… since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear. Introduction to introduction to introduction to … Optimizat ion Leonhard Euler
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Introduction to introduction to introduction to … Optimization
Introduction to introduction to introduction to … Optimization . Leonhard Euler. … since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear. Boredom. - PowerPoint PPT Presentation
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… since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear.
Introduction to introduction to introduction to …
Optimization
Leonhard Euler
Boredom
Lecture time
Understanding
1 min 10 mins 30 mins 1 hour
Optimal listening time for a talk: 8 minutes 25 seconds *
height
time
Action at a point := Kinetic Energy – Potential Energy.Action for a path := Integrate action at points over the path.
Nature chooses the path of “least” action! Pierre Louis Moreau de Maupertuis
The path taken between two points by a ray of light is the path that can be traversed in the least time
Fermat
For all thermodynamic processes between the same initial and final state, the delivery of work is a maximum for a reversible process
Gibbs
William of Ockham
Among competing hypotheses, the hypothesis with the fewest assumptions should be selected.
Travelling Salesman Problem (TSP)
Courtesy: xkcd
• A hungry cow is at position (2,2) in a open field.• It is tied to a rope that is 1 unit long.• Grass is at position (4,3)• A perpendicular electric fence passes through the point (2.5,2)• How close can the cow get to the fodder?
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• What do we want to find?• Position of cow: Let (x,y)
be the solution.
• What do we want to be solution to satisfy?• Grass: min (x-4)^2 + (y-
3)^2
• What restrictions does the cow have?• Rope: (x-2)^2 + (y-2)^2
<= 1• Fence: x <= 2.5
Variables: (x,y) (position of cow)
Objective : (x-4)^2 + (y-3)^2 (distance from grass)
Constraints:
(x-2)^2 + (y-2)^2 <= 1 (rope) x <= 2.5 (fence)
Framework
minimize/maximize Objective (a function of Variables) subject to Constraints (functions of Variables)
How?Unconstrained case:
min (x-4)^2 + (y-3)^2
- Cow starts at (2,2)- Does not know where grass is. Knows only distance from
grass.- Needs ‘good’ direction to move from current point.
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The key question in optimization is ‘What is a good direction?’