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A descent framework for linked signal system with network flows
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Applied Mathematics and Computation 194 (2007) 441–452 www.elsevier.com/locate/amc
A descent framework for linked signal system with network flows Suh-Wen Chiou Department of Information Management, National Dong Hwa University 1, Sec. 2, Da Hsueh Road, Shou-Feng, Hualien 97401, Taiwan
Abstract Optimization of linked signal system in urban traffic road networks is considered in this paper. This problem can be formulated as a nonlinear program subject to network flows following Wardrop’s first principle in which traffic rerouting effects are taken into account. We propose a descent framework to effectively solve a linked signal system with network flows. Numerical calculations are implemented on well-known example road networks and good results are reported. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Linked signal system; Traffic assignment; Descent framework; Network flows
1. Introduction Optimization of linked signal system (LSS) in urban traffic road networks is considered in this paper. This problem can be formulated as a special case of the Mathematical Program with Equilibrium Constraints (MPEC), which has been extensively studied [1–4]. For an urban road network of signal-controlled junctions with short spacing between adjacent links, a linked signal system can be established. The well-known TRANSYT (TRAffic Network StudY Tool) model [5] has been widely recognized as one of the most useful tools in studying the optimization of linked signal system. As it has been indicated by Allsop [6], the network flows and travel costs are strongly influenced by the operation of signals in a linked signal system. For a LSS, a good performance value can be achieved by optimizing a chosen objective function with respect to the signal settings, in which the users’ behavior of choice of routes is supposed to follow Wardrop’s first principle. That is, a road user will choose his route between a specified origin-destination pair with minimal travel cost, which is in turn dependent on the choice of the linked signal settings. Therefore, in the optimization process for the LSS, not only the signal settings themselves need to be considered but also the consequential effects on the network flows caused by the signal design need to be taken into account. Ways of using mathematical programming to solve the linked signal setting problem (LSSP) and network flow have been well researched. Allsop and Charlesworth [7] reported a mutually consistent calculation for E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.04.046
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linked signal settings and equilibrium flow. The resulting mutually consistent signal settings and link flows will, however, in general be a non-optimal solution as has been discussed by Gershwin and Tan [8] and Dickson [9]. Smith [10] also presented a capacity-maximizing policy where a conventional signal setting problem can be formulated as a Wardropean equilibrium. As for the development of solution methods for network design with general applications, [11–18] have employed the bilevel programming technique to tackle a road network design problem. Recently, Yang and Yagar [19] proposed a sensitivity analysis based (SAB) algorithm to solve signal settings in a simple road network where a linearized sub-problem is formulated at current signal settings and solved by the simplex method. Ceylan and Bell [20], on the other hand, proposed a genetic algorithm (GA) to calculate the linked signal settings and network flows where numerical illustrations were performed on a small-scale road network. Because of the non-convexity of the LSSP, solution algorithms aforementioned can solve the LSSP only locally. In this paper, we propose a descent framework for a coordinated fixed time linked signal system, which can be formulated as one LSSP with respect to the common cycle time, the start and duration of greens. The performance index is defined as the sum of a weighted linear combination of rate of delay and number of stops per unit time for all traffic streams. Following the technique of sensitivity analysis proposed by Patriksson [21], the directional derivatives and gradients are obtained by solving an affine variational inequality problem with respect to the changes in the signal settings. A Quasi-Newton projection (QP) method is presented to effectively solve the LSSP. A new class of hybrid heuristics combining the locally optimal search and global search in the signal settings is thus developed. The Karush–Kuhn–Tucker (KKT) points for the LSSP can be easily found with global convergence. Numerical calculations are conducted on well-known example road networks. As it shows, the proposed QP method achieved substantially better performance than did those other approaches when solving the LSSP. The rest of the paper is organized as follows. In next section the LSSP with respect to signal setting variables is formulated, where the sensitivity analysis of traffic equilibrium is conducted. In Section 3, a descent framework is presented where a globally convergent Quasi-Newton projection (QP) method is proposed. A new class of hybrid heuristics combining the locally optimal search and global search in signal settings is therefore developed. In Section 4, numerical computations are conducted on well-known example road networks with distinct sets of initial data. As it shown, the proposed QP method has achieved substantially better performance than did those other approaches when solving a LSSP. Conclusions for this paper and further points of interests are made in Section 5. 2. Problem formulation The problem for solving the linked signal system (LSS) with respect to signal setting variables while the travelers’ route choice is taken into account is formulated. A generalized traffic assignment with signal settings in terms of a parametric variational inequality is presented. The sensitivity analysis of traffic equilibrium is performed for which the computations of the directional derivatives and the gradients can be solved by an affine variational inequality. 2.1. Notation GðN ; LÞ a signal-controlled network with node set N and link set L W set of origin-destination (OD) pairs Rw set of paths between OD pair w, 8w 2 W T ¼ ½T w matrix of travel demands for OD pair w 8w 2 W W ¼ ðf; h; /Þ set of signal setting variables, respectively for the reciprocal of cycle time, start and duration of greens, where h ¼ ½hjm and / ¼ ½/jm respectively represent the vector of starts hjm and durations of green /jm for signal group j at junction m as proportions of common cycle time ka the duration of effective green for link a 8a 2 L gjm the minimum green for signal group j at junction m cjlm the clearance time between the end of green for signal group j and the start of green for incompatible signal group l at junction m
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Xm ðj; lÞ a collection of numbers 0 and 1 for each pair of incompatible signal groups at junction m; where Xm ðj; lÞ ¼ 0 if the start of green for signal group j proceeds that of l and Xm ðj; lÞ ¼ 1, otherwise Da the rate of delay on link a 8a 2 L Sa the number of stops per unit time on link a 8a 2 L pa the maximum degree of saturation for link a 8a 2 L sa the saturation flow on link a 8a 2 L q ¼ ½qa vector of average link flows 8a 2 L f ¼ ½fp vector of path flows between points of entry to points of exit from the network 8p 2 Rw ; 8w 2 W d the link-path incidence matrix D the OD-path incidence matrix c ¼ ½ca vector of link flow travel cost 8a 2 L C ¼ ½C p vector of path flow travel cost 8p 2 Rw ; 8w 2 W p vector of minimum travel cost 2.2. A linked signal system problem (LSSP) By using the traffic model from TRANSYT for evaluating the performance of linked signal settings, the objective function for the LSSP can be formulated as a non-linear mathematical program as follows. As for the performance index used in TRANSYT, it can be expressed as a weighted linear combination of travel delays and number of stops at downstream links at signal-controlled junctions. Let MD and MS respectively denote the monetary factors for rate of delay and number of stops, we have X Z ¼ Z 0 ðW; q ðWÞÞ ¼ M D Da W 1aD þ M S S a W 1aS ; ð1Þ Min W
a2L
subject to
ð2Þ
fmin 6 f 6 fmax ; gjm f 6 /jm 6 1;
8j; m;
qa 6 pa sa ka ; 8a 2 L; hjm þ /jm þ cjlm f 6 hlm þ Xm ðj; lÞ; 1 aD
ð3Þ j 6¼ l; 8j; l; m;
ð4Þ ð5Þ
1 aS
where W and W are respectively link-specific weighting factors for the rate of delay and the number of stops per unit time used in TRANSYT. The constraint (2) is the bound for the common cycle time and the constraints (3)–(5) are for the green phase, link capacity and clearance time. Also the equilibrium flows q ðWÞ can be found by solving the following traffic assignment problem. 2.3. User equilibrium traffic assignment A general formulation for a user equilibrium traffic assignment problem can be expressed in terms of a variational inequality as shown by Smith [22] and identified by Dafermos [23]. Find values q such that ct ðqÞðz qÞ P 0
ð6Þ
for all z 2 K ¼ fq : q ¼ df ; T ¼ Df ; f P 0g where the superscript t denotes matrix transpose. The equivalent variational inequality in terms of path flow for (6) can be given as follows: Find values f such that C t ðf Þðz f Þ P 0
ð7Þ
for all z 2 K 1 ¼ ff : T ¼ Df ; f P 0g. 2.4. A parametric variational inequality For user equilibrium traffic assignment problem with signal settings, W ¼ ðf; h; /Þ, the parametric variational inequality for (6) can be expressed as follows: ct ðW; qÞðz qðWÞÞ P 0 for all z 2 KðWÞ ¼ fq : qðWÞ ¼ df ðWÞ; T ¼ Df ðWÞ; f ðWÞ P 0g.
