pool management.

Flexible parameter interface: command line, parm file, direct call overrides.

Visualization tools for graph problems (linked to graphviz).

DECOMP Framework

Features:

DECOMP Framework

Features:

DECOMP Framework

DECOMP - TSP ExampleTSP DecompApp

c l a s s TSP DecompApp : pub l i c DecompApp {// De f i n e modelCore as 2−deg r ee c o n s t r a i n t s , mode lRe lax empty .vo id APPcreateModel ( double ∗& ob jCoe f f ,

v e c to r<DecompConst ra intSet∗> & modelCore ,v ec to r<DecompConst ra intSet∗> & modelRe lax ) ;

//A 2−matching s o l v e r , a 1−t r e e s o l v e r .decompStat APPso lveRe laxed ( const i n t whichModel ,

const double ∗ redCostX ,const double ∗ o r i gCos t ,l i s t <DecompVar∗> & va r s ) ;

// S t r u c t u r e d s e p a r a t i o n r o u t i n e s :// connected components f o r s ub t ou r s// BFS f o r b lo s soms and g e n e r a l combsv i r t u a l i n t gene r a t eCu t s ( const double ∗ x ,

const DecompVar & var ,DecompCutList & new cuts ) ;

// Standard s e p a r a t i o n r o u t i n e s :// 1 . ) min−cut f o r s ub t ou r s// 2 . ) padberg−rao f o r b lo s soms// 3 . ) h e u r i s t i c s f o r g e n e r a l combsi n t gene r a t eCu t s ( const double ∗ x ,

const DecompConst ra intSet & modelCore ,const DecompConst ra intSet & modelRelax ,DecompCutList & newCuts ) ;

//Does x d e f i n e a tou r ?boo l i s F e a s i b l e ( const double ∗ x ,

const i n t nCols ,DecompCutList & newCuts ) ;

. . .}

DecompApp

TSP_DecompApp

DECOMP Framework

DECOMP - Algorithms

The base class DecompAlgo provides the shell (init / master / subproblem / update).

Each of the methods described have derived default implementations DecompAlgoX :

public DecompAlgo which are accessible by any application class, allowing full flexibility.

New, hybrid or extended methods can be easily derived by overriding the varioussubroutines, which are called from the base class. For example,

Alternative methods for solving the master LP in DW, such as interior point methods or ACCPM.

The user might choose to add some advanced stabilizing factor to the dual updates in LD, as inbundle methods.

The user might choose the Volume algorithm for solving the LD, which provides an approximateprimal solution, for which cuts can be generated.

Hybrid methods like using LD to initialize the columns of the DW master.

During PC, adding cuts to both Pt+1O and Pt+1

I simultaneously (Vanderbeck).

DecompAlgo

DecompAlgoUSERDecompAlgoRCDecompAlgoPCDecompAlgoC

DECOMP Framework

DECOMP - TSP Example

TSP Main

i n t main ( i n t argc , char ∗∗ a rgv ){Ut i lApp u t i lApp ( argc , a rgv ) ;

// c r e a t e the u s e r a p p l i c a t i o n ( a DecompApp)TSP DecompApp t sp ( u t i lApp ) ;t s p . c r ea teMode l ( ) ;

// c r e a t e i n s t a n c e s o f each a l g o r i t hm and s o l v eDecompAlgoC algoC(& t sp ) ;a lgoC . i n i t S e t u p (TSP DecompApp : : BASIC ) ;a lgoC . processNode ( ) ;

DecompAlgoPC algoPC(& t sp ) ;algoPC . i n i t S e t u p (TSP DecompApp : :TWO MATCH) ;algoPC . processNode ( ) ;algoPC . r e f r e s h ( ) ;algoPC . i n i t S e t u p (TSP DecompApp : : ONE TREE ) ;algoPC . processNode ( ) ;

DecompAlgoRC algoRC(& t sp ) ;algoRC . i n i t S e t u p (TSP DecompApp : :TWO MATCH) ;algoRC . processNode ( ) ;algoRC . r e f r e s h ( ) ;algoRC . i n i t S e t u p (TSP DecompApp : : ONE TREE ) ;algoRC . processNode ( ) ;

DECOMP Framework

DECOMP - Interface to ALPS/BiCePs

XXX DecompApp is also built as a callable library

BcpsDecompModel : public BcpsModel

a wrapper class that calls (data access) methods from DecompApp

BcpsDecompTreeNode : public BcpsDecompTreeNode

a wrapper class that calls (algorithmic) methods from DecompAlgo

Minimal additional user responsibilities:

(optional) branching decisions - defaults are straightforward since we work in the original x-space

each built-in DecompAlgoXXX provides a default BiCePs branching routine

default RC branching assumes use of volume or bundle (need primal solutions)

(not optional) enforcing branching decisions in subproblem oracle

(not optional) encoding/decoding Decomp objects for disributed processing

DECOMP Framework

Summary

Traditional Decomposition Methods approximate P as P ′ ∩ Q′′.P′ ⊃ P may have a large description.

Integrated Decomposition Methods approximate P as PI ∩ PO.Both PI ⊂ P′ and PO ⊃ P may have a large description.

Structured separation can be much easier than general separation.Two new techniques based on SS: revised-PC and DC

A new paradigm for column generation using multiple oracles.

DECOMP provides an easy-to-use framework for comparing and developing variousdecomposition-based bounding methods.

The key to ease-of-use, is that the user can stay in the original (intuitive) space.

The interface to ALPS allows us to investigate large-scale problems on distributed networks.

The code is open-source, currently released under CPL and will soon be available throughthe COIN-OR project repository www.coin-or.org.

DECOMP Framework

Summary

DECOMP Framework

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DECOMP Framework

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DECOMP Framework

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