Optimisation

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Optimisation
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Contents

• Integer Programming
• Branch and Bound
• Dynamic Programming
• Heuristics
• Travelling Salesman
• Stochastic Optimisation
• Simulated Annealing
• Genetic Algorithms
• Distributed Optimisation
• Sum Product Algorithm

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Integer Programming
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Integer Programming
Objective Function Linear or Quadratic
Decision Variables Upper and Lower
Bounds Integer, Linear or Mixed
Constraints Linear
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Integer Programming

• Since (LR) is less constrained than (IP), the
  following are immediate
• If (IP) is a maximisation, the optimal
  objective value for (LR) is more than or equal to
  the optimal objective for (IP).
• If (LR) is infeasible, then so is (IP).
• If (LR) is optimized by integer variables, then
  that solution is feasible and optimal for (IP).
• If the objective function coefficients are
  integer, then for maximisation, the optimal
  objective for (IP) is less than or equal to the
  round down'' of the optimal objective for (LR).
  

Integer Program
Linear Relaxation
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Dynamic Programming
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s0
sK
i1
i2
i3
i4
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Heuristics
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Traveling Saleman (TSP)
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Stochastic Optimisation
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Fitness Landscapes
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Simulated Annealing
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Genetic Algorithms
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Genetic Algorithms
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Representation
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Genetic Operators
Mutation Flip bits with probability 1/L
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Selection
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Selection
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Graph Colouring
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Distributed Optimisation
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Sum Product Algorithm
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Message Parsing
Messages
From functions to variables.
From variables to functions.
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Message Parsing
Variables
Functions
Messages capture how strongly the functions want
the variables to be in any particular state and
allows it to flow around the network.
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Loops
Tree Exact proved convergence
Loops Approximate solutions May cycle But Very
Fast Actually works well in real applications
(coding)