CALCULUS FOR THE MORNING COMMUTE

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1 CALCULUS FOR THE MORNING COMMUTE Prof. David Bernstein Department of Computer Science JMU Department of Mathematics and Statistics Fall 2007 Colloquia

2 Calculus for the Morning Commute -1 Motivation Some History: The shortest path problemis easy to solve 25 years ago it became clear that we could develop route guidance systems if we only had network data The Breakthrough (in the U.S.): The Census Bureau digitized the entire country Today: A few companies maintain and market (very similar) street databases Many companies develop and market route guidance systems A Thought Experiment: What would happen if everyone used such a system to determine their route from home to work?

3 Calculus for the Morning Commute -2 Notation The Network: N denotes the set of nodes A denotes the set of arcs (or links) W N 2 denotes the set of origin-destinat pairs R w denotes the set of routes connecting w W (with R w = R w ) R = w W R w (with R = w W R w ) c : Z R + R R + denotes the vector of route cost functions Demand and Flows: D =(D w : w W) Z R ++ denotes the demand vector h =(h r : r R) Z R + denotes the route flow vector H D = { h Z R + : r R w h r = D w,w W } denotes a feasible route flow pattern

4 Calculus for the Morning Commute -3 Behavioral Model The Key Behavioral Assumption: Each commuter attempts to minimize her/his own travel cost, given the behavior of other commuters The EquilibriumModel: A flow pattern, h H D is said to be a (Nash) network equilibriumiff: h r > 0 c r (h) c s (h +1 s 1 r ) s R w for all r R w and w W. An Interpretation: In equilibrium, no commuter has any incentive to change her/his route

5 Calculus for the Morning Commute -4 The Important Questions Existence: When will such an equilibriumexist? Uniqueness: When will there be exactly one equilibrium? Stability: When will the equilibrium(or set of equilibria) be stable? Computation: Can we find equilibria efficiently (using a numerical algorithm)?

6 Calculus for the Morning Commute -5 A Simplifying Assumption Something a Computer Scientist Should Never Do: Consider the limiting case Operationalizing this Simplification: The equilibriumcondition: h r > 0 c r (h) c s (h +1 s 1 r ) s R w becomes: h r > 0 c r (h) c s (h + ɛ1 s ɛ1 r ) s R w and we consider the limit as ɛ 0

7 Calculus for the Morning Commute -6 Starting Again Notation: Replace Z with R above Let c w (h) =min{c s (h) :s R w } The EquilibriumModel: A flow pattern, h H D is said to be a continuous network equilibriumiff: h r > 0 c r (h) =c w (h) for all r R w and w W. The set of such flow patterns is denoted by CNE(c, D) Interpretation: In equilibrium, the cost on all used routes (connecting an OD-pair) must be minimal (and, hence, equal)

8 Calculus for the Morning Commute -7 A Graphical Approach The Network: Origin Destination A Simple Case: c 1 c2( h2) c1( h1) c 2 h 1 =0 h1= D h2= D h 2 =0

9 Calculus for the Morning Commute -8 Some Interesting Cases No Intersection: c1( h1) c 1 c 2 c2( h2) h 1 =0 h1= D h2= D h 2 =0 Discontinuous Costs: c 1 c2( h2) c1( h1) c 2 h 1 =0 h1= D h2= D h 2 =0

10 Calculus for the Morning Commute -9 Some Interesting Cases (cont.) Many Points of Equal Cost: c 1 c2( h2) c1( h1) c 2 h 1 =0 h1= D h2= D h 2 =0 No Intersection: c 1 c1( h1) c2( h2) c 2 h 1 =0 h1= D h2= D h 2 =0

11 Calculus for the Morning Commute -10 Lagrange Multipliers: A Review The Theorem: Let f : R n R and h : R m R be appropriately continuous and differentiable functions. Let x R n with h(x) =c, and let S denote the level set for h with value c. If f restricted to S has an optimum at x then there exists a λ R such that: f(x) λ h(x) =0 An Interpretation: Let L(x) =f(x) λh(x) and consider the critical points of L

