MULTICOMMODITY NETWORK FLOWS WITH NONCONVEX ARC COSTS

We present a survey on nonconvex models and algorithms for multicommodity network design problems. We put in perspective the alternative modelling of these problems, traditionally represented by mixed-integer linear programs, by separable nonconvex arc cost functions. We show in particular that some problems take profit of a continuous modelling and explore the case of capacity expansion of communications networks.

Keywords:
network design; capacity and flow assignment problem; capacity expansion

Authorship

Philippe Mahey

LIMOS-CNRS and ISIMA, Université de Clermont Auvergne, France. E-mail: philippe.mahey@isima.frUniversité de Clermont AuvergneFranceFranceLIMOS-CNRS and ISIMA, Université de Clermont Auvergne, France. E-mail: philippe.mahey@isima.fr

Mauricio C. de Souza ^**Corresponding author.

DEP, Universidade Federal de Minas Gerais, Brazil. E-mail: mauricio.souza@dep.ufmg.brUniversidade Federal de Minas GeraisBrazilBrazilDEP, Universidade Federal de Minas Gerais, Brazil. E-mail: mauricio.souza@dep.ufmg.br

*Corresponding author.

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables | Formulas

Figures (5)
Tables (2)
Formulas (23)

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Figure 1:
The linear case does not work.

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Figure 2:
Convex and concave arc costs.

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Figure 3:
Piecewise affine concave approximation.

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Figure 4:
Piecewise affine increasing function.

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Figure 5:
The integrated function of congestion and expansion costs.

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Table 1:
Network characteristics.

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Table 2:
Results for larger networks with c _1e /c _0e =4 and γ = 50%.

(1)

(2)

(3)

Figure 1: The linear case does not work.

Figure 2: Convex and concave arc costs.

Figure 3: Piecewise affine concave approximation.

Figure 4: Piecewise affine increasing function.

Figure 5: The integrated function of congestion and expansion costs.

Table 1: Network characteristics.

Instance	Topology	\|V\|	\|E\|	K
att	AT&T Worldnet backbone	90	274	272
fr250	Frame-relay	60	688	250
fr500	Frame-relay	60	906	500
hier50	2-level hierarchical	50	148	245

Table 2: Results for larger networks with c _1e /c _0e =4 and γ = 50%.

			Initial Solution		Local Optimum
	b _k	c _0e	d(%)	it	d(%)	exp	it (CA-FA)	it (CCA)
att	1	10	27.5	2302	21.0	122	426 (2)	0
fr250	1	4	69.0	191	42.3	55	718 (4)	0
fr250	2	4	61.8	177	33.7	189	1933 (5)	120
fr500	1	4	65.8	255	39.8	162	902 (3)	0
fr500	2	4	41.8	234	27.2	387	6796 (11)	0
hier50	1	10	36.6	1270	29.4	86	3715 (4)	53
hier50	2	10	9.3	2827	7.2	119	3175 (3)	0

Minimise \sum_{e \in E} f_{e} (x_{e}) (MCF) subject to x_{e} - \sum_{k} x_{e}^{k} = 0, \forall e \in E x^{k} \in F_{k}, \forall k \in K

f_{e} (x_{e}) = \{\begin{cases} 0 if x_{e} = 0 \\ F_{e} if x_{e} > 0 \end{cases}

λ (x, Θ) = \sum_{e \in Θ^{+}} f_{e}^{' +} (x_{e}) - \sum_{e \in Θ^{-}} f_{e}^{' -} (x_{e})

(1)

f_{e} (x_{e}) = \min {Φ_{e l} (c_{e l}, x_{e}) + π_{e l}, l = 0, \dots, L}

E_{0} = {e \in E | x_{e}^{*} \in [0, γ_{e} c_{0 e})} E_{1} = {e \in E | x_{e}^{*} \in (γ_{e} c_{0 e}, c_{1 e})} G = {e \in E | x_{e}^{*} = γ_{e} c_{0 e}}

C_{i} = {x feasible to MCF | x_{e} \in [0, γ_{e} c_{0 e}] for e \in E_{0} \cup G_{0 i} x_{e} \in [γ_{e} c_{0 e}, c_{1 e}] for e \in E_{1} \cup G_{1 i}}

