# A surrogate-based optimization algorithm for network design problems

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## Abstract

Network design problems (NDPs) have long been regarded as one of the most challenging problems in the field of transportation planning due to the intrinsic non-convexity of their bi-level programming form. Furthermore, a mixture of continuous/discrete decision variables makes the mixed network design problem (MNDP) more complicated and difficult to solve. We adopt a surrogate-based optimization (SBO) framework to solve three featured categories of NDPs (continuous, discrete, and mixed-integer). We prove that the method is asymptotically completely convergent when solving continuous NDPs, guaranteeing a global optimum with probability one through an indefinitely long run. To demonstrate the practical performance of the proposed framework, numerical examples are provided to compare SBO with some existing solving algorithms and other heuristics in the literature for NDP. The results show that SBO is one of the best algorithms in terms of both accuracy and efficiency, and it is efficient for solving large-scale problems with more than 20 decision variables. The SBO approach presented in this paper is a general algorithm of solving other optimization problems in the transportation field.

## Key words

Network design problem Surrogate-based optimization Transportation planning Heuristics## CLC number

U491 TP202## 1 Introduction

The rapid and worldwide growth in population has necessitated building numerous roads to satisfy the increasing traffic demands. It is natural and crucial that decision-makers continuously try to maximize the system performance by building new links or expanding existing ones. Network design problems (NDPs) determine how to allocate a limited budget to the expansion of capacity for the existing links or the construction of new ones to maximize social welfare or minimize general travel costs. Generally, NDPs can be categorized into three classes: continuous NDPs (CNDPs) where decision variables are continuous, discrete network design problems (DNDPs) where decision variables are discrete (usually binary), and mixed network design problems (MNDPs) which contain both types of decision variables.

NDPs can be formulated as a bi-level programming problem (BLPP), or more specifically, mathematical programming with equilibrium constraints (MPEC) (Luathep *et al.*, 2011). In the BLPP formulation, the upper-level program seeks to optimize the network performance index within the budget, and the lower-level program is the user equilibrium (UE) problem (e.g., Yang and Bell, 1998; Meng *et al.*, 2001) or stochastic user equilibrium (SUE) problem (Davis, 1994). Furthermore, some expansions have been made to incorporate the uncertainty of travel demand (Yin *et al.*, 2009) and time-dependent (Lo and Szeto, 2009) effects.

Given the intrinsic non-convexity of the problem and the nonlinearity of its objective function, NDPs have been viewed as one of the most challenging problems in transportation. Methods for solving NDP will be reviewed in Sections 1.1–1.3.

### 1.1 Algorithms for solving continuous network design problems

CNDP has received the greatest attention in transportation literature. Generally, traditional approaches for solving CNDP can be categorized into two major groups: gradient-based algorithms and derivative-free algorithms.

Gradient-based algorithms generally use the sensitivity information of the lower-level UE programming to search local solutions (Meng *et al.*, 2001). Sensitivity analysis-based methods are often incorporated into the optimization procedure of the upper-level objective function (Yang and Yagar, 1995). Chiou (2005) developed a series of methods based on the gradient analysis scheme. These methods have been reported to be efficient for finding a local solution; however, a saticfactory solution quality could not be guaranteed as exhibited by subsequent work.

Derivative-free algorithms include iterative optimization-assignment procedures (Allsop, 1974), Hooke-Jeeves algorithms (Abdulaal and LeBlanc, 1979), equilibrium decomposed optimization (Suwansirikul *et al.*, 1987), simulated annealing (Friesz *et al.*, 1992), and genetic algorithm-based approaches (Yin, 2000). These derivative-free algorithms are incapable of guaranteeing convergence, though it has been reported that some of them can find a good solution after a lengthy computation.