ð8Þ
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2.5. Sensitivity analysis by directional derivatives Following the technique employed by Patriksson [21], the sensitivity analysis of traffic equilibrium of (8) can be established in the following way. Introduce e ðWÞ ¼ fdqðWÞ : 9df ðWÞ such that dqðWÞ ¼ ddf ðWÞ; Ddf ðWÞ ¼ 0; and df ðWÞ 2 K 0 ðWÞg K and
K 0 ðWÞ ¼
8 > > > <
ðiÞ dfp ðWÞ free;
> df : > > :
if f p ðWÞ > 0
ðiiÞ dfp ðWÞ ¼ 0; if C p > pw ðiiiÞ dfp ðWÞ ¼ 0; if C p ¼ pw and f p ðWÞ ¼ 0 with dC p > 0; ðivÞ dfp ðWÞ > 0;
if C p ¼ pw ; and f p ðWÞ ¼ 0 with dC p 6 0
9 > > > = 8p 2 Rw ; w 2 W > > > ;
;
where the changes in link or path flows with respect to the changes in signal settings are denoted by dqðWÞ or df ðWÞ and the corresponding changes in path flow travel time denoted by dC p . Therefore the directional derive ðWÞ, atives of (8) dq can be obtained by solving the following affine variational inequality. For all z 2 K t
ðrW cðW; qÞdW þ rq cðW; qÞdqÞ ðz dqÞ P 0;
ð9Þ
where rW c and rq c are gradients evaluated at ðaW; qÞ when the changes in signal settings dW are specified. 2.6. A MPEC problem In constraints (2)–(5), let A and b be the coefficient matrix and corresponding constant vector associated with the signal settings thus the constraints (2)–(5) can be re-expressed as the following form. AWt 6 b: The LSSP (1)–(5) can be briefly addressed as follows: Min
Z ¼ Z 0 ðW; q ðWÞÞ;
subject to
AWt 6 b
W
and
q 2 S 0 ðWÞ;
ð10Þ
where S 0 ðÞ denotes the solution set for (8). 2.7. A single-level problem Following the results in sensitivity analysis for the parametric variational inequality (9), the LSSP (10) now can be rewritten as the following single-level problem. Min
Z ¼ Z 1 ðWÞ;
subject to
AWt 6 b:
W
ð11Þ
3. Solution method As it is known that the LSSP (11) is one instance of mathematical program with equilibrium constraints (MPEC), subject to the non-convexity of MPEC, the generalized gradients of (11) in terms of Clarke’s definition [24] can be expressed in the following manner. oZ 1 ðW Þ ¼ cof lim rZ 1 ðWk Þ : Wk ! W ; rZ 1 ðWk Þexistsg: k!1
ð12Þ
According to [24], the generalized gradient is a convex hull (abbreviated to co) of all points of the form lim rZ 1 ðWk Þ where the subsequence fWk g converges to the limit value W . And the gradients in (12) evaluated at ðWk ; qk Þ are expressed in the following way.
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rZ 1 ðWk Þ ¼ rW Z 0 ðWk ; qk Þ þ rq Z 0 ðWk ; qk ÞdqðWk Þ;
445
ð13Þ
where the gradient of equilibrium flow with respect to signal settings can be obtained by solving (9). 3.1. A descent framework Regarding the solution method for LSSP (11), in the followings we propose a descent framework via the implementation of a Quasi-Newton projection method (QP) to determine the search direction along which the value of the objective function decreases monotonically. The search process is either terminated at a KKT point or a new search direction can be generated. Theorem 1 (Quasi-Newton method). Let W1 locally solve LSSP (11), and Q1 the corresponding positive definite symmetric matrix. For k ¼ 1; . . . ; n, let Wkþ1 ¼ Wk þ ad k ;
ð14Þ
where d k ¼ Qk rZ t1 ðWk Þ
ð15Þ
is a descent direction at Wk with a for LSSP (11). For k ¼ 1; . . . ; n 1, Qk is determined by Qkþ1 ¼ Qk
Qk uk utk Qk vk vtk þ t ; utk Qk uk v k uk
ð16Þ
where uk ¼ Wkþ1 Wk and vtk ¼ rZ 1 ðWkþ1 Þ rZ 1 ðWk Þ. If rZ 1 ðWk Þ 6¼ 0 for k ¼ 1; . . . ; n, then Q1 ; . . . ; Qn are symmetric and positive definite so that d 1 ; . . . ; d n are descent directions. Proof. See [25].
h
The search direction generated by Quasi-Newton method for an unconstrained nonlinear problem is a descent direction, which strictly decreases the objective function value at each iteration provided that the corresponding gradient is not zero. In the followings, we apply the Quasi-Newton method to the linear constraint set as in (11) by using the concept of gradient projection method. Definition 2 (Projection matrix). A n n matrix H is called a projection matrix if H ¼ H t and H t H ¼ H . Thus the Quasi-Newton Projection (QP) method can be presented in the following way. Theorem 3 (Quasi-Newton Projection method, QP). In problem (11), a sequence of iterates fWk g can be generated according to Wkþ1 ¼ Wk þ aH k d k ;
ð17Þ
where dk is the descent direction determined by (15) and a is the step length which minimize Z1 along dk from Wk and the projection matrix Hk is of the following form. H k ¼ I M tk ðM k M tk Þ1 M k ;
ð18Þ
where Mk is the gradient of active constraints in (11) at Wk , where the active constraint gradients are linearly independent and thus Mk has full rank. The search direction hk can be determined in the following form. hk ¼ H k d k :
ð19Þ
Then the sequence of points fWk g generated by the Quasi-Newton projection method, Z 1 ðWkþ1 Þ < Z 1 ðWkþ1 Þ;
k ¼ 0; 1; 2; . . . ;
ð20Þ
whenever H k rZ 1 ðWk Þ 6¼ 0. Proof. Following the results of Theorem 1 in (15), we have rZ t1 ðWk Þd k ¼ rZ t1 ðWk ÞQk rZ 1 ðWk Þ < 0;
k ¼ 1; 2; 3; . . . ;
ð21Þ
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since Qk is positive definite, for all rZ 1 ðWk Þ 6¼ 0. Multiply equation (21) by projection matrix Hk, it becomes rZ t1 ðWk ÞH k d k ¼ rZ t1 ðWk ÞH k Qk rZ 1 ðWk Þ ¼ rZ t1 ðWk ÞH tk H k Qk rZ 1 ðWk Þ ¼ kH k k2 rZ 1 ðWk ÞQk rZ 1 ðWk Þ < 0:
ð22Þ
Thus for sufficiently small l, l > 0, we have Z 1 ðWk Þ > Z 1 ðWk þ lhk Þ:
ð23Þ
Because by definition a is the step length which minimize P1 along hk from Wk , it implies Z 1 ðWk Þ > Z 1 ðWk þ lhk Þ P Z 1 ðWk þ ahk Þ ¼ Z 1 ðWkþ1 Þ; which completes this proof.
