ESI 6448 Discrete Optimization Theory

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ESI 6448Discrete Optimization Theory

• Lecture 31

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Last class

• Decomposition
• Dantzig-Wolfe reformulation
• Column generation
• Solve LPM
• Example (STSP)

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Dantzig-Wolfe reformulation

• IP Master Problem(IPM)

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Linear Programming Master Problem

• LP relaxation of IPM (LPM)

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Solving LPM

• Use column generation or Branch and price
  algorithm
• to solve LPM by the primal simplex algorithm
• pricing step of choosing a column to enter the
  basis is modified
• instead of pricing the columns one by one, the
  problem of finding a column with the largest
  reduced price is itself a set of K optimization
  problems
• Initialization Restricted LPM (RLPM)

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Solving LPM (contd)

• Price Feasibility
• Any feasible solution of RLPM is feasible for LPM
• optimal primal solution ? of RLPM is feasible in
  LPM, and so for optimal dual solution (?, ?),
• Optimality Check for LPM
• (?, ?) is dual feasible for LPM?
• check for each column (for each k) and for each x
  ? Xk whether the reduced price ckx ?Akx ?k ?
  0
• rather than examining each point, solve an
  optimization subproblem

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Solving LPM (contd)

• Stopping Criterion
• If ?k 0 for k 1, , K, the solution (?, ?) is
  dual feasible for LPM, and so
• Generating a new column

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Solving LPM (contd)

• A Dual (Upper) Bound
• From the subproblem, we have ?k ? (ck ?Ak)x
  ?k for all x ? Xk.
• Setting ? (?1, , ?k), (?, ??) is dual
  feasible in LPM
• An Alternative Stopping Criterion

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STSP by Column Generation

• LPM
• single subproblem

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Strength of LPM

• Choice of approach depends on the relative
  difficulty in solving the two problems and on the
  convergence of the column generation and cutting
  plane algorithms in practice.

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IP column generation

• Branch-and-price (IP column generation) algorithm
• If optimal solution vector
  is not integer, IPM is not yet solved.
• zLPM ? z, so it provides an upper bound to be
  used in a branch-and-bound algorithm
• IP
• IPM

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IP column generation for 0-1 IP

• IPM(Si)

S
S0
S1
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IP column generation for 0-1 IP

• Branching on some fractional ?k,t variable
• On the branch in which ?k,t 0
• just one column corresponding to the tth solution
  ofsubproblem k is excluded
• resulting enumeration tree will be highly
  unbalanced
• Often difficult to impose the condition ?k,t 0
  and to prevent the same solution being generated
  again as optimal solution after branching
• Potential advantage of column generation
• optimal solutions to RLPM are often integral or
  close to integral
• can provide a feasible solution

S
S0
S1
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Implicit partitioning/packing problems

• partitioning/packing problems
• Given a finite set M 1, ..., m, there are K
  implicitly described sets of feasible subsets,
  and the problem is to find a maximum value
  packing or partition of M consisting of certain
  of these subsets.
• Set xk (y k, wk) with y k ? Bm the incidence
  vector of subset k of M, ck (ek, f k), Ak
  (I, 0) and b 1 and formulate
• IPM

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Multi-Item Lot-Sizing

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Clustering

• Given a graph G (V, E), edge costs ce for e ?
  E, node weights di for i ? V, and a cluster
  capacity C, partition the node set V into K
  clusters satisfying
• the sum of the node weights in each cluster ? C
• the sum of the costs of edges between clusters is
  minimized orthe sum of the costs of edges within
  clusters is maximized

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Capacitated Vehicle Routing

• Given a graph G (V, E), a depot node 0, edge
  costs ce for e ? E, K identical vehicles of
  capacity C, and client orders di for i ? V\0,
  find a set of subtours (cycles) for each vehicle
  satisfying
• each subtours contain the depot
• together the subtours contain all the nodes
• the subtours are disjoint on the node set V \ 0
• the total demand on each subtour (total amount
  delivered by each vehicle) ? C

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Partitioning with identical subsets

• clustering and vehicle routing
• clusters or vehicles are interchangeable
  (independent of k)
• Xk X, (ek, f k) (e, f ), Tk T for all k,
• IPM
• subproblem
• branching?

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Branching rules

S
S0
S1
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Example

• Clustering problem with a complete graph G (V,
  E), V 3, K 3 clusters, the objective of
  choosing as many as possible within the clusters,
  and at most 2 nodes allowed per cluster, node
  weights di 1 for all i ? V, edge weights ce 1
  for all e ? E, cluster capacity C 2.
• RLPM

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Example (Contd)

• subproblem of selecting a feasible cluster of
  maximum reduced price
• ? 1, w12 y1 y2 1

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Example (Contd)

• solution of LPM
• Branching
• i 1, j 2
• S1 nodes 1 and 2 in different subsets ? ?4 0
• S2 nodes 1 and 2 in the same subset ? ?1 ?2
  ?5 ?6 0

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Example (Contd)

• Reoptimizing for S1

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Example (Contd)

• subproblem for S1
• ? 0, LPM(S1) is solved with zLPM(S1) 1
• pruned by bound

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Example (Contd)

• Node S2
• ?1 ?2 ?5 ?6 0 ? ?3 ?4 1
• Eventually zLPM(S2) 1, pruned by bound
• ?1 ?6 1, corresponds to one cluster
  containing node 1 and another containing nodes 2,
  3 and the third empty

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Today

• Strength of LPM
• IP column generation for 0-1 IP
• Implicit partitioning/packing problems
• Partitioning with identical subsets