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Unformatted text preview: 16.410/413 Principles of Autonomy and Decision Making Lecture 16: Mathematical Programming I Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology November 8, 2010 E. Frazzoli (MIT) L07: Mathematical Programming I November 8, 2010 1 / 23 Assignments Readings Lecture notes [IOR] Chapters 2, 3, 9.13. [PA] Chapter 6.12 E. Frazzoli (MIT) L07: Mathematical Programming I November 8, 2010 2 / 23 Shortest Path Problems on Graphs Input: V , E , w , s , G : V : set of vertices (finite, or in some cases countably infinite). E V V : set of edges. w : E R + , e w ( e ): a function that associates to each edge a strictly positive weight (cost, length, time, fuel, prob. of detection). S , G V : respectively, start and end sets. Either S or G , or both, contain only one element. For a pointtopoint problem, both S and G contain only one element. Output: T , W T is a weighted tree (graph with no cycles) containing one minimumweight path for each pair of startgoal vertices ( s , g ) S G . W : S G R + is a function that returns, for each pair of startgoal vertices ( s , g ) S G , the weight W ( s , g ) of the minimumweight path from s to g . The weight of a path is the sum of the weights of its edges. E. Frazzoli (MIT) L07: Mathematical Programming I November 8, 2010 3 / 23 Example: pointtopoint shortest path Find the minimumweight path from s to g in the graph below: Solution: a simple path P = s , a , d , g ( P = s , b , d , g would be acceptable, too), and its weight W ( s , g ) = 8. E. Frazzoli (MIT) L07: Mathematical Programming I November 8, 2010 4 / 23 Another look at shortest path problems Cost formulation The cost of a path P is the sum of the cost of the edges on the path. Can we express this as a simple mathematical formula? Label all the edges in the graph with consecutive integers, e.g., E = { e 1 , e 2 , . . . , e n E } . Define w i = w ( e i ), for all i 1 , . . . , n E . Associate with each edge a variable x i , such that: x i = 1 if e i P , otherwise. Then, the cost of a path can be written as: Cost ( P ) = n E i =1 w i x i . Notice that the cost is a linear function of the unknowns { x i } E. Frazzoli (MIT) L07: Mathematical Programming I November 8, 2010 5 / 23 Another look at shortest path problems (2) Constraints formulation Clearly, if we just wanted to minimize the cost, we would choose x i = 0, for all i = 1 , . . . , n E : this would not be a path connecting the start and goal vertices (in fact, it is the empty path). Add these constraints: There must be an edge in P that goes out of the start vertex. There must be an edge in P that goes into the goal vertex. Every (non start/goal) node with an incoming edge must have an outgoing edge A neater formulation is obtained by adding a virtual edge e from the goal to the start vertex: x = 1, i.e., the virtual edge is always chosen....
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 Fall '10
 Prof.BrianWilliams
 Mass

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