ð24Þ
h
Theorem 4. Following Theorem 3, when H k rZ 1 ðWk Þ ¼ 0, if all the Lagrange multipliers corresponding to the active constraint gradients in (11) are positive or zeros, it implies the current Wk is a KKT point. Otherwise choose ^ k of the active constraint gradients by deleting the one negative Lagrange multiplier, say lj , and construct a new M jth row of Mk, which corresponds to the negative component lj , and make the projection matrix of the following form ^ t Þ1 M ^ k: ^ t ðM ^ kM Hk ¼ I M k k
ð25Þ
The search direction then can be determined by (19) and the results of Theorem 3 hold. Proof. This proof is separated as in the following two parts. First, suppose all the Lagrange multipliers are positive or zeros, and by definition we have 1
0 ¼ H k rZ 1 ðWk Þ ¼ ðI M tk ðM k M tk Þ M k ÞrZ 1 ðWk Þ ¼ rZ 1 ðWk Þ þ M tk w;
ð26Þ
1 ðM k M tk Þ M k rZ 1 ðWk Þ
where w ¼ corresponds to the Lagrange multipliers l matching the active constraint gradients Mk. If l P 0, and the active constraint gradients are linearly independent, then ðWk ; lÞ is a KKT pair to the problem (11) and the current Wk is a KKT point. Or choose one negative Lagrange multiplier, ^ k of the active constraint gradients by deleting the jth row of Mk, and make say lj , and construct another M a new projection matrix as in (25). The new search direction can be determined by (19). Provided that H k rZ 1 ðWk Þ 6¼ 0, then the results of Theorem 3 hold. We only need to show that H k rZ 1 ðWk Þ 6¼ 0. By contra^ t Þ1 M ^ k rZ 1 ðWk Þ, we ^ kM ^ ¼ ðM diction, suppose that H k rZ 1 ðWk Þ ¼ 0. By the definition of Hk and letting w k have ^ t Þ1 M ^ k ÞrZ 1 ðWk Þ ¼ rZ 1 ðWk Þ þ M ^ t ðM ^ kM ^ tw 0 ¼ H k rZ 1 ðWk Þ ¼ ðI M k k k ^:
ð27Þ
^ tw ^ t þ lj At where Aj is the jth row of active constraint gradients, consider (26), we get Since M j k ^ ¼ Mkw t ^ tw 0 ¼ rZ 1 ðWk Þ þ M k þ l j Aj :
ð28Þ
Subtracting (28) from (27), it follows: ^ t ð^ Þ lj Atj : 0¼M k ww
ð29Þ
^ k of full rank. Therefore H k rZ 1 ðWk Þ 6¼ 0 which completes this Since lj ¼ 6 0, it violates that the assumption M proof. h According to the results of Theorems 3 and 4, the search process for solving the problem (11) either can be terminated at a KKT point, which satisfies the following convergent theorem of QP method or a new search direction can be generated. Theorem 5 (Convergence of QP method). Suppose in (11) Z1 function is lower semi-continuous on the domain set X. Given a W1 such that Z 1 ðW1 Þ ¼ b and the level set Pb ¼ fW : W 2 X; Z 1 ðWÞ 6 bg is bounded and Z1 is
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continuous differentiable on the convex hull of Pb . Let fWk g be the sequence of points generated by QP method as described above. Then every cluster point W satisfies 0 2 @Z 1 ðW Þ:
ð30Þ
Proof. We proof this theorem by contradiction. Supposing 0 62 @Z 1 ðW Þ, by definition in (12) there is no subgradient rZ 1 ðWk Þ ¼ 0 at Wk in the convex hull of Pb , whose cluster point is W . Then there is a a > 0 minimizing Z 1 ðW þ ah Þ and a e > 0 such that Z 1 ðW Þ ¼ Z 1 ðW þ a h Þ þ e and W þ a h is an interior point of Pb . By the mean value theorem, for any Wk we have Z 1 ðWk þ a hk Þ ¼ Z 1 ðW þ a h Þ þ rZ t1 ðnk ÞðWk W þ a ðhk h ÞÞ;
ð31Þ
where nk ¼ W þ a h þ lðW Wk þ a ðh hk ÞÞ for some 0 < l < 1. Following Bozano theorem that there is a subsequence fWkn g of fWk g that converges to W , then frZ 1 ðnkn Þg converges to rZ 1 ðW þ a h Þ and fWkn W þ a ðhkn h Þg converges to zero. For sufficiently large kn, the vector nkn belongs to the convex hull of Pb and e e ð32Þ Z 1 ðWkn þ a hkn Þ 6 Z 1 ðW þ a h Þ þ ¼ Z 1 ðW Þ : 2 2 Let akn be the minimizing point of Z 1 ðWkn þ akn hkn Þ. Since fZ 1 ðWkn Þg is monotone decreasing and converges to Z 1 ðW Þ, we have e ð33Þ Z 1 ðW Þ < Z 1 ðWkn þ akn hkn Þ 6 Z 1 ðWkn þ a hkn Þ 6 Z 1 ðW Þ ; 2 a contradiction. Therefore every cluster point W satisfies 0 2 @Z 1 ðW Þ. h Corollary 6 (Stopping condition). If Wk is a KKT point for LSSP (11) satisfying Theorem 5 then the search process may stop; otherwise a new descent direction at Wk can be generated according to Theorems 3 and 4. 3.2. Heuristics for solving LSSP (11) In order to obtain a near global optimum for LSSP (11), a new class of hybrid heuristics in terms of H1–H4 can be developed in the following way. 3.2.1. Heuristic 1 (H1) Heuristic 1 is a locally optimal search with respect to the common cycle time, the start and duration of green at each junction for LSSP (11) and is carried out in the following steps. Step 1. Start with Wk and set index k = 1. Step 2. Solve a user equilibrium traffic assignment problem (8) with signal settings Wk , find the first order derivatives by (9). Step 3. Use the QP method to determine a search direction by (19). Step 4. If H k rZ 1 ðWk Þ 6¼ 0, find a new Wkþ1 in (17) by deciding a the step length, which minimize Z1 along hk and letting k k þ 1. Go to Step 2. If H k rZ 1 ðWk Þ ¼ 0 and all the Lagrange multipliers corresponding to the active constraint gradients are positive or zeros, Wk is a KKT point and stop. Otherwise, follow the results of Theorem 4 and find a new projection matrix and go to Step 3. 3.2.2. Heuristic 2 (H2) Given a common cycle time, a locally optimal search for partial optimization with respect to the start and duration of green at each junction can be conducted as the steps given in Heuristic 1, where the common cycle time is fixed. 3.2.3. Heuristic 3 (H3) Given a common cycle time and duration of green at each junction, an unconstrained optimization problem with respect to the offsets is constructed for LSSP (11). Since there is no physical constraint taken into account
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for the starts of greens, i.e. the offsets, a global search is implemented to find a better KKT point in another part of the feasible region. After such a KKT point is found, conduct Heuristic1 again and locate new signal settings where the common cycle time and green durations are fixed. 3.2.4. Heuristic 4 (H4) Analogous to the Heuristic 3, in Heuristic 4, the Heuristics 1 and 2 are alternately employed after a good KKT point is found. The Heuristic 4 is conducted in the mixture of Heuristics 1–3 until the difference of the values of the performance index between successive iterations is negligible. 4. Numerical calculations In this section, numerical experiments are conducted for the proposed heuristics, H1–H4, and the previous methods: SAB and GA on example road networks in the following way. Firstly, numerical computations are carried out at a two-junction signal-controlled road network, as shown in Fig. 1, where the proposed heuristics, H1–H4, are compared to the lower bound system-optimal (SO) solution for the LSSP. Next, a real data Sioux-Falls aggregated network [26] consisting of 24 nodes, 76 links and 528 OD trips, where 6 signal-controlled junctions are taken into account as shown in Fig. 2, is illustrated for numerical calculations. Using typical values found in practice, the minimum green time is 7 s, and the clearance times are 5 s between incompatible signal groups. The maximum cycle time is set 180 s. 4.1. Computational results for two-junction road network Computational results for conducting the solution methods for LSSP at two-junction road network are summarized in Table 1. The performance index for LSSP in (11) is measured by total travel cost in dollars. The corresponding computational efforts are measured by the numbers of solving traffic assignments. As it seen in Table 1, the initial value of PI (short for performance index) for LSSP is $78 and the best value of PI is $30.8 which is achieved by the system optimum (SO). The proposed heuristics: H1–H4 respectively improved the PI values by 63%, 64%, 65% and 65% while the SAB method and recently proposed GA approach improved the PI values only by 60%. The relative difference ratios in the final values of PI between H1–H4 and SO were 9%, 4%, 3% and 2% while the relative difference ratio in the final value of PI between SAB and SO was nearly 17%. Regarding the numbers of solving traffic assignments for LSSP, it took H4 9 iterations while it took SAB 37 iterations.
Fig. 1. Layout for two-junction road network.
