Mathematical Programming for Optimisation

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Mathematical Programming for Optimisation
CAIR Seminar Programme, Wednesday 28th May 2008

• Mike Morgan and Vic Grout
• Centre for Applied Internet Research (CAIR)
• University of Wales
• NEWI Plas Coch Campus, Mold Road
• Wrexham, LL11 2AW, UK
• mi.morganv.grout_at_newi.ac.uk
• www.cair-uk.org

NEWI North East Wales Institute of Higher
Education - Centre for Applied Internet Research
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Mathematical Programming for Optimisation

• Optimisation generally involves maximising or
  minimising something
• The something is the objective function
• e.g., Maximise x or just max x (no
  solution)
• Also, there are usually constraints
• e.g., max x subject to (s.t.) x ? 2
  (trivially x 2)
• A better example, max 3x 2y s.t. 3x - 2y
  ? 4 7y 3x ? 9
• Not so obvious!

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A Better Example

• max 3x 2y s.t. 3x - 2y
  ? 4 7y 3x ? 9

Solution
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An Even Better Example
Algorithms such as the Simplex Method find (as in
home-in on) the optimal solution
This is Mathematical Programming
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Common Notation

• Used to simplify/generalise large expressions
• in many dimensions/variables
• 3x 4y 2z
• ax by cz
• a1x1 a2x2 a3x3 anxn

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The General Form

• max/min S xi
• s.t. S ? / / ?
• S ? / / ?
• S ? / / ?
• xi ? 0,1 etc.
• Or, it can be expressed using linear algebra
• Vectors and matrices can save space

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Example The Diet Problem

• One of the first problems ever formulated in this
  way
• Consider buying the weekly shopping. Choice of
• n foods, containing
• m nutrients
• A (aij) is the amount matrix (2-d array (m x
  n))
• aij is the quantity of the ith nutrient in the
  jth food
• r (ri) is the requirement vector (1-d array
  (m))
• ri is the weekly requirement of nutrient i
• c (cj) is the cost vector (1-d array (n))
• cj is the cost of food j
• x (xj) is the consumption vector (1-d array
  (n))
• xj is the weekly consumption of food j

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Example The Diet Problem

• We want to find a diet, of minimal cost, that
  satisfies the dietary requirement of all
  nutrients
• min cx
• s.t. Ax ? r
• x ? 0

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Simple Network/Graph Problem

• Minimum Dominating Set (MDS)

Find the minimum subset of nodes S, such that
each node not in S has a neighbour in S
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MDS as an Integer Program

• Vector s (si), where si 1 if node i is a
  relay and 0 otherwise
• Adjacency matrix A (aij), where aij 1 if
  there is an edge between nodes i and j and 0
  otherwise.

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A more complex problem

• Minimum connected dominating set (MCDS)
• Same as MDS only the subgraph induced by S must
  be connected
• Solve using surplus network flow variables
• Increases complexity (runtime) for solving
  problem

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Testing Connectivity with Network Flows (almost)

• Flow matrix F (fij) records flow across edges
• Flows are directional
• To test connectivity, choose one node as source
  and all the others as sinks. The source
  introduces n-1 units of flow onto the network and
  each sink must absorb 1.
• Flow may only be transmitted from a relay or from
  the source

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Well, thats lovely, but

• it doesnt work! Consider using a different node
  as source
• Need to restrict problem so that the source node
  can only send flow along one edge, unless it is a
  relay
• In other words, if the source is not a relay, it
  gives all its flow to a neighbouring relay

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What does this look like as an integer program?

• First of all, set the source node as node 1
• Node 1 must introduce n-1 units of flow onto the
  network
• All other nodes must consume 1 unit

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What does this look like as an integer program?
(contd)

• For all nodes other than the source, flows must
  originate from a relay and traverse a valid edge
• Flow can originate from the source node if it
  isnt a relay, but must travel along a valid edge
  to a neighbouring relay

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What does this look like as an integer program?
(contd)

• Binary source vector q(qi), defined as qi1 if
  flow is transmitted from the src node to i and 0
  otherwise
• If source is not a relay, q must sum to 1

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How do we solve the problem?

• Solving a mathematical program with binary or
  integer variables is NP-complete
• Powerful method for finding optimal solutions
• Particularly if you use 3rd party software!

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Useful results

• Solving problems this way involves little
  software development
• Express your problem in this form and let the
  solver do the work
• Cplex solver originally developed by B. Bixby but
  now taken over by ILOG
• For MCDS (with a small alteration from model
  outlined) weve obtained optimal solutions for
  problems with n lt 100 nodes.
• Test accuracy of heuristics
• More complex variants of problem solved using
  combination of IP and heuristics

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Subsets of Constraints

• Suppose we require 2 node-disjoint paths between
  all node pairs on the network.
• In this example, it is only necessary to
  constrain one node pair to get a feasible
  solution
• Number of flow variables is reduced from n2m to m
• Runtime of LP solver is exponentially related to
  number of variables!!

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Subsets of constraints (contd)

• First require that every node have 2 relay
  neighbours
• minimise
• subject to

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Subsets of constraints (contd)

• Introduce flow variables incrementally
• Each constrained node pair (u,v) represents a
  commodity
• u creates 2 units of flow and v absorbs 2. All
  other nodes conserve flow.
• Transient nodes may only carry one unit of flow
  for a given commodity (ensures 2 disjoint paths)

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Lower bounds and heuristics

• After each IP is solved, solution is checked for
  feasibility.
• If it is not feasible, the size of S represents a
  lower bound on the size of the optimal solution.
• Use heuristic after each IP, if it finds solution
  with size lower bound, that solution is
  optimal.
• The more constraints are added, the more accurate
  the lower bound becomes
• Eventually, either the IP will yield a feasible
  solution or the heuristic will hit the lower bound

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Good news and bad news

• Problems solved to optimality for nlt100 and up
  to four node-disjoint paths between all node
  pairs
• Unfortunately this does involve some software
  development

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Any questions?
CAIR Seminar Programme, Wednesday 28th May 2008
Thank you

• Mike Morgan and Vic Grout
• Centre for Applied Internet Research (CAIR)
• University of Wales
• NEWI Plas Coch Campus, Mold Road
• Wrexham, LL11 2AW, UK
• mi.morganv.grout_at_newi.ac.uk
• www.cair-uk.org

NEWI North East Wales Institute of Higher
Education - Centre for Applied Internet Research