Recently, a mixed-integer linear programming (MILP) approximation approach was introduced to solve CNDP. Wang and Lo (2010) proposed a path-based MILP to solve CNDP and it guarantees a global convergence. Luathep *et al.* (2011) further extended the approach to a link-based MILP formulation to reduce its computational effort while maintaining the convergence property. Liu and Wang (2015) further proposed a linearized technique to solve CNDP with a stochastic user equilibrium. All of these methods obtained good convergence results on some widely used test networks; however, the computational efforts required by these methods were not that appealing compared to their reported theoretical perfection; i.e., none of those algorithms were tested on networks with more than 20 decision variables.

Thus, this challenge inspired us to develop a solution framework incorporating properties that were good from both theoretical and computational perspectives.

### 1.2 Algorithms for solving discrete network design problems

LeBlanc (1975) first developed DNDP and proposed a branch-and-bound algorithm to efficiently solve it. The algorithm was tested on the Sioux Falls network with only five binary variables. Poorzahedy and Turnquist (1982) further developed the branch-and-bound algorithm to solve DNDP by approximating the bi-level program through a single-level program; however, the accuracy of the approximation was questioned by other researchers (e.g., Gao *et al.*, 2005). Gao *et al.* (2005) then proposed a generalized Benders’ decomposition algorithm to solve DNDP. The algorithm translates the original bi-level problem into a series of master problems and subproblems using a supporting function, and solves some small-scale problems efficiently.

Recently, as in CNDP, an MILP approximation approach has also been introduced to solve DNDP globally. Wang *et al.* (2013) proposed two techniques, i.e., system-optimum (SO) relaxation and UE-reduction, to solve DNDP. The two techniques were solved by using a MILP approach globally. Wang *et al.* (2015) further proposed a linearized approach, which considers optimal new link additions and their optimal capacities simultaneously, to solve DNDP. Farvaresh and Sepehri (2011) also proposed a single-level MI programming approach to solve DNDP.

Many heuristics have also been developed to solve DNDP. This type of algorithm includes genetic algorithms (Cantarella and Vitetta, 2006), ant system heuristics (Poorzahedy and Abulghasemi, 2005), and hybrid meta-heuristics (Poorzahedy and Rouhani, 2007).

A major challenge for solving DNDPs nowadays lies in searching for an efficient way to deal with large-scale problems, i.e., problems with more than 20 decision variables. Most of the algorithms mentioned above were tested on problems with no more than 10 decision variables in their original papers.

### 1.3 Algorithms for solving mixed network design problems

Compared with CNDP and DNDP, MNDP draws much less attention in the literature. To our best knowledge, the only well-formulated algorithm to solve MNDP is the LMILP proposed by Luathep *et al.* (2011), where a numerical example was presented to illustrate its solution properties for MNDP on a problem with 10 continuous variables and 10 discrete variables. Other than that, some heuristics (e.g., a scatter search (Gallo *et al.*, 2010)), were proposed in the field.

- 1.
Theoretically, while solving CNDP, SBO can guarantee asymptotically complete convergence.

- 2.
Practically, the proposed SBO framework is able to solve a large-scale problem with satisfactory solution qualities for all of these kinds of problems as mentioned above, i.e., CNDP, DNDP, and MNDP.

In recent years, SBO has been applied to transportation research (Chen *et al.*, 2014a; 2014b; 2015a; 2015b; Chow and Regan, 2014). Specifically, Chen *et al.* (2015b) applied an SBO framework to solve NDPs under a dynamic traffic assignment paradigm. Chow and Regan (2014) studied a probabilistic convergent SBO for solving multi-objective robust road pricing, a problem similar to CNDP. However, less work has been carried out in comparing the efficiency and solution quality of SBO with those of existing NDP algorithms.

SBO can be tracked back to the work of efficient global optimization (EGO) proposed by Jones *et al.* (1998). An expected improvement (EI) scheme was proposed to balance local and global searches. However, the original EGO takes a long time to optimize the response surface, and the global convergence is not guaranteed.

