Optimization

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Optimization
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Types of Optimization

• function / trial and error
• single variable / multiple variable
• static / dynamic
• discrete / continuous
• constrained / unconstrained
• random / minimum seeking

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Minimum Seeking Algorithms

• Exhaustive search
• Analytical Optimization
• f(x,y)x sin(4x)1.1y sin(2y) 0lt x,y lt 10
• recall f(g(x))' f'(g(x)) g'(x)
• recall (f(x) g(x))' f'(x)g(x) f(x) g'(x)
• df / dx 4cos(4x) sin(4x)
• df / dy 2.2 cos(2y) 1.1sin(2y)
• d2 f /d2 x -16sin(4x)-4cos(4x)
• d2 f /d2 y -4.4sin(2y)-2.2sin(2y)
• minimum when d2 f /d2 x d2 f /d2 y gt0

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Lagrange Multipliers

• find extrema of f(x,y,...)
• subject to gi(x,y,...)0 for i1 to a
• solution
• find extrema of f(x,y,...)g1(x,y,...)k1...gk(x,
  y,...)ka
• find the minimum of fu22v2w2x2
• subject to u3v-wx2, 2u-vw2x4

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Simplex Method

• Starts with a simplex (e.g. triangle in 2D)
• move the simplex in the direction of the minimum
• contract the simplex around the minimum
• gets stuck in local minima

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Line minimization

• optimize on x
• optimize on y
• optimize on z
• loop

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Steepest Gradient

• Start at an arbitrary point
• Go in direction of Of(x,...) df/dx ...
• (xn1,yn1)(xn,yn)-? Of(xn,yn)
• draw picture

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Newton's method

• Extension of the gradient method using Taylor
  series expansion

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Greedy Algorithm

• store n books on shelves
• book b_i has thickness t_i
• books must be in order

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Greedy Algorithm

• job processing problem
• a server receives n jobs and must process them in
  optimal fashion
• minimize ? i1n (wipi)
• if all jobs arrive at the same time - greedy does
  the job
• process shortest job first
• If not, greedy doesn't do the job

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Depth-first search

• searching in a maze
• searching for a gift
• always finds a solution only if the search tree
  is finite
• breath-first tree - good for infinite search trees

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The 8 - puzzle

• 7 6
• 4 3 1
• 2 5 8
• 1 2 3
• 8 4
• 7 6 5
• use heuristics to tell us how far is the current
  state from the goal state.
• admissible heuristic doesn't over estimate the
  number of moves

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The 8-puzlle problem

• count how many tiles are in the wrong place -
  h18
• h2 - how far each tile needs to move to get in
  its place (Manhattan distance) h218
• h2gt h1
• h3 - extra difficulty to swap tiles
• h3(n)h2(n) 2 k(n), where k(n) is the number of
  tiles that need to be swapped

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Hill Climbing

• check in each possible direction
• find best solution
• improvement
• check all direction and choose the highest jump

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Best-First Search

• find best possibility from all visited nodes

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Beam Search

• use breath-first search
• but only follow the best few paths

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A algorithm

• extension of best-first search algorithm
• adds the cost to go to the current node
• f(node)g(node)h(node)
• g(node) - cost so far
• h(node) - underestimated cost to best solution

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Uniform search

• also known as branch and bound
• same as A, but set h0
• good, provided the cost of a path increases
  monotonically

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Greedy search

• g(node) is set to 0 - i.e. best first strategy

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Backtracking

• similar to depth-first search
• start from the tree root
• explore paths till the end
• 8th queen problem

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Forward checking

• look ahead
• remove possible future choice that cannot be part
  of a solution
• e.g. 8 queen problem

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Most-Constrained Variable

• e.g. 8 queen problem
• 1 3 5
• other example map coloring
• most constraining variable
• most constraints on future variables
• least-constraint value
• leaves the greatest number of choices for other
  variables

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Heuristic Repair Approach

• e.g. min-conflicting heuristic
• move queen so it conflicts with fewer queens

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Local Search - Exchange Heuristics

• change the value of one or more variables
• TSP
• use k-exchange k 3

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Iterated Local Search

• Start at different locations
• Find local optima
• e.g., TSP
• Other searches
• Tabu Search - avoid cycles
• Ant Colony Optimization
• pheromones - good for networks, dynamic
  optimization

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Simulated Annealing

• Annealing - heating glass/metal and slow cool-off
• uses Metropolis Monte Carlo simulation
• start with a bunch of random state
• start changing them for better, e.g. lower energy
• if energy is higher, use a dice )
• e -dE/T
• where T is current temperature
• dE - increase in temperature
• idea allow system to escape local minima at high
  temperature
• cooling schedule TnewTold - dT or TnewCTold
• explain how system is cooled off
• end condition T0

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Iterative Deepening A (IDA)

• Iterative Deepening, but of f(node)
• each sequential search imposes a greater limit on
  f
• f(node)g(node)h(node)
• but this time g(node) distance from current root

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Bidirectional Search

• e.g., shortest distance, maze
• also known as wave search
• do breath first from both ends

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Nondeterministic search

• choose which node to expand at random
• good deep solution
• combine with hill climbing

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Island-Driven Search

• assumes an island exists somewhere between start
  and end
• if a solution can not be found, try new island

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Nonchronological Backtracking

• don't undo decisions in chronological order
• look at last few decisions and see where a good
  opportunity exists