12 Calculus for the Morning Commute -11 Karush-Kuhn-Tucker Conditions Motivation: L involves problems of the form: min x f(x) s.t. h i (x) =0fori =1,...,l KKT deal with problems of the form: min x f(x) s.t. h i (x) =0fori =1,...,l g i (x) 0fori =1,...,m The Necessary Conditions: For appropriately continuous and differentiable functions, if x solves the problemabove then there exist scalars µ i and λ i such that: f(x)+ m i=1 µ i g i (x)+ l i=1 λ i h i (x) =0 µ i g i (x) =0fori =1,...,m µ i 0fori =1,...,m The Sufficient Conditions: Require appropriate convexity of f, g i and h i

13 Calculus for the Morning Commute -12 Back to the Continuous EquilibriumProblem An Optimization Problem: min h V (h) s.t. r Rw h r = D w h r 0 w W r R The KKT Conditions: r V (h) µ w η r 0 for all r R w and w W η r h r =0forallr R η r 0 for all r R

14 Calculus for the Morning Commute -13 Using the KKT Conditions r V (h) µ w η r 0 for all r R w and w W η r h r =0forallr R η r 0 for all r R Some Simple Results: It follows from η r 0 and r V (h) µ w η r 0 that r V (h) µ w It follows from η r h r = 0 that h r > 0 η r =0 It follows that h r > 0 r V (h) =µ w It Would Be Nice If: r (h) =c r (h) for all r R (i.e., V (h) =c(h)) Because Then: µ w would be c w (h) A minimizer of V (h) subject to h H D would be an equilibrium

15 Calculus for the Morning Commute -14 In Order to Move Ahead An Important Question: Given a function, c : R R + R R +, under what conditions does there exist a function, V C 1 (R R +), with V = c? This Question Arises Elsewhere: In physics V is called a potential function for the gradient vector field, c

16 Calculus for the Morning Commute -15 In Order to Move Ahead (cont.) One Important Sufficient Condition: Path Independence [Stoke s Theorem] A Potential Function: V (h) = h 0 c(ω)dω A More Easily Verified Sufficient Condition: Lipschitz continuous costs (i.e., c(x) c(y) K x y where denotes the max norm) Symmetry (i.e., r c s (h) = s c r (h) for almost all h where both gradients exist) [Frobenius Theorem] A Usable Formulation: The Rectilinear Path of Integration: V (h) = R h i i=1 0 c i (h 1,h 2,...,h i 1,x,0,...,0)dx

17 Calculus for the Morning Commute -16 How Does This Help? Existence: An equilibriumwill exist whenever a minimum exists Uniqueness: The objective function is, in general, not strictly convex so equilibriumroute flows are not unique However, if route costs are additive we can use the following function of link flows: L i=1 f i 0 t i(f 1,f 2,...,f i 1,x,0,...,0)dx which is strictly convex if the link cost function, t, is strictly monotone

18 Calculus for the Morning Commute -17 How Does This Help? (cont.) Calculation: Consider feasible descent algorithms we need to calculate an initial feasible solution and then we need to iteratively find feasible descent directions. Can we? Stability: We need a behaviorally meaningful adjustment process

19 Calculus for the Morning Commute -18 A Behavioral Adjustment Mechanism Notation: ḣ rs denotes the switching rate from r to s (where r, s R w ) and ḣr s R w r(ḣsr ḣrs). ḣ rs is assumed to be determined by some route switching process a rs (h) 0 with an associated adjustment operator a r (h) s R w r(a sr a rs ). A function p : R + R R satisfying ṗ(t) = a[p(t)],t R + is called an adjustment path for a. Behavioral Assumptions: A c-adjustment operator, a, must satisfy: (Rationality) a rs (h) > 0 c r (h) >c s (h) (Feasibility) p(0) H D p H D (Persistency) a(h) =0 h CNE(c, D) for all D and r, s R w

20 Calculus for the Morning Commute -19 Adjustments and Stability Notation: The Hausdorff distance froma point, h R R, to a nonempty set, S R R,isgivenby ρ(h, S) =inf{ h x : x S}. The neighborhood of S in H D is given by H D (S, ɛ) ={h H D : ρ(h, S) ɛ}. The set of minima is given by MIN(V c,s)={h S : V c (h) =min g S V c (g)}. Definitions: A set S H D is locally V c -minimal iff there exists some ɛ>0 such that S =MIN[V c,h D (S, ɛ)] and S is isolated iff S = KKT(V c,h D ) H D (S, ɛ). A set, S H D is asymptotically a-stable iff there exists some ɛ>0 such that p(0) H D (S, ɛ) lim t ρ[p(t),s]=0 and for each ɛ>0 there is some α ɛ (0,ɛ]such that p(0) H D (s, α ɛ ) p(r + ) H D (S, ɛ)