To improve the theoretical properties of SBO, Regis and Shoemaker (2007) proposed a stochastic response surface (SRS) framework for global optimization of the expensive functions. In their work, a candidate point scheme was used as a stochastic iterative process and incorporated to prove that the SBO framework was asymptotically completely convergent. Müller *et al.* (2013) further extended the SRS framework to an SO for a mixed integer (SO-MI) problem framework, which could be used in discrete and MI optimization.

This study combines the merits of both EGO and SRS to propose a framework based on projected candidate point heuristics using the Kriging approximation and EI as the infill criterion. In such a framework, the asymptotically complete convergence is guaranteed when solving continuous problems, and the computational effort will be small for each iteration. This framework is then applied to solve CNDP, DNDP, and MNDP. Numerical experiments are conducted to illustrate the performance of the proposed method. The results show that SBO is among the most advanced algorithms for solving NDPs.

## 2 Methodology

### 2.1 Problem formulation

Notations

Notation | Explanation |
---|---|

| The set of unchanged links in the network |

| The set of project links for improvement in the network |

| The set of new project links in the network |

| The minimum of set |

| Monetary cost function of link |

| The cost to build a new link |

| The set of objective function values obtained by |

\(f_l^{{\rm{rs}}}\) | Path flow on route |

| Implicit function form of the lower-level UE problem |

)U | Kriging approximation of the lower-level UE in iteration |

| The set of routes between OD pair rs |

| An arbitrarily large positive number |

| Number of initially evaluated points |

| Total demand between OD pair rs |

| The set of origins |

| The set of destinations |

\(\hat s_i^2(y,U)\) | Mean square error of the Kriging approximation in iteration |

| Link travel time function w.r.t. link flow and capacity expansion |

| A binary scalar representing the decision whether to build candidate link |

| Vector form of |

| The set of OD pairs |

| Flow on link |

| Vector form of |

| The value of capacity expansion on link |

| Vector form of |

\({\underline y_a}\) | Lower bound of the capacity enhancement of link |

| Upper bound of the capacity enhancement of link |

\(\delta_{a,l}^{{\rm{rs}}}\) | 1, if route |

| The set of evaluation points in iteration |

| The set of inputs for the Kriging approximation in iteration |

| The set of inputs for the Kriging approximation in iteration |

For upper-level optimizations (1a)–(1e), *Z*_{ u } of Eq. (1a) is the objective function which is intended to minimize the total travel cost. Note that when the problem is a CNDP, *A*_{2} is an empty set, and when the problem is a DNDP, *A*_{1} is an empty set. Constraint (1b) is the construction budget constraint. Constraint (1c) defines the bounds of the continuous capacity expansion of link *a*. Constraint (1d) states that only candidate links with *U*_{a}>0 are allowed to load a positive link flow, and Eq. (1e) states that *U*_{ a } is a binary variable.

For the lower-level programming (1f)–(1i), constraint (1g) is the demand conservation condition. The sum of any path flow connecting OD pair rs should be equal to demand *q*^{rs}. Constraint (1h) is the link-path flow conservation condition. Constraint (1i) limits all path flows to be nonnegative. The Karush-Kuhn-Tucker (KKT) conditions for problem (1f)–(1i) are equivalent to Wardrop’s first principle (Wardrop, 1952).

*is the vector of the continuous capacity expansion,*

**y***is the binary vector for the construction of new links, and*

**U***(*

**G***,*

**y***) is a mathematically difficult-to-solve function. Though it is too complicated to formulate analytically, some properties (e.g., the uniqueness of*

**U***given*

**v***and*

**y***and the continuity of*

**U***(*

**G***,*

**y***) w.r.t.*

**U***) can be proven.*

**y****Proposition 1** Given * y* and

*,*

**U***=*

**v***(*

**G***,*

**y***) is unique if the link travel time function is monotonically increasing.*

**U****Proof** According to the definition of * G*(

*,*

**y***),*

**U***is the solution of program (1f)–(1i). If the link travel time function is monotonically increasing, program (1f)–(1i) is convex. Obviously, the solution of a convex program is unique.*