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Fig. 2. Sioux-Falls road network
4.2. Computational results on Sioux Falls city road network Computational results on Sioux Falls city road network are summarized in Tables 2 and 3 for two arbitrary sets of initial signal settings. Starting with two distinct sets of initial signal settings, the proposed Heuristic H4, combining the locally optimal search and global search heuristic achieved at respective local optima with very close values to the SO solutions. As it seen in Table 2, the initial value of PI (short for performance index) for LSSP is $1520 and the best value of PI is $1181.1 which is achieved by the system optimum (SO). The proposed heuristics: H1–H4 respectively improved the PI values by 21%, 22%, 22% and 22% while the SAB method and recently proposed GA approach improved the PI values by 13% and 15% only. The relative difference ratios in the final values of PI between H1–H4 and SO were less 1.5% while the relative difference ratios in the final values of PI between SAB and SO and between GA and SO were nearly 12.5% and 9%. Regarding the numbers of solving traffic assignments for LSSP, it took H4 25 iterations while it took SAB 98 iterations. Computational results at the second set signal settings are given in Table 3 and can be summarized in the following similar way. As it seen in Table 3, the initial value of PI for LSSP is $1650 and the best value of PI is $1181.2 which is achieved by the system optimum (SO). The proposed heuristics: H1–H4 respectively improved the PI values by 28% on average while the SAB method and recently proposed GA approach improved the PI values by 18% and 21% only. The relative difference ratios in the final values of PI between H1 and H4 and SO were less 1% while the relative difference ratios in the final values of PI between SAB and
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Table 1 Computational results for two-junction road network Variable/algorithm Initial 1=f Initial PI
SO 90 78
SAB 90 78
GA 90 78
H1 90 78
H2 90 78
H3 90 78
H4 90 78
/11 =f /21 =f /12 =f /22 =f /32 =f q1 q2 q3 q4 q5 q6 1=f PI #
43 7 43 7 60 750 0 0 750 0 0 60 30.8 5
43 7 43 7 60 750 0 0 750 0 0 60 36.1 37
58 7 58 7 75 750 0 0 750 0 0 75 35.9 11
42 8 42 8 60 750 0 0 750 0 0 60 33.5 7
43 7 43 7 60 750 0 0 750 0 0 60 32.1 9
43 7 43 7 60 750.3 0.1 0.1 749.6 0.1 0.1 60 31.7 10
43 7 43 7 60 750 0 0 750 0 0 60 31.3 9
Where /jm =f denotes the duration of greens for signal group j at junction m measured in sec and 1=f denotes the common cycle time measured in sec. qa is measured in veh/h, PI denotes the performance index value measured in $. # denotes the number of traffic assignments solved.
Table 2 Computational results for Sioux Falls network at 1st set signal settings Variable/algorithm Initial 1=f Initial PI
SO 120 1520
SAB 120 1520
GA 120 1520
H1 120 1520
H2 120 1520
H3 120 1520
H4 120 1520
/17 =f /18 =f /19 =f /1;10 =f /1;16 =f /1;18 =f 1=f PI #
44 52 49 47 53 51 105 1181.1 20
48 46 48 52 58 50 102 1328.5 98
38 57 52 49 45 47 116 1287.2 54
38 57 52 49 45 47 106 1194.3 61
44 50 49 48 51 50 102 1192.7 50
50 44 51 52 53 49 96 1186.7 25
45 48 50 51 55 52 104 1183.5 25
Table 3 Computational results for Sioux Falls network at 2nd set signal settings Variable/algorithm Initial 1=f Initial PI
SO 150 1650
SAB 150 1650
GA 150 1650
H1 150 1650
H2 150 1650
H3 150 1650
H4 150 1650
/17 =f /18 =f /19 =f /1;10 =f /1;16 =f /1;18 =f 1=f PI #
60 55 50 50 55 60 105 1181.2 20
60 58 52 50 57 54 106 1350.2 95
41 59 55 49 53 48 130 1300.7 70
51 56 54 48 54 49 118 1192.1 71
56 53 57 51 55 51 120 1191.2 55
61 58 55 52 55 52 115 1185.2 29
61 58 55 52 55 52 100 1182.1 25
SO and between GA and SO were 14.3% and 10.1%. Regarding the numbers of solving traffic assignments for LSSP, it took H4 25 iterations while it took SAB 95 iterations.
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Implementations for carrying out the computational efforts on the solution methods for LSSP were conducted on SUN SPARC SUNW, 900 MHZ processor with 4 Gb RAM under operating system Unix SunOS 5.8 using C++ compiler gnu g++ 2.8.1. The stopping criterion for these solutions is set when the relative difference in the performance index value between the consecutive iterations less than 0.15%. Computational efforts in solving the LSSP for numerical examples mentioned above were no greater than 10 min CPU time on average. Total computation efforts for complete run of H4 did not exceed 1 min of CPU time in all cases. 5. Conclusions and discussion In this paper, a linked signal setting problem (LSSP) is considered while the route choices of road users are properly taken into account. A mathematical program with equilibrium constraints (MPEC) is formulated for the LSSP where the equilibrium constraint is expressed as a parametric variational inequality with respect to signal settings. Sensitivity analysis by directional derivatives has been conducted by solving an affine variational inequality problem where the prior knowledge of positive path flows is not necessary. A descent framework of a globally convergent Quasi-Newton projection method (QP) has been proposed to effectively solve the LSSP. A new class of hybrid heuristics has been developed. Numerical computations have been carried out on example networks and good results have been obtained. In comparison with other feasible optimization based algorithms, the proposed heuristics have shown promising results in the effectiveness in solving the equilibrium assignment problems and the robustness to various sets of initial signal settings on two test networks. Regarding the development of solution methods for various applications of the MPEC problems, investigations are being undertaken via a non-smooth optimization approach and we will discuss this issue in a subsequent paper. Acknowledgement Many thanks go to Taiwan National Science Council via Grant NSC-95-2416-H-259-014. References [1] Z.Q. Luo, J.-S. Pang, D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, UK, 1996. [2] P. Marcotte, D.-L. Zhu, Exact and inexact penalty methods for generalized bilevel programming problems, Mathematical Programming 74 (1996) 141–157. [3] K. Shimizu, Y. Ishizuka, J. Bard, Nondifferentiable and Two-level Mathematical Programming, Kluwer Academic Publishers, Boston, 1997. [4] J. Outrata, M. Kocvara, J. Zowe, Nonsmooth approach to optimization problems with equilibrium constraints: theory, Applications and Numerical Results, Kluwer Academic Publishers, Boston, 1998. [5] D.I. Robertson, TRANSYT: a traffic network study tool, RRL Report, LR 253, Transport and Road Research Laboratory, Crowthrone, 1969. [6] R.E. Allsop, Some possibilities for using traffic control to influence trip distribution and route choice, in: D.J. Buckley (Ed.), The Proceeding of the 6th International Symposium on Transportation and Traffic Theory, Elsevier, New York, 1974, pp. 345–374. [7] R.E. Allsop, J.A. Charlesworth, Traffic in a signal-controlled road network: an example of different signal timings inducing different routeings, Traffic Engineering Control 18 (5) (1977) 262–264. [8] S.B. Gershwin, H.N. Tan, Hybrid optimization: optimal static traffic control constrained by drivers’ route choice behavior, Massachusetts Institute of Technology, Laboratory for Information and Decision System Report LIDS-p-870, 1979. [9] T.J. Dickson, A note on traffic assignment and signal timings in a signal-controlled road network, Transportation Research B 15 (1981) 267–271. [10] M.J. Smith, A local traffic control policy which automatically maximizes the overall travel capacity of an urban road network, Traffic Engineering and Control 21 (1980) 298–302. [11] P. Marcotte, Network design problem with congestion effects: a case of bilevel programming, Mathematical Programming 34 (1986) 142–162. [12] T.L. Friesz, R.L. Tobin, H.-L. Cho, N.J. Mehta, Sensitivity analysis based algorithms for mathematical programs with variational inequality constraints, Mathematical Programming 48 (1990) 265–284. [13] S. Suh, T.J. Kim, Solving nonlinear bilevel programming models of the equilibrium network design problem: a comparative review, Annals of Operations Research 34 (1992) 203–218.