21 Calculus for the Morning Commute -20 A Stability Property Theorem: If c is a gradient cost structure with cost potential V c, then for all c-adjustment processes a and demand patterns D each isolated, locally V c - minimal set, S H D is asymptotically a-stable. Sketch of the Proof: 1. Show that V c is a strict Liapunov function for every c-adjustment process a (i.e., that h H D V c (h) 0 and that V c (h) =0 a(h) =0). 2. Show that if there exists a Liapunov function V c for a on H D then every locally V c -minimal set is a-stable. 3. Show that all c-adjustment processes, a, for gradient cost structures eventually converge to network equilibria.

22 Calculus for the Morning Commute -21 An Interesting Adjustment Process Notation: For any x R let x + =max{0,x} δ 0 (0) = 1 and δ 0 (x) = 0 for all x>0 The Process: a rs(h) =h r [c r (h) c s (h)] + δ 0 [c s (h) c w (h)] a r (h) = s R w r[a sr(h + ) a rs (h + )] What s Interesting About It? It involves discontinuities at every point where the set of minimal cost routes in R w changes (for any w W)

23 Calculus for the Morning Commute -22 A Solution Concept for Discontinuous Dif. Eqs. Notation: Given x R n and a nonempty set S R n, the Hausdorff distance from x to S is defined as ρ(x, S) =inf{ x y : y S} For any ɛ>0, closed set X R n and nonempty set S X, theɛ-neghborhood of S in X is defined as X(S, ɛ) ={x X : ρ(x, S) ɛ} conv(s) denotes the convex hull of the set S R n cconv(s) denotes the closure of the convex hull of the set S R n Getting Started: If the image set of a is denoted by a [H D (h, ɛ)] = {a (g) :g H D (h, ɛ)} then cconv(a [H D (h, ɛ)]) contains all of the convex combinations of vectors in a [H D (h, ɛ)] ɛ>0 cconv(a [H D (h, ɛ)]) must contain the limits of such convex combinations as the size of the neighborhood goes to zero

24 Calculus for the Morning Commute -23 A Solution Concept for Disc. Dif. Eqs. (cont.) A Reminder A function, f :[a, b] R n is said to be absolutely continuous on [a, b] if given ɛ>0 there exists a δ>0 such that n i=1 f(y i ) f(z i ) <ɛ for every finite collextion, {(y i,z i ):i =1,...,n} of nonoverlapping intervals with n i=1 y i z i <δ Krasovkij Solutions to a : An absolutely continuous function, p : R + R R + is a solution to a iff for almost all t R + ṗ(t) ɛ>0 cconv(a [H D (p(t),ɛ)]) An Observation: Over regions of H D where a is continuous, this reduces to ṗ(t) =a [p(t)] (i.e., the classical Carathéodory solution)

25 Calculus for the Morning Commute -24 An Example One OD with three Routes: D =12 c 1 (h) =1+h 2 1 c 2 (h) =1+h 2 c 3 (h) =15+h 3 Solving: The unique equilibriumis h =(3, 9, 0)

26 Calculus for the Morning Commute -25 An Example h s 12 9 C 2. p(0) 1 4 p(0) C, is the set of points with c 1 (h) =c 2 (h) At p(0) commuters on 3 will siwtch to 1 and 2 so p moves in direction 1 This movement collapses the set of minimum cost routes from {1, 2} to {2} so commuters instantly stop switching to 1, moving p in direction 2 This shifts the set of minimum cost routes to {1} so that commuters instantly stop switching to 2 and start switchingto1,movingp in direction 3 12 h 1

27 Calculus for the Morning Commute -26 An Example (cont.) Conclusion: No direction of movement can persist for more than an instant (called a chattering regime) Commuters are continuously switching to 1 or 2 (or both) so the p is moving up and to the right Interpretation: The cone of feasible direction vectors corresponds to ɛ>0 cconv(a [H D (p(0),ɛ)]) (i.e., the Krasovskij solutions) Observations: The only absoutely continuous adjustment path starting at a point on C is the path with trajectory along C ṗ(t) must be almost everywhere tangent to C a [p(t)] is always parallel to direction 1 for any point p(t) C So ṗ(t) isnowhere equal to a [p(t)]