**v****Proposition 2** * G*(

*,*

**y***) is continuous w.r.t.*

**U***if the link travel time function is continuously differentiable and monotonically increasing.*

**y****Proof**If

** solves program (1f)–(1i), then*

**v**** also solves the following variational inequality problem:*

**v***satisfying constraints (1g)–(1i).*

**v**Since the polyhedron defined by constraints (1g)–(1i) is a convex and closed set, and the travel time function is assumed to be continuously differentiable, according to the continuity theorem of the parametric variational inequality problem proposed by Harker and Pang (1990), * v** is Lipschitz continuous w.r.t. the variation of

*.*

**y**Propositions 1 and 2 prove the uniqueness and continuity of * G*(

*,*

**y***), which are very important in the proof of an asymptotically complete convergence of SBO. The differentiability of*

**U***(*

**G***,*

**y***) cannot be guaranteed.*

**U**### 2.2 Solution procedure

The description of this framework is shown in Algorithm

.The explanations of the SBO-NDP procedure above are briefly listed in Sections 2.2.1–2.2.4.

#### 2.2.1 Latin hypercube sampling strategy

A proper scheme to carefully allocate initial evaluation points is needed to enhance the accuracy of the approximation. The Latin hypercube sampling strategy (Nielsen *et al.*, 2002) is one of the most widely used methods. See Chen *et al.* (2014b) for a detailed discussion.

#### 2.2.2 Kriging model

The Kriging model is a powerful tool for the complicated function evaluation. The core idea is to approximate an unknown function by introducing a Gaussian process. In this model, a linear function with Gaussian white noise is used as a metamodel for the true but unknown function. Parameters can be estimated via a maximum likelihood estimation (MLE). The Kriging model provides not only an approximation result but also MSE at each point in the feasible domain.

#### 2.2.3 Expected improvement

Jones *et al.* (1998) claimed that EI was an attractive way to balance the local and global searches. The objective of EI is given by Eq. (3), which is to maximize the expectation of a given point to become a potential global optimum.

#### 2.2.4 Projected candidate point heuristics

In the stochastic response surface framework, a candidate point strategy is used to find a near-optimal solution based on the surrogate model. Candidate point heuristics generate several groups of candidate points that belong to a normal distribution or uniform distribution, to capture both local and global properties.

- 1.
When generating candidate points for DNDP, deleting infeasible ones still guarantees convergence.

- 2.
When generating candidate points for MNDP, discrete variables are checked first. If the construction cost of discrete variables violates the budget constraint, the deletion still guarantees convergence; otherwise, continuous variables are projected, so that the total construction cost is within the limit.

### 2.3 Proof of convergence

We now prove the asymptotically complete convergence of the SBO-NDP framework. Most existing algorithms for solving NDPs are actually incomplete except some newly developed linear relaxation ones.

Regis and Shoemaker (2007) provided a proof by discussing the convergence properties of SBO for box-constrained problems. In this paper, we extend the convergence result by projecting candidate points into a convex feasible domain; thus, the asymptotically complete convergence of the SBO-NDP framework on any convex set can be proved. However, some lemmas are required before proving the final results. They are put in Appendix because the proof procedure is quite complicated.

**Proposition 3** SBO-NDP is an asymptotically complete convergence for CNDP.

**Proof** According to the continuity of * G*(

*,*

**y***), which is proven in Proposition 2, the proof of the asymptotically complete convergence could be directly obtained from Lemma A2 in Appendix, considering that the feasible set defined by constraints (1b)–(1e) is convex.*

**U**For DNDP, we note that step 5 in the SBO-NDP framework (Algorithm

) is a safeguard to prevent repeated solutions. It is straightforward to conclude that SBO reaches the global optimum within finite iterations because the cardinality of the feasible set is finite.For MNDP, the asymptotically complete convergence cannot be guaranteed due to its complexity. However, since the number of combinations of feasible discrete variables is finite, then MNDP can be viewed as a finite number of CNDPs. According to Proposition 3, given an indefinitely long run, the solution given by SBO-NDP is the global optimum.