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[14] P. Marcotte, G. Marquis, Efficient implementation of heuristics for the continuous network design problem, Annals of Operations Research 34 (1992) 163–176. [15] H. Yang, MG.H. Bell, Models and algorithms for road network design: a review and some new developments, Transport Reviews 18 (1998) 257–278. [16] H. Yang, Transportation bilevel programming problems: recent methodological advances, Transportation Research B 35 (1) (2001) 1–106 (special issue). [17] M.J. Smith, Y. Xiang, R.A. Yarrow, M. Ghali, Bilevel and other modelling approaches to urban traffic management and control, in: P. Marcotte, S. Nguyen (Eds.), Equilibrium and Advanced Transportation Modelling, Kluwer Academic Publishers, Boston, 1998, pp. 283–325. [18] M. Patriksson, R.T. Rockafellar, A mathematical model and descent algorithm for bilevel traffic management, Transportation Science 36 (2002) 271–291. [19] H. Yang, S. Yagar, Traffic assignment and signal control in saturated road networks, Transportation Research A 29 (2) (1995) 125– 139. [20] H. Ceylan, M.G.H. Bell, Traffic signal timing optimization based on genetic algorithm approach, including drivers’ routing, Transportation Research B 38 (2004) 329–342. [21] M. Patriksson, Sensitivity analysis of traffic equilibria, Transportation Science 38 (2004) 258–281. [22] M.J. Smith, The existence, uniqueness and stability of traffic equilibria, Transportation Research B 13 (1979) 295–304. [23] S.C. Dafermos, Traffic equilibrium and variational inequalities, Transportation Science 14 (1) (1980) 42–54. [24] F.H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983. [25] M.S. Bazaraa, H.D. S herali, C.M. Shetty, Nonlinear Programming Theory and Algorithms, second ed., John Wiley & Sons, New York, 1993. [26] L.J. LeBlanc, E.K. Morlok, W.P. Pierskalla, An efficient approach to solving the road network equilibrium traffic assignment problem, Transportation Research 9 (1975) 308–318.
Applied Mathematics and Computation 194 (2007) 441–452 www.elsevier.com/locate/amc
A descent framework for linked signal system with network flows Suh-Wen Chiou Department of Information Management, National Dong Hwa University 1, Sec. 2, Da Hsueh Road, Shou-Feng, Hualien 97401, Taiwan
Abstract Optimization of linked signal system in urban traffic road networks is considered in this paper. This problem can be formulated as a nonlinear program subject to network flows following Wardrop’s first principle in which traffic rerouting effects are taken into account. We propose a descent framework to effectively solve a linked signal system with network flows. Numerical calculations are implemented on well-known example road networks and good results are reported. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Linked signal system; Traffic assignment; Descent framework; Network flows
1. Introduction Optimization of linked signal system (LSS) in urban traffic road networks is considered in this paper. This problem can be formulated as a special case of the Mathematical Program with Equilibrium Constraints (MPEC), which has been extensively studied [1–4]. For an urban road network of signal-controlled junctions with short spacing between adjacent links, a linked signal system can be established. The well-known TRANSYT (TRAffic Network StudY Tool) model [5] has been widely recognized as one of the most useful tools in studying the optimization of linked signal system. As it has been indicated by Allsop [6], the network flows and travel costs are strongly influenced by the operation of signals in a linked signal system. For a LSS, a good performance value can be achieved by optimizing a chosen objective function with respect to the signal settings, in which the users’ behavior of choice of routes is supposed to follow Wardrop’s first principle. That is, a road user will choose his route between a specified origin-destination pair with minimal travel cost, which is in turn dependent on the choice of the linked signal settings. Therefore, in the optimization process for the LSS, not only the signal settings themselves need to be considered but also the consequential effects on the network flows caused by the signal design need to be taken into account. Ways of using mathematical programming to solve the linked signal setting problem (LSSP) and network flow have been well researched. Allsop and Charlesworth [7] reported a mutually consistent calculation for E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.04.046
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linked signal settings and equilibrium flow. The resulting mutually consistent signal settings and link flows will, however, in general be a non-optimal solution as has been discussed by Gershwin and Tan [8] and Dickson [9]. Smith [10] also presented a capacity-maximizing policy where a conventional signal setting problem can be formulated as a Wardropean equilibrium. As for the development of solution methods for network design with general applications, [11–18] have employed the bilevel programming technique to tackle a road network design problem. Recently, Yang and Yagar [19] proposed a sensitivity analysis based (SAB) algorithm to solve signal settings in a simple road network where a linearized sub-problem is formulated at current signal settings and solved by the simplex method. Ceylan and Bell [20], on the other hand, proposed a genetic algorithm (GA) to calculate the linked signal settings and network flows where numerical illustrations were performed on a small-scale road network. Because of the non-convexity of the LSSP, solution algorithms aforementioned can solve the LSSP only locally. In this paper, we propose a descent framework for a coordinated fixed time linked signal system, which can be formulated as one LSSP with respect to the common cycle time, the start and duration of greens. The performance index is defined as the sum of a weighted linear combination of rate of delay and number of stops per unit time for all traffic streams. Following the technique of sensitivity analysis proposed by Patriksson [21], the directional derivatives and gradients are obtained by solving an affine variational inequality problem with respect to the changes in the signal settings. A Quasi-Newton projection (QP) method is presented to effectively solve the LSSP. A new class of hybrid heuristics combining the locally optimal search and global search in the signal settings is thus developed. The Karush–Kuhn–Tucker (KKT) points for the LSSP can be easily found with global convergence. Numerical calculations are conducted on well-known example road networks. As it shows, the proposed QP method achieved substantially better performance than did those other approaches when solving the LSSP. The rest of the paper is organized as follows. In next section the LSSP with respect to signal setting variables is formulated, where the sensitivity analysis of traffic equilibrium is conducted. In Section 3, a descent framework is presented where a globally convergent Quasi-Newton projection (QP) method is proposed. A new class of hybrid heuristics combining the locally optimal search and global search in signal settings is therefore developed. In Section 4, numerical computations are conducted on well-known example road networks with distinct sets of initial data. As it shown, the proposed QP method has achieved substantially better performance than did those other approaches when solving a LSSP. Conclusions for this paper and further points of interests are made in Section 5. 2. Problem formulation The problem for solving the linked signal system (LSS) with respect to signal setting variables while the travelers’ route choice is taken into account is formulated. A generalized traffic assignment with signal settings in terms of a parametric variational inequality is presented. The sensitivity analysis of traffic equilibrium is performed for which the computations of the directional derivatives and the gradients can be solved by an affine variational inequality. 2.1. Notation GðN ; LÞ a signal-controlled network with node set N and link set L W set of origin-destination (OD) pairs Rw set of paths between OD pair w, 8w 2 W T ¼ ½T w matrix of travel demands for OD pair w 8w 2 W W ¼ ðf; h; /Þ set of signal setting variables, respectively for the reciprocal of cycle time, start and duration of greens, where h ¼ ½hjm and / ¼ ½/jm respectively represent the vector of starts hjm and durations of green /jm for signal group j at junction m as proportions of common cycle time ka the duration of effective green for link a 8a 2 L gjm the minimum green for signal group j at junction m cjlm the clearance time between the end of green for signal group j and the start of green for incompatible signal group l at junction m
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Xm ðj; lÞ a collection of numbers 0 and 1 for each pair of incompatible signal groups at junction m; where Xm ðj; lÞ ¼ 0 if the start of green for signal group j proceeds that of l and Xm ðj; lÞ ¼ 1, otherwise Da the rate of delay on link a 8a 2 L Sa the number of stops per unit time on link a 8a 2 L pa the maximum degree of saturation for link a 8a 2 L sa the saturation flow on link a 8a 2 L q ¼ ½qa vector of average link flows 8a 2 L f ¼ ½fp vector of path flows between points of entry to points of exit from the network 8p 2 Rw ; 8w 2 W d the link-path incidence matrix D the OD-path incidence matrix c ¼ ½ca vector of link flow travel cost 8a 2 L C ¼ ½C p vector of path flow travel cost 8p 2 Rw ; 8w 2 W p vector of minimum travel cost 2.2. A linked signal system problem (LSSP) By using the traffic model from TRANSYT for evaluating the performance of linked signal settings, the objective function for the LSSP can be formulated as a non-linear mathematical program as follows. As for the performance index used in TRANSYT, it can be expressed as a weighted linear combination of travel delays and number of stops at downstream links at signal-controlled junctions. Let MD and MS respectively denote the monetary factors for rate of delay and number of stops, we have X Z ¼ Z 0 ðW; q ðWÞÞ ¼ M D Da W 1aD þ M S S a W 1aS ; ð1Þ Min W
a2L
subject to
ð2Þ
fmin 6 f 6 fmax ; gjm f 6 /jm 6 1;
8j; m;
qa 6 pa sa ka ; 8a 2 L; hjm þ /jm þ cjlm f 6 hlm þ Xm ðj; lÞ; 1 aD
ð3Þ j 6¼ l; 8j; l; m;
ð4Þ ð5Þ
1 aS
where W and W are respectively link-specific weighting factors for the rate of delay and the number of stops per unit time used in TRANSYT. The constraint (2) is the bound for the common cycle time and the constraints (3)–(5) are for the green phase, link capacity and clearance time. Also the equilibrium flows q ðWÞ can be found by solving the following traffic assignment problem. 2.3. User equilibrium traffic assignment A general formulation for a user equilibrium traffic assignment problem can be expressed in terms of a variational inequality as shown by Smith [22] and identified by Dafermos [23]. Find values q such that ct ðqÞðz qÞ P 0
ð6Þ
for all z 2 K ¼ fq : q ¼ df ; T ¼ Df ; f P 0g where the superscript t denotes matrix transpose. The equivalent variational inequality in terms of path flow for (6) can be given as follows: Find values f such that C t ðf Þðz f Þ P 0
ð7Þ
for all z 2 K 1 ¼ ff : T ¼ Df ; f P 0g. 2.4. A parametric variational inequality For user equilibrium traffic assignment problem with signal settings, W ¼ ðf; h; /Þ, the parametric variational inequality for (6) can be expressed as follows: ct ðW; qÞðz qðWÞÞ P 0 for all z 2 KðWÞ ¼ fq : qðWÞ ¼ df ðWÞ; T ¼ Df ðWÞ; f ðWÞ P 0g.
