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# CS483/683 Multi-Agent Systems PowerPoint PPT Presentation

Computer Science & Engineering, University of Nevada, Reno. CS483/683 Multi-Agent Systems. Lectures 5-6: From Satisfaction to Optimization ADOPT: Asynchronous Distributed Optimization. 2-4 February 2010 Instructor: Kostas Bekris. Distributed Constrained Optimization. DCOP:

CS483/683 Multi-Agent Systems

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Computer Science & Engineering, University of Nevada, Reno

### CS483/683 Multi-Agent Systems

• Lectures 5-6:

• From Satisfaction to Optimization

2-4 February 2010

Instructor: Kostas Bekris

### Distributed Constrained Optimization

• DCOP:

• A set of n variables V={X1, ..., Xn}

• Each variable has a discrete domain: D1, ..., Dn

• Each variable is assigned to an agent

• Only the agent who is assigned a variable has control over its value

• And knowledge of its domain

• Assign values to variable so as to optimize a global objective function

• The optimization of the function satisfies a set of constraints

• Requirements:

• Distributed computation using only local communication

• Fast, asynchronous computation - agents should work in parallel

• Quality guarantees are needed

• Provably optimal solutions whenever possible

• Trade-off between computation and solution quality

• Bounded-error approximation: Guarantee solution within a distance from optimal, less time than the optimal

• The objective function is a summation over a set of cost functions

• F(A)=∑Xi,Xj∈V fij(di,dj)

• Xi←di, Xj←dj in A

• We want to find A* that minimizes F(A)

• e.g. F( { (X1,0), (X2,0), (X3,0), (X4,0) } ) = 4

• F( { (X1,1), (X2,1), (X3,1), (X4,1) } ) = 0

X1

X2

X3

X4

Constraint Graph

### Assumptions

• 1. Summation operation over cost function

• Associative

• Commutative

• Monotonic

• Cost of a solution can only increase as more costs are aggregated (i.e., we cannot have negative cost)

• 2. Constraints are at most binary

• There are ways to extend to constraints that involve a larger number of variables

• 3. Each agent is assigned a single variable

• There is a way to extend to the case that an agent must handle multiple variables

• 1. Opportunistic best-first search

• Agents are prioritized in a tree structure

• an agent has a single parent and multiple children

• Each agent keeps on choosing the best value based on the current available information

• i.e., chooses the variable which implies the smallest lower bound

• lower bounds do not need global information to be estimated

• Each agent maintains a lower and an upper bound for the cost of its subtrees

• and informs its parent about its own bounds

• Strategy allows agents to abandon partial solutions which have not been proven to be suboptimal

• they may have to reconsider the same assignments into the future

X1

X2

X3

X4

Communication Graph

• 2. Backtrack Threshold

• When an agent knows from previous search experiences that lb is a lower bound for its subtree

• inform the subtree agents not to bother searching for a solution whose cost is lower than lb

• In the general case, remembering these lower bounds for past assignments requires exponential space

• Approach remembers only one value and then cost is subdivided to children arbitrarily and adapted on the fly as new computations are executed

• 3. Built-in Termination Detection

• Keeping track of bounds (lower and upper bound for the cost function) on each agent

• allows to keep track of the progress towards the optimum solution

• and automatically terminates when necessary

### Messages and Data Structures

X1

• VALUE message (like ok?)

• Send selected value to children along the

• constraint graph

• COST message (like NoGood)

• Send to parents along the communication graph

• THRESHOLD message

• Send to children along the communication graph

• Each agent maintains the “context”

• (like the “agent_view”)

• A recorf of higher priority neighbors’ current variable assignment

• Two contexts are compatible if they do not disagree on any variable assignment

X2

X3

X4

Constraint Graph

X1

X2

X3

X4

Communication Graph

### COST message

• Xk transmits COST message to Xi

• Message contains

• context of Xk

• lb of Xk

• up of Xk

• When Xi receives the message it stores

• lb(d,Xk)

• ub(d,Xk)

• where d is the assignment of Xi in Xk’s context

• If context of Xi is incompatible with the context of Xk:

• lb(d,Xk) = 0

• ub(d,Xk) = ∞

Xi

Xk

### Costs and Bounds

X1

• Local cost: δ(di) = ∑(Xj,dj) fij(di,dj)

• di : assignment of agent Xi

• Xj : higher priority neighbors than Xi

• Lower bound for value d:

• ∀ d ∈ Di:

• LB(d) = δ(d) + ∑ Xk ∈ children lb(d,Xk)

• similarly for the upper bound for value d

• Lower bound:

• LB = min d ∈ Di LB(d)

• similarly for the upper bound

• For leaves: LB(d) = UB(d) = δ(d)