## 3 Numerical examples

This section shows the tests of SBO-NDP’s performance. The CPU time for the computation of a single UE assignment on the Sioux Falls network is about 0.3–0.4 s. In each scenario, a numerical example with no less than 20 decision variables is presented.

### 3.1 Continuous network design problem

#### 3.1.1 A 16-link network

*et al.*(1987). In this study, a demand of 10 is used to illustrate the algorithm performance. The upper-level objective function of CNDP is \(Z = \sum\limits_{a \in A\bigcup {{A_1}}} {{x_a}{t_a}({x_a}) + \theta \sum\limits_{a \in {A_1}} {{c_a}{y_a}}}\), which is the sum of the total network travel time and equivalent link construction costs, where

*θ*is the conversion factor, and in this case was set to 1. To cope with the existing results of other algorithms, the budget constraint was transferred to a penalty term in the objective function. There were 16 continuous decision variables. The maximum iteration of SBO was set to 100.

*Z*’s of the existing algorithms were recomputed to guarantee consistency. The optimal objective function values reported in the literature are denoted by

*Z*

_{ o }, which can be slightly different from

*Z*.

Continuous network design problem solving results using different algorithms on a 16-link network

Abbr eviation | Algorithm | Source | | |
---|---|---|---|---|

IOA | Iterative optimization assignment | Allsop (1974) | 557.84 | 556.61 |

HJ | Hooke-Jeeves algorithm | Abdulaal and LeBlanc (1979) | 562.15 | 557.22 |

EDO | Equilibrium decomposed optimization | Suwansirikul | 540.20 | 540.74 |

MINOS | Modular in-core nonlinear system | Suwansirikul | 557.18 | 557.14 |

SA | Simulated annealing | Friesz | 526.75 | 528.50 |

SAB | Sensitivity analysis based | Yang and Yagar (1995) | 536.88 | 536.08 |

AL | Augmented Lagrangian | Meng | 532.57 | 532.71 |

GP | Gradient projection | Cantarella and Vitetta (2006) | 533.08 | 534.02 |

CGP | Conjugated gradient projection | Chiou (2005) | 535.27 | 534.11 |

QNEW | Quasi-Newton | Chiou (2005) | 535.64 | 534.08 |

PT | PARTAN GP | Chiou (2005) | 534.49 | 534.02 |

PMILP | Path-based MILP | Wang and Lo (2010) | 522.31 | 523.63 |

LMILP | Link-based MILP | Luathep | 526.08 | 526.49 |

PMC | Penalty with multicutting plane | Li | 522.73 | 522.74 |

SBO (be st) | Surrogate-based optimization | This paper | | — |

SBO (medium) | Surrogate-based optimization | This paper | 522.40 | — |

SBO (worst) | Surrogate-based optimization | This paper | 525.42 | — |

In Table 2, the best result among 20 random runs of SBO is 521.24, which is also the best among all the algorithms listed and even better than the results given by PMILP (522.31) and PMC (522.73). The 10th best result for SBO (522.40) is only slightly worse than that of PMILP, and the worst SBO result (525.42) is only worse than those of PMILP and PMC.