ð8Þ
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2.5. Sensitivity analysis by directional derivatives Following the technique employed by Patriksson [21], the sensitivity analysis of traffic equilibrium of (8) can be established in the following way. Introduce e ðWÞ ¼ fdqðWÞ : 9df ðWÞ such that dqðWÞ ¼ ddf ðWÞ; Ddf ðWÞ ¼ 0; and df ðWÞ 2 K 0 ðWÞg K and
K 0 ðWÞ ¼
8 > > > <
ðiÞ dfp ðWÞ free;
> df : > > :
if f p ðWÞ > 0
ðiiÞ dfp ðWÞ ¼ 0; if C p > pw ðiiiÞ dfp ðWÞ ¼ 0; if C p ¼ pw and f p ðWÞ ¼ 0 with dC p > 0; ðivÞ dfp ðWÞ > 0;
if C p ¼ pw ; and f p ðWÞ ¼ 0 with dC p 6 0
9 > > > = 8p 2 Rw ; w 2 W > > > ;
;
where the changes in link or path flows with respect to the changes in signal settings are denoted by dqðWÞ or df ðWÞ and the corresponding changes in path flow travel time denoted by dC p . Therefore the directional derive ðWÞ, atives of (8) dq can be obtained by solving the following affine variational inequality. For all z 2 K t
ðrW cðW; qÞdW þ rq cðW; qÞdqÞ ðz dqÞ P 0;
ð9Þ
where rW c and rq c are gradients evaluated at ðaW; qÞ when the changes in signal settings dW are specified. 2.6. A MPEC problem In constraints (2)–(5), let A and b be the coefficient matrix and corresponding constant vector associated with the signal settings thus the constraints (2)–(5) can be re-expressed as the following form. AWt 6 b: The LSSP (1)–(5) can be briefly addressed as follows: Min
Z ¼ Z 0 ðW; q ðWÞÞ;
subject to
AWt 6 b
W
and
q 2 S 0 ðWÞ;
ð10Þ
where S 0 ðÞ denotes the solution set for (8). 2.7. A single-level problem Following the results in sensitivity analysis for the parametric variational inequality (9), the LSSP (10) now can be rewritten as the following single-level problem. Min
Z ¼ Z 1 ðWÞ;
subject to
AWt 6 b:
W
ð11Þ
3. Solution method As it is known that the LSSP (11) is one instance of mathematical program with equilibrium constraints (MPEC), subject to the non-convexity of MPEC, the generalized gradients of (11) in terms of Clarke’s definition [24] can be expressed in the following manner. oZ 1 ðW Þ ¼ cof lim rZ 1 ðWk Þ : Wk ! W ; rZ 1 ðWk Þexistsg: k!1
ð12Þ
According to [24], the generalized gradient is a convex hull (abbreviated to co) of all points of the form lim rZ 1 ðWk Þ where the subsequence fWk g converges to the limit value W . And the gradients in (12) evaluated at ðWk ; qk Þ are expressed in the following way.
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rZ 1 ðWk Þ ¼ rW Z 0 ðWk ; qk Þ þ rq Z 0 ðWk ; qk ÞdqðWk Þ;
445
ð13Þ
where the gradient of equilibrium flow with respect to signal settings can be obtained by solving (9). 3.1. A descent framework Regarding the solution method for LSSP (11), in the followings we propose a descent framework via the implementation of a Quasi-Newton projection method (QP) to determine the search direction along which the value of the objective function decreases monotonically. The search process is either terminated at a KKT point or a new search direction can be generated. Theorem 1 (Quasi-Newton method). Let W1 locally solve LSSP (11), and Q1 the corresponding positive definite symmetric matrix. For k ¼ 1; . . . ; n, let Wkþ1 ¼ Wk þ ad k ;
ð14Þ
where d k ¼ Qk rZ t1 ðWk Þ
ð15Þ
is a descent direction at Wk with a for LSSP (11). For k ¼ 1; . . . ; n 1, Qk is determined by Qkþ1 ¼ Qk
Qk uk utk Qk vk vtk þ t ; utk Qk uk v k uk
ð16Þ
where uk ¼ Wkþ1 Wk and vtk ¼ rZ 1 ðWkþ1 Þ rZ 1 ðWk Þ. If rZ 1 ðWk Þ 6¼ 0 for k ¼ 1; . . . ; n, then Q1 ; . . . ; Qn are symmetric and positive definite so that d 1 ; . . . ; d n are descent directions. Proof. See [25].
h
The search direction generated by Quasi-Newton method for an unconstrained nonlinear problem is a descent direction, which strictly decreases the objective function value at each iteration provided that the corresponding gradient is not zero. In the followings, we apply the Quasi-Newton method to the linear constraint set as in (11) by using the concept of gradient projection method. Definition 2 (Projection matrix). A n n matrix H is called a projection matrix if H ¼ H t and H t H ¼ H . Thus the Quasi-Newton Projection (QP) method can be presented in the following way. Theorem 3 (Quasi-Newton Projection method, QP). In problem (11), a sequence of iterates fWk g can be generated according to Wkþ1 ¼ Wk þ aH k d k ;
ð17Þ
where dk is the descent direction determined by (15) and a is the step length which minimize Z1 along dk from Wk and the projection matrix Hk is of the following form. H k ¼ I M tk ðM k M tk Þ1 M k ;
ð18Þ
where Mk is the gradient of active constraints in (11) at Wk , where the active constraint gradients are linearly independent and thus Mk has full rank. The search direction hk can be determined in the following form. hk ¼ H k d k :
ð19Þ
Then the sequence of points fWk g generated by the Quasi-Newton projection method, Z 1 ðWkþ1 Þ < Z 1 ðWkþ1 Þ;
k ¼ 0; 1; 2; . . . ;
ð20Þ
whenever H k rZ 1 ðWk Þ 6¼ 0. Proof. Following the results of Theorem 1 in (15), we have rZ t1 ðWk Þd k ¼ rZ t1 ðWk ÞQk rZ 1 ðWk Þ < 0;
k ¼ 1; 2; 3; . . . ;
ð21Þ
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since Qk is positive definite, for all rZ 1 ðWk Þ 6¼ 0. Multiply equation (21) by projection matrix Hk, it becomes rZ t1 ðWk ÞH k d k ¼ rZ t1 ðWk ÞH k Qk rZ 1 ðWk Þ ¼ rZ t1 ðWk ÞH tk H k Qk rZ 1 ðWk Þ ¼ kH k k2 rZ 1 ðWk ÞQk rZ 1 ðWk Þ < 0:
ð22Þ
Thus for sufficiently small l, l > 0, we have Z 1 ðWk Þ > Z 1 ðWk þ lhk Þ:
ð23Þ
Because by definition a is the step length which minimize P1 along hk from Wk , it implies Z 1 ðWk Þ > Z 1 ðWk þ lhk Þ P Z 1 ðWk þ ahk Þ ¼ Z 1 ðWkþ1 Þ; which completes this proof.
