{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

MIT16_410F10_lec16

# MIT16_410F10_lec16 - 16.410/413 Principles of Autonomy and...

This preview shows pages 1–8. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Assignments Readings Lecture notes [IOR] Chapters 2, 3, 9.1-3. [PA] Chapter 6.1-2 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 point-to-point problem, both S and G contain only one element. Output: T , W T is a weighted tree (graph with no cycles) containing one minimum-weight path for each pair of start-goal vertices ( s , g ) S × G . W : S × G R + is a function that returns, for each pair of start-goal vertices ( s , g ) S × G , the weight W ( s , g ) of the minimum-weight 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Example: point-to-point shortest path Find the minimum-weight 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 , 0 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 0 from the goal to the start vertex: x 0 = 1, i.e., the virtual edge is always chosen. Every node with an incoming edge must have an outgoing edge E. Frazzoli (MIT) L07: Mathematical Programming I November 8, 2010 6 / 23
Another look at shortest path problems (3) Summarizing, what we want to do is: minimize n E i =1 w i x i subject to: e i In ( s ) x i e j Out ( s ) x j = 0 , s V ; x i 0 , i = 1 , . . . , n E ; x 0 = 1 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 28

MIT16_410F10_lec16 - 16.410/413 Principles of Autonomy and...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online