Computation time comparison for solving for the continuous network design problem

Algorithm | Test computer | Computation effort |
---|---|---|

SA | — | More than 10 000 UE assignments |

PMILP | Dell GX260 3 GB RAM | 22.8 h |

PMC | Intel core 2 duo, 2.53 GHz CPU, 4 GB RAM | 12.0 h |

SBO | 2.4 GHz intel core, 4 GB RAM | 30 s; 100 UE assignments |

#### 3.1.2 Sioux Falls network

*et al.*(1987). Fig. 2 illustrates the topology of the Sioux Falls network and links to be expanded. The objective function is \(Z = \sum\limits_{a \in A\bigcup {{A_1}}} {{x_a}{t_a}({x_a}) + 0{.}001 \cdot \sum\limits_{a \in {A_1}} {{c_a}y_a^2}}\), which is the sum of the total network travel time and a nonlinear transformation of the link construction costs. To illustrate the SBO-NDP capability, we set 30 continuous decision variables as shown in Fig. 2. As a comparison, two well-known heuristics, i.e., the genetic algorithm (GA) and the simulated annealing (SA), were tested.

Algorithm performance comparison for solving continuous design problems on the Sioux Falls network

Algorithm | | Number of UE assignments | Average CPU time (min) |
---|---|---|---|

GA (best) | 72.98 | 5000 | 78 |

GA (worst) | 74.62 | 5000 | 78 |

SA (best) | 78.62 | 5000 | 75 |

SA (worst) | 81.43 | 5000 | 75 |

SBO (best) | | 200 | 5 |

SBO (worst) | 73.24 | 200 | 5 |

In Table 4, it is clear that SBO achieves better solutions than GA and SA. The worst solution for SBO is only 1% worse than the best solution, indicating that SBO can obtain a robust solution within only 200 iterations for 30 decision variables.

### 3.2 Discrete network design problem

Discrete network design problem solving results using different algorithms on a 24-candidate-link network

Budget | | Probability of reaching the optimum |
---|---|---|

3000 | 120.410 | 1.00 |

4000 | 87.790 | 0.85 |

5000 | 67.813 | 0.80 |

6000 | 66.050 | 0.65 |

7000 | 65.020 | 0.85 |

Algorithm performance comparison for solving discrete network design problem on a 24-candidate-link network

Algorithm | | Number of UE assignments | Average CPU time (min) |
---|---|---|---|

GA (best) | | 5000 | 25 |

GA (worst) | 82.43 | 5000 | 25 |

SA (best) | 69.39 | 5000 | 23 |

SA (worst) | 84.37 | 5000 | 23 |

SBO (best) | | 200 | 1 |

SBO (worst) | 67.23 | 200 | 1 |

### 3.3 Mixed network design problem

*et al.*(2011). Table 7 illustrates the comparison results of LMILP and SBO, showing that the best and medium solutions for SBO outperform LMILP in terms of quality, and the worst solution still has some acceptance.

Mixed network design problem solving results using different algorithms on the Sioux Falls network

Algorithm | |
---|---|

LMILP | 67.27 |

SBO (best) | |

SBO (medium) | 67.10 |

SBO (worst) | 67.64 |

Algorithm performance comparison for mixed network design problem on the Sioux Falls network

Algorithm | | Number of UE assignments | Average CPU time (min) |
---|---|---|---|

GA (best) | 67.40 | 2500 | 43 |

GA (worst) | 74.12 | 2500 | 43 |

SA (best) | 72.19 | 2500 | 40 |

SA (worst) | 80.29 | 2500 | 40 |

SBO (best) | | 100 | 3 |

SBO (worst) | 67.64 | 100 | 3 |

## 4 Conclusions

- 1.
Theoretically, SBO for solving CNDP is asymptotically completely convergent. Given an indefinitely long run time, the algorithm was proven to reach the global optimum with probability one.

- 2.
Practically, SBO works well and outperforms other existing algorithms in different numerical tests. In the problem with 10–20 decision variables, the SBO solution quality is comparable to the best results obtained by existing algorithms but with a relatively low computation cost. In the problem with more than 20 decision variables, SBO performs much better than other heuristics (e.g., GA and SA).

There is still room for future research. First, though guaranteeing asymptotically complete convergence, SBO is still intrinsically heuristic because it cannot estimate the gap between the current solution and the true global optimum. Second, SBO assumes a specific surrogate form, which may be inappropriate when the true function is highly singular.

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