ð24Þ
h
Theorem 4. Following Theorem 3, when H k rZ 1 ðWk Þ ¼ 0, if all the Lagrange multipliers corresponding to the active constraint gradients in (11) are positive or zeros, it implies the current Wk is a KKT point. Otherwise choose ^ k of the active constraint gradients by deleting the one negative Lagrange multiplier, say lj , and construct a new M jth row of Mk, which corresponds to the negative component lj , and make the projection matrix of the following form ^ t Þ1 M ^ k: ^ t ðM ^ kM Hk ¼ I M k k
ð25Þ
The search direction then can be determined by (19) and the results of Theorem 3 hold. Proof. This proof is separated as in the following two parts. First, suppose all the Lagrange multipliers are positive or zeros, and by definition we have 1
0 ¼ H k rZ 1 ðWk Þ ¼ ðI M tk ðM k M tk Þ M k ÞrZ 1 ðWk Þ ¼ rZ 1 ðWk Þ þ M tk w;
ð26Þ
1 ðM k M tk Þ M k rZ 1 ðWk Þ
where w ¼ corresponds to the Lagrange multipliers l matching the active constraint gradients Mk. If l P 0, and the active constraint gradients are linearly independent, then ðWk ; lÞ is a KKT pair to the problem (11) and the current Wk is a KKT point. Or choose one negative Lagrange multiplier, ^ k of the active constraint gradients by deleting the jth row of Mk, and make say lj , and construct another M a new projection matrix as in (25). The new search direction can be determined by (19). Provided that H k rZ 1 ðWk Þ 6¼ 0, then the results of Theorem 3 hold. We only need to show that H k rZ 1 ðWk Þ 6¼ 0. By contra^ t Þ1 M ^ k rZ 1 ðWk Þ, we ^ kM ^ ¼ ðM diction, suppose that H k rZ 1 ðWk Þ ¼ 0. By the definition of Hk and letting w k have ^ t Þ1 M ^ k ÞrZ 1 ðWk Þ ¼ rZ 1 ðWk Þ þ M ^ t ðM ^ kM ^ tw 0 ¼ H k rZ 1 ðWk Þ ¼ ðI M k k k ^:
ð27Þ
^ tw ^ t þ lj At where Aj is the jth row of active constraint gradients, consider (26), we get Since M j k ^ ¼ Mkw t ^ tw 0 ¼ rZ 1 ðWk Þ þ M k þ l j Aj :
ð28Þ
Subtracting (28) from (27), it follows: ^ t ð^ Þ lj Atj : 0¼M k ww
ð29Þ
^ k of full rank. Therefore H k rZ 1 ðWk Þ 6¼ 0 which completes this Since lj ¼ 6 0, it violates that the assumption M proof. h According to the results of Theorems 3 and 4, the search process for solving the problem (11) either can be terminated at a KKT point, which satisfies the following convergent theorem of QP method or a new search direction can be generated. Theorem 5 (Convergence of QP method). Suppose in (11) Z1 function is lower semi-continuous on the domain set X. Given a W1 such that Z 1 ðW1 Þ ¼ b and the level set Pb ¼ fW : W 2 X; Z 1 ðWÞ 6 bg is bounded and Z1 is
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continuous differentiable on the convex hull of Pb . Let fWk g be the sequence of points generated by QP method as described above. Then every cluster point W satisfies 0 2 @Z 1 ðW Þ:
ð30Þ
Proof. We proof this theorem by contradiction. Supposing 0 62 @Z 1 ðW Þ, by definition in (12) there is no subgradient rZ 1 ðWk Þ ¼ 0 at Wk in the convex hull of Pb , whose cluster point is W . Then there is a a > 0 minimizing Z 1 ðW þ ah Þ and a e > 0 such that Z 1 ðW Þ ¼ Z 1 ðW þ a h Þ þ e and W þ a h is an interior point of Pb . By the mean value theorem, for any Wk we have Z 1 ðWk þ a hk Þ ¼ Z 1 ðW þ a h Þ þ rZ t1 ðnk ÞðWk W þ a ðhk h ÞÞ;
ð31Þ
where nk ¼ W þ a h þ lðW Wk þ a ðh hk ÞÞ for some 0 < l < 1. Following Bozano theorem that there is a subsequence fWkn g of fWk g that converges to W , then frZ 1 ðnkn Þg converges to rZ 1 ðW þ a h Þ and fWkn W þ a ðhkn h Þg converges to zero. For sufficiently large kn, the vector nkn belongs to the convex hull of Pb and e e ð32Þ Z 1 ðWkn þ a hkn Þ 6 Z 1 ðW þ a h Þ þ ¼ Z 1 ðW Þ : 2 2 Let akn be the minimizing point of Z 1 ðWkn þ akn hkn Þ. Since fZ 1 ðWkn Þg is monotone decreasing and converges to Z 1 ðW Þ, we have e ð33Þ Z 1 ðW Þ < Z 1 ðWkn þ akn hkn Þ 6 Z 1 ðWkn þ a hkn Þ 6 Z 1 ðW Þ ; 2 a contradiction. Therefore every cluster point W satisfies 0 2 @Z 1 ðW Þ. h Corollary 6 (Stopping condition). If Wk is a KKT point for LSSP (11) satisfying Theorem 5 then the search process may stop; otherwise a new descent direction at Wk can be generated according to Theorems 3 and 4. 3.2. Heuristics for solving LSSP (11) In order to obtain a near global optimum for LSSP (11), a new class of hybrid heuristics in terms of H1–H4 can be developed in the following way. 3.2.1. Heuristic 1 (H1) Heuristic 1 is a locally optimal search with respect to the common cycle time, the start and duration of green at each junction for LSSP (11) and is carried out in the following steps. Step 1. Start with Wk and set index k = 1. Step 2. Solve a user equilibrium traffic assignment problem (8) with signal settings Wk , find the first order derivatives by (9). Step 3. Use the QP method to determine a search direction by (19). Step 4. If H k rZ 1 ðWk Þ 6¼ 0, find a new Wkþ1 in (17) by deciding a the step length, which minimize Z1 along hk and letting k k þ 1. Go to Step 2. If H k rZ 1 ðWk Þ ¼ 0 and all the Lagrange multipliers corresponding to the active constraint gradients are positive or zeros, Wk is a KKT point and stop. Otherwise, follow the results of Theorem 4 and find a new projection matrix and go to Step 3. 3.2.2. Heuristic 2 (H2) Given a common cycle time, a locally optimal search for partial optimization with respect to the start and duration of green at each junction can be conducted as the steps given in Heuristic 1, where the common cycle time is fixed. 3.2.3. Heuristic 3 (H3) Given a common cycle time and duration of green at each junction, an unconstrained optimization problem with respect to the offsets is constructed for LSSP (11). Since there is no physical constraint taken into account
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for the starts of greens, i.e. the offsets, a global search is implemented to find a better KKT point in another part of the feasible region. After such a KKT point is found, conduct Heuristic1 again and locate new signal settings where the common cycle time and green durations are fixed. 3.2.4. Heuristic 4 (H4) Analogous to the Heuristic 3, in Heuristic 4, the Heuristics 1 and 2 are alternately employed after a good KKT point is found. The Heuristic 4 is conducted in the mixture of Heuristics 1–3 until the difference of the values of the performance index between successive iterations is negligible. 4. Numerical calculations In this section, numerical experiments are conducted for the proposed heuristics, H1–H4, and the previous methods: SAB and GA on example road networks in the following way. Firstly, numerical computations are carried out at a two-junction signal-controlled road network, as shown in Fig. 1, where the proposed heuristics, H1–H4, are compared to the lower bound system-optimal (SO) solution for the LSSP. Next, a real data Sioux-Falls aggregated network [26] consisting of 24 nodes, 76 links and 528 OD trips, where 6 signal-controlled junctions are taken into account as shown in Fig. 2, is illustrated for numerical calculations. Using typical values found in practice, the minimum green time is 7 s, and the clearance times are 5 s between incompatible signal groups. The maximum cycle time is set 180 s. 4.1. Computational results for two-junction road network Computational results for conducting the solution methods for LSSP at two-junction road network are summarized in Table 1. The performance index for LSSP in (11) is measured by total travel cost in dollars. The corresponding computational efforts are measured by the numbers of solving traffic assignments. As it seen in Table 1, the initial value of PI (short for performance index) for LSSP is $78 and the best value of PI is $30.8 which is achieved by the system optimum (SO). The proposed heuristics: H1–H4 respectively improved the PI values by 63%, 64%, 65% and 65% while the SAB method and recently proposed GA approach improved the PI values only by 60%. The relative difference ratios in the final values of PI between H1–H4 and SO were 9%, 4%, 3% and 2% while the relative difference ratio in the final value of PI between SAB and SO was nearly 17%. Regarding the numbers of solving traffic assignments for LSSP, it took H4 9 iterations while it took SAB 37 iterations.