28 Calculus for the Morning Commute -27 Lower-Semicontinuous Costs Definition of Equilibrium: The classical definition (Wardrop) doesn t work Other definitions (Dafermos, Heydecker) allow collaboration The most obvious alternative (Nash) doesn t work Modeling Smallness : liminf ɛ 0 c r (h + ɛ1 r ɛ1 s )= lim ɛ 0 inf{c r (h+α1 r α1 s ):0<α<min(ɛ, h s )} New Definition: h is a user equilibrium iff: h r > 0 c r (h) liminf ɛ 0 c s (h + ɛ1 s ɛ1 r )

29 Calculus for the Morning Commute -28 Lower-Semicontinuous Costs (cont.) One Result: If c is lower semicontinuous then every Wardrop eqmis a user eqm If c is upper semicontinuous then every user eqm is a Wardrop eqm Another Result: If c is lower semicontinuous and flow shifts between routes only create discontinuities on those links where the flow changes then a user equilibriumexists

30 Calculus for the Morning Commute -29 SRD Equilibrium The Issue: Drivers simultaneously choose both a path, p, and a departure time, t [0,T]. Fluid Approximation: The departure rate on path p at time t is denoted by h p (t) R +

31 Calculus for the Morning Commute -30 SRD Equilibrium(cont.) Notation: h(t) =(h r (t) :r R) H D = { h : r R w T 0 h r (t)dν(t) =D w,w W } (where ν(t) is a Lebesgue measure on [0,T]). Assumptions: Each departure rate pattern, h, gives rise to a (time varying) traffic pattern, x(t) =(x a (t) : a A). The relationship between h and x is driven by the time needed to traverse arc a when entered at time t, d a (t). Link travel times are determined by the link occupancies at the time the link is entered (i.e., the number of vehicles ahead of you on a link when you enter).

32 Calculus for the Morning Commute -31 SRD Equilibrium(cont.) The time needed to traverse path r when departing from the origin at time t is given by: d p (t, h) =d a r 1 [x a r 1 (t)] + d a r 2 [x a r 2 (τ a r 1 (t))] where + + d a r m(r) [x a r m(r) (τ a r m(r) 1 (t))] τ a r 1 (t) =t + d a r 1 [x a r 1 (t)] r R τ a r i (t) =τ a r i 1 (t)+d a r i [x a r i (τ a r i 1 (t))] r R,i [2,m(r)].

33 Calculus for the Morning Commute -32 SRD Equilibrium(cont.) Travelers may arrive early or late but incur a schedule cost. α is the dollar penalty for early arrival. β is the dollar penalty for late arrival. [T,T + ] is the set of equally acceptable arrival times. The schedule cost is given by Φ r (t, h) = α[(t ) (t + d r (t, h))] if (T ) > [t + d r (t, h)] 0 if (T ) [t + d r (t, h)] (T + ) β[(t + d r (t, h)) (T + )] if (T + )< [t + d r (t, h)], Letting γ denote the value of travel time, total cost is given by c r (t, h) =γd r (t, h)+φ r (t, h) r R

34 Calculus for the Morning Commute -33 SRD Equilibrium(cont.) Letting µ r (h) = ess inf{c r (t, h) :t [0,T]} the relevant lower bound on achievable costs for a w-commuter is given by µ w (h) =min{µ r (h) :r R w } we can define an equilibriumas follows: Definition 1 (SRD Equilibrium) A departure rate pattern, h H D is said to be a simultaneous route and departure-time choice equilibrium (SRD equilibrium) for D if and only if (iff) h satisfies the following condition for all w W,and r R w : h r (t) > 0= c r (t, h) =µ w (h) for ν-almost all t [0,T]

35 Calculus for the Morning Commute -34 SRD Equilibrium(cont.) It is possible to formulate the SRD equilibrium problemas an infinite dimensional variational inequality: Theorem 1 A departure rate pattern, ĥ H D is an SRD equilibrium for c if and only if T r R for all h H D. 0 c r(t, ĥ)[h r(t) ĥr(t)]dν(t) 0 (1) This result is quite general, and does not depend on the formof d a or c r. This result allows us to consider questions of existence and solution algorithms.

36 Calculus for the Morning Commute -35 The Big Open Questions SRD Equilibrium: Existence and Uniqueness Efficient Algorithms Information Provision: How will users respond to traffic forecasts? How do we incorporate their reponses into our forecasts?