A convergent and efficient decomposition method for the traffic assignment problem

Abstract

In this work we consider the network equilibrium problem formulated as convex minimization problem whose variables are the path flows. In order to take into account the difficulties related to the large dimension of real network problems we adopt a decomposition-based approach suitably combined with a column generation strategy. We present an inexact block-coordinate descent method and we prove the global convergence of the algorithm. The results of computational experiments performed on medium-large dimensional problems show that the proposed algorithm is at least competitive with state of the art algorithms.

Appendix: A quadratic line search

The properties of an Armijo-type line search do not guarantee, without further assumptions on the search direction $d^k$, that the distance between successive points tends to zero, which is a usual requirement of decomposition methods. Such a property can be satisfied by the line search algorithm described below and based on the acceptance condition

$$\begin{aligned} f(x^k+\alpha ^kd^k)\le f(x^k)-\gamma \left( \alpha ^k\Vert d^k\Vert \right) ^2, \end{aligned}$$

which replaces the Armijo’s condition

$$\begin{aligned} f(x^k+\alpha ^kd^k)\le f(x^k)+\gamma \alpha ^k\nabla f(x^k)^Td^k \end{aligned}$$

Before describing the line search procedure, we state a formal assumption on the sequence of search directions.

Let $d^k\in R^n$ be a feasible direction at $x^k\in \mathcal {F}$. Let $\beta ^k$ be the maximum feasible step length along $d^k$. Taking, for instance, $\hat{x}^k\in \mathcal {F}$ and $d^k=\hat{x}^k-x^k\ne 0,$ it follows $\beta ^k\ge 1$.

Assumption 1

$\{d^k\}$ is a sequence of feasible search directions such that

• (a) For all $k$ we have $\Vert d^k\Vert \le M$ and $\beta ^k\le \bar{\beta }$ for given numbers $M>0$ and $\bar{\beta }>0$;

• (b) For all $k$ we have $\nabla f(x^k)^Td^k<0$.

The Quadratic Line Search algorithm can be described as follows.

The properties of Algorithm QLS are stated in the next proposition.

Proposition 5

Let $\{x^k\}$ be a sequence of points belonging to the feasible set $\mathcal {F}$, and let $\{d^k\}$ be a sequence of search directions satisfying Assumption 1. Then:

• (i) Algorithm QLS determines, in a finite number of iterations, a scalar $\alpha ^k$ such that

  $$\begin{aligned} f(x^k+\alpha ^k d^k)\le f(x^k)-\gamma (\alpha ^k\Vert d^k\Vert )^2; \end{aligned}$$

  (29)

• (ii) If $\{x^k\}$ converges to $\bar{x}$ and

  $$\begin{aligned} \lim _{k\rightarrow \infty } \left( f(x^k)-f(x^k+\alpha ^kd^k)\right) =0, \end{aligned}$$

  (30)

  then we have

  $$\begin{aligned} \lim _{k\rightarrow \infty }\alpha ^k\Vert d^k\Vert =0, \end{aligned}$$

  (31)

  and

  $$\begin{aligned} \lim _{k\rightarrow \infty }\beta ^k\nabla f(x^k)^Td^k=0. \end{aligned}$$

  (32)

Proof

In order to prove assertion (i), let us assume, by contradiction, that condition of the while cycle is violated for every $j=0,1,\ldots$. Then, we have for $j=0,1,\ldots$

$$\begin{aligned} f(x^k+\bar{\alpha }\delta ^jd^k)>f(x^k)-\gamma \left( \bar{\alpha }\delta ^j\right) ^2\Vert d^k\Vert ^2, \end{aligned}$$

where $\bar{\alpha }=\min \lbrace \beta ^k,\lambda \rbrace$. Taking limits for $j\rightarrow \infty$ we obtain:

$$\begin{aligned} \nabla f(x^k)^Td^k\ge 0, \end{aligned}$$

which contradicts (b) of Assumption 1.

From (29) and (30) it follows that (31) holds.

In order to prove (32), assume by contradiction that there exists an infinite subset $K\subseteq \{0,1,\ldots \}$ such that

$$\begin{aligned} \lim _{k\in K,\,\,k\rightarrow \infty } \beta ^k\nabla f(x^k)^Td^k=\beta ^\star \nabla f(\bar{x})^T\bar{d}<0. \end{aligned}$$

(33)

From (33) we get, for $k\in K$ and $k$ sufficiently large,

$$\begin{aligned} \beta ^k\ge \eta _1, \end{aligned}$$

(34)

and

$$\begin{aligned} \Vert d^k\Vert \ge \eta _2, \end{aligned}$$

(35)

for some numbers $\eta _1,\eta _2 >0$.

Then, (31) and (35) imply that we can write

$$\begin{aligned} \lim _{k\in K,k\rightarrow \infty }\alpha ^k=0. \end{aligned}$$

(36)

From (36) and (34) we obtain, for $k\in K$ and $k$ sufficiently large,

$$\begin{aligned} \alpha ^k<\bar{\alpha }=\min \{\beta ^k,\lambda \}, \end{aligned}$$

which implies, recalling the instructions of the algorithm,

$$\begin{aligned} f\left( x^k+{{\alpha ^k}\over {\delta }}d^k\right) >f(x^k)- \gamma \left( {{\alpha ^k}\over {\delta }}\Vert d^k\Vert \right) ^2. \end{aligned}$$

(37)

From (37), using the Mean Value Theorem, it follows

$$\begin{aligned} \nabla f(\xi ^k)^Td^k>-\gamma \left( {{\alpha ^k}\over {\delta }}\right) \Vert d^k\Vert ^2, \end{aligned}$$

(38)

where $\xi ^k=x^k+t^k{{\alpha ^k}\over {\delta }}d^k$ and $t^k\in (0,1)$. Taking the limits for $k\in K$ and $k\rightarrow \infty$, recalling (36), we obtain

$$\begin{aligned} \nabla f(\bar{x})^T\bar{d}\ge 0, \end{aligned}$$

which contradicts (33).