Fig. 1. Layout for two-junction road network.
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Fig. 2. Sioux-Falls road network
4.2. Computational results on Sioux Falls city road network Computational results on Sioux Falls city road network are summarized in Tables 2 and 3 for two arbitrary sets of initial signal settings. Starting with two distinct sets of initial signal settings, the proposed Heuristic H4, combining the locally optimal search and global search heuristic achieved at respective local optima with very close values to the SO solutions. As it seen in Table 2, the initial value of PI (short for performance index) for LSSP is $1520 and the best value of PI is $1181.1 which is achieved by the system optimum (SO). The proposed heuristics: H1–H4 respectively improved the PI values by 21%, 22%, 22% and 22% while the SAB method and recently proposed GA approach improved the PI values by 13% and 15% only. The relative difference ratios in the final values of PI between H1–H4 and SO were less 1.5% while the relative difference ratios in the final values of PI between SAB and SO and between GA and SO were nearly 12.5% and 9%. Regarding the numbers of solving traffic assignments for LSSP, it took H4 25 iterations while it took SAB 98 iterations. Computational results at the second set signal settings are given in Table 3 and can be summarized in the following similar way. As it seen in Table 3, the initial value of PI for LSSP is $1650 and the best value of PI is $1181.2 which is achieved by the system optimum (SO). The proposed heuristics: H1–H4 respectively improved the PI values by 28% on average while the SAB method and recently proposed GA approach improved the PI values by 18% and 21% only. The relative difference ratios in the final values of PI between H1 and H4 and SO were less 1% while the relative difference ratios in the final values of PI between SAB and
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Table 1 Computational results for two-junction road network Variable/algorithm Initial 1=f Initial PI
SO 90 78
SAB 90 78
GA 90 78
H1 90 78
H2 90 78
H3 90 78
H4 90 78
/11 =f /21 =f /12 =f /22 =f /32 =f q1 q2 q3 q4 q5 q6 1=f PI #
43 7 43 7 60 750 0 0 750 0 0 60 30.8 5
43 7 43 7 60 750 0 0 750 0 0 60 36.1 37
58 7 58 7 75 750 0 0 750 0 0 75 35.9 11
42 8 42 8 60 750 0 0 750 0 0 60 33.5 7
43 7 43 7 60 750 0 0 750 0 0 60 32.1 9
43 7 43 7 60 750.3 0.1 0.1 749.6 0.1 0.1 60 31.7 10
43 7 43 7 60 750 0 0 750 0 0 60 31.3 9
Where /jm =f denotes the duration of greens for signal group j at junction m measured in sec and 1=f denotes the common cycle time measured in sec. qa is measured in veh/h, PI denotes the performance index value measured in $. # denotes the number of traffic assignments solved.
Table 2 Computational results for Sioux Falls network at 1st set signal settings Variable/algorithm Initial 1=f Initial PI
SO 120 1520
SAB 120 1520
GA 120 1520
H1 120 1520
H2 120 1520
H3 120 1520
H4 120 1520
/17 =f /18 =f /19 =f /1;10 =f /1;16 =f /1;18 =f 1=f PI #
44 52 49 47 53 51 105 1181.1 20
48 46 48 52 58 50 102 1328.5 98
38 57 52 49 45 47 116 1287.2 54
38 57 52 49 45 47 106 1194.3 61
44 50 49 48 51 50 102 1192.7 50
50 44 51 52 53 49 96 1186.7 25
45 48 50 51 55 52 104 1183.5 25
Table 3 Computational results for Sioux Falls network at 2nd set signal settings Variable/algorithm Initial 1=f Initial PI
SO 150 1650
SAB 150 1650
GA 150 1650
H1 150 1650
H2 150 1650
H3 150 1650
H4 150 1650
/17 =f /18 =f /19 =f /1;10 =f /1;16 =f /1;18 =f 1=f PI #
60 55 50 50 55 60 105 1181.2 20
60 58 52 50 57 54 106 1350.2 95
41 59 55 49 53 48 130 1300.7 70
51 56 54 48 54 49 118 1192.1 71
56 53 57 51 55 51 120 1191.2 55
61 58 55 52 55 52 115 1185.2 29
61 58 55 52 55 52 100 1182.1 25
SO and between GA and SO were 14.3% and 10.1%. Regarding the numbers of solving traffic assignments for LSSP, it took H4 25 iterations while it took SAB 95 iterations.
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Implementations for carrying out the computational efforts on the solution methods for LSSP were conducted on SUN SPARC SUNW, 900 MHZ processor with 4 Gb RAM under operating system Unix SunOS 5.8 using C++ compiler gnu g++ 2.8.1. The stopping criterion for these solutions is set when the relative difference in the performance index value between the consecutive iterations less than 0.15%. Computational efforts in solving the LSSP for numerical examples mentioned above were no greater than 10 min CPU time on average. Total computation efforts for complete run of H4 did not exceed 1 min of CPU time in all cases. 5. Conclusions and discussion In this paper, a linked signal setting problem (LSSP) is considered while the route choices of road users are properly taken into account. A mathematical program with equilibrium constraints (MPEC) is formulated for the LSSP where the equilibrium constraint is expressed as a parametric variational inequality with respect to signal settings. Sensitivity analysis by directional derivatives has been conducted by solving an affine variational inequality problem where the prior knowledge of positive path flows is not necessary. A descent framework of a globally convergent Quasi-Newton projection method (QP) has been proposed to effectively solve the LSSP. A new class of hybrid heuristics has been developed. Numerical computations have been carried out on example networks and good results have been obtained. In comparison with other feasible optimization based algorithms, the proposed heuristics have shown promising results in the effectiveness in solving the equilibrium assignment problems and the robustness to various sets of initial signal settings on two test networks. Regarding the development of solution methods for various applications of the MPEC problems, investigations are being undertaken via a non-smooth optimization approach and we will discuss this issue in a subsequent paper. Acknowledgement Many thanks go to Taiwan National Science Council via Grant NSC-95-2416-H-259-014. References [1] Z.Q. Luo, J.-S. Pang, D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, UK, 1996. [2] P. Marcotte, D.-L. Zhu, Exact and inexact penalty methods for generalized bilevel programming problems, Mathematical Programming 74 (1996) 141–157. [3] K. Shimizu, Y. Ishizuka, J. Bard, Nondifferentiable and Two-level Mathematical Programming, Kluwer Academic Publishers, Boston, 1997. [4] J. Outrata, M. Kocvara, J. Zowe, Nonsmooth approach to optimization problems with equilibrium constraints: theory, Applications and Numerical Results, Kluwer Academic Publishers, Boston, 1998. [5] D.I. Robertson, TRANSYT: a traffic network study tool, RRL Report, LR 253, Transport and Road Research Laboratory, Crowthrone, 1969. [6] R.E. Allsop, Some possibilities for using traffic control to influence trip distribution and route choice, in: D.J. Buckley (Ed.), The Proceeding of the 6th International Symposium on Transportation and Traffic Theory, Elsevier, New York, 1974, pp. 345–374. [7] R.E. Allsop, J.A. Charlesworth, Traffic in a signal-controlled road network: an example of different signal timings inducing different routeings, Traffic Engineering Control 18 (5) (1977) 262–264. [8] S.B. Gershwin, H.N. Tan, Hybrid optimization: optimal static traffic control constrained by drivers’ route choice behavior, Massachusetts Institute of Technology, Laboratory for Information and Decision System Report LIDS-p-870, 1979. [9] T.J. Dickson, A note on traffic assignment and signal timings in a signal-controlled road network, Transportation Research B 15 (1981) 267–271. [10] M.J. Smith, A local traffic control policy which automatically maximizes the overall travel capacity of an urban road network, Traffic Engineering and Control 21 (1980) 298–302. [11] P. Marcotte, Network design problem with congestion effects: a case of bilevel programming, Mathematical Programming 34 (1986) 142–162. [12] T.L. Friesz, R.L. Tobin, H.-L. Cho, N.J. Mehta, Sensitivity analysis based algorithms for mathematical programs with variational inequality constraints, Mathematical Programming 48 (1990) 265–284. [13] S. Suh, T.J. Kim, Solving nonlinear bilevel programming models of the equilibrium network design problem: a comparative review, Annals of Operations Research 34 (1992) 203–218.
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