• If not a leaf but has not get received a COST message: LB = δ(d) and UB = ∞

X2

X3

X4

Constraint Graph

X1

X2

X3

X4

Communication Graph

### When does Xi change value?

• Whenever LB(di) exceeds the backtrack threshold value, Xi changes its variable value to one with smaller lower bound

• The threshold is updated with the following three ways:

• Its value can increase whenever Xi determines that LB is greater than the current threshold

• guarantees that there is always a variable with a lower bound than the threshold

• Its value can decrease whenever Xi determines that UB is lower than the current threshold

• Invariant: LB ≤ threshold ≤ UB

• Its value is also updated whenever a THRESHOLD message is received from a parent

• a parent subdivides its own threshold value among its children

• t(d,Xk): the threshold on cost allocated by parent Xi to child Xk

• then the value of t(d,Xk) respects the following invariants:

• threshold = δ(di) + ∑ Xk ∈ children t(di,Xk)

• ∀ d ∈ Di, ∀ Xk ∈ children: lb(d,Xk) ≤ t(d,Xk) ≤ ub(d,Xk)

### Example

• All agents begin concurrently choosing 0.

• Each agents send a VALUE message to lower priority agents along the constraint graph

• We will follow one specific execution path - there are many possible

X1

X1

X2

X2

X3

X4

X3

X4

Constraint Graph

Communication Graph

### Example

X1=0

• X2 receives X1’s VALUE message

• and records this value to its context

• X2’s context: {X1=0}

• Then it computes bounds:

• LB(0) = δ(0) + lb(0,X3) + lb(0,X4) = 1

• LB(1) = δ(1) + lb(1,X3) + lb(1,X4) = 2

• LB(0) < LB(1) ⇒ LB = LB(0) = 1

• Similarly: UB = ∞

• threshold was set to LB(0), equal to 1 so the invariant holds

• Transmits a COST message to X1:

• COST( {X1=0}, 1, ∞ )

X2=0

X4=0

X3=0

### Example

X1=0

• X3 receives X1’s and X2’s VALUE messages

• and records these values to its context

• X3’s context: {X1=0, X2=0}

• Then it computes bounds:

• LB(0) = δ(0) = 1 + 1 = 2

• LB(1) = δ(1) = 2 + 2 = 4

• LB(0) < LB(1) ⇒ LB = LB(0) = 2

• Similarly: UB = 2

• threshold was LB(0) = 2 so the invariant holds

• Transmits a COST message to X2:

• COST( {X1=0, X2=0}, 2, 2 )

• Similarly with X4... but no reference to X1

X2=0

X4=0

X3=0

### Example

X1=1

• X1 receives X2’s COST message

• COST( {X1=0}, 1, ∞ )

• test if compatible with its current context

• store: lb(0,X2) = 1 and ub(0,X2) = ∞

• Then it computes bounds:

• LB(0) = δ(0) + lb(0,X2) = 0 + 1 = 1

• LB(1) = δ(1) + lb(1,X2) = 0 + 0 = 0

• LB(1) < LB(0) ⇒ LB = LB(1) = 0

• Similarly: UB = ∞

• threshold was 0, but LB(0) = 1:

• violation of the invariant, change assignment

• Send VALUE messages to children

X2=0

X4=0

X3=0

### Example

X1=1

• Assume COST messages from X3 and X4 are delayed... instead VALUE message from X1 arrives first at X2

• Current context at X2: {X1=1}

• When X2 receives the COST messages from X3 its context will be incompatible with X2’s

• the bounds in the message will not be stored

• The message from X4 is not incompatible:

• store lb(0,x4) = 1 and up(0,x4) = 1

• Then it computes bounds:

• LB(0) = δ(0) + lb(0,X3) + lb(0,X4) = 2+ 0 +1 = 3

• LB(1) = δ(1) + lb(1,X3) + lb(1,X4) = 0 +0 +0 = 0

• LB(1) < LB(0) ⇒ LB = LB(1) = 1

• Similarly: UB = ∞

X2=1

X4=0

X3=0

### Example

X1=1

• X2 will inform X3 and X4 about the changes.

• Similar changes will take place on X3 and X4:

• 1 will be selected as the value

• COST messages will be transmitted:

• from X4 to X2: ( {X2=1}, 0, 0 )

• from X3 to X1 and X2: ( {X1=1,X2=1}, 0, 0 )

• from X2 to X1: ( {X1=1}, 0, 0)

• this is after receiving the COST messages from X3 and X4

• Upon receipt of the COST message of X2 at X1:

• LB = UB = threshold = 0

• X1 sends TERMINATE messages to other agents.

X2=1

X4=1

X3=1