\{\begin{cases} \begin{array}{l} u_{e}^{i} = \partial Φ (c_{0 e}, x_{e}^{*}) / \partial x_{e}, 0 < x_{e}^{*} < γ_{e} c_{0 e}, e \in E_{0} u_{e}^{i} \leq \partial Φ (c_{0 e}, x_{e}^{*}) / \partial x_{e}, x_{e}^{*} = 0, e \in E_{0} \\ u_{e}^{i} \geq \partial Φ (c_{0 e}, x_{e}^{*}) / \partial x_{e}, x_{e}^{*} = γ_{e} c_{0 e}, e \in G_{0 i} \end{array} \\ u_{e}^{i} = \partial Φ (c_{1 e}, x_{e}^{*}) / \partial x_{e}, γ_{e} c_{0 e} < x_{e}^{*} < c_{1 e}, e \in E_{1} u_{e}^{i} \leq \partial Φ (c_{1 e}, x_{e}^{*}) / \partial x_{e}, x_{e}^{*} < γ_{e} c_{0 e}, e \in G_{1 i} \end{cases}

\forall p \in P_{k}, x_{k p} > 0, v_{k}^{i} = \sum_{e \in p} u_{e}^{i} \forall p \in P_{k}, x_{k p} = 0, v_{k}^{i} \leq \sum_{e \in p} u_{e}^{i}

f_{e} (x_{e}) = \{\begin{cases} F_{e} + h_{e} x_{e} for x_{e} > 0 \\ 0 for x_{e} = 0 \end{cases}

\sum_{k} x_{e}^{k} \leq c_{e} y_{e}, \forall e \in E

x_{e}^{k} \leq q_{e k} y_{e}, \forall e \in E, \forall k \in K

a_{k p}^{e} = \{\begin{cases} 1 if arc e \in p \\ 0 otherwise \end{cases}

x_{e} = \sum_{k \in K} \sum_{p \in P_{k}} a_{k p}^{e} x_{k p} .

(DCE) \begin{array}{l} Minimize \sum_{e} [Φ (c_{0 e} + δ_{e} y_{e}, x_{e}) + π_{e} y_{e}] \\ subject to x \in M (G, T) \\ x_{e} \leq c_{0 e} + δ_{e} y_{e}, \forall e \in E \\ y_{e} \in {0, 1}, \forall e \in E \end{array}

(CCE) \begin{array}{l} Minimize \sum_{e} f_{e} (x_{e}) = min {Φ (c_{0 e} + x_{e}), Φ (c_{1 e} + x_{e}) + π_{e}} \\ subject to x \in M (G, T) \\ x_{e} \leq c_{1 e}, \forall e \in E \end{array}

x_{e}^{*} > γ_{e} c_{0 e} ⟹ y_{e}^{*} = 1 x_{e}^{*} < γ_{e} c_{0 e} ⟹ y_{e}^{*} = 0

Φ (c_{0 e}, γ_{e} c_{0 e}) = Φ (c_{1 e}, γ_{e} c_{0 e}) + π_{e}

y_{e}^{*} \{\begin{cases} = 0 i f 0 \leq x_{e}^{*} < γ_{e} c_{0 e} = 1 i f γ_{e} c_{0 e} < x_{e}^{*} < c_{1 e} \\ \in {0, 1} i f x_{e}^{*} = γ_{e} c_{0 e} \end{cases}

(x_{i j} - γ_{i j} c_{0 i j}) (x_{j i} - γ_{j i} c_{0 j i}) \geq 0

\max_{e \in C_{A}} \frac{x_{e}}{γ_{e} c_{0 e}} \geq 1

(2)

\{\begin{cases} f_{e}^{' -} (x_{e}^{t} + α_{e}) \leq f_{e}^{' +} (x_{e}^{t}) - λ (x^{t}, Θ_{t}) if e Θ_{t}^{+} and x_{e}^{t} \geq γ c_{0 e} f_{e}^{' -} (x_{e}^{t} + α_{e}) \leq f_{e}^{' +} (x_{e}^{t}) - λ (x^{t}, Θ_{t}) and α_{e} \leq γ c_{0 e} - x_{e}^{t} if e \in Θ_{t}^{+} and x_{e}^{t} < γ c_{0 e} \\ f_{e}^{' +} (x_{e}^{t} - α_{e}) \geq f_{e}^{' -} (x_{e}^{t}) + λ (x^{t}, Θ_{t}) and α_{e} \leq \min {x_{e}^{k t}, x_{e}^{t} - γ c_{0 e}}, if e \in Θ_{t}^{-} and x_{e}^{t} > γ c_{0 e} f_{e}^{' +} (x_{e}^{t} - α_{e}) \geq f_{e}^{' -} (x_{e}^{t}) + λ (x^{t}, Θ_{t}) and α_{e} \leq x_{e}^{k t}, if e \in Θ_{t}^{-} and x_{e}^{t} \leq γ c_{0 e} \end{cases}

(3)

x^{k, t + 1} = x^{k t} + α^{t} θ_{t}

\sum_{e \in Θ^{+}} f^{' +} ({\bar{x}}_{e}) < \sum_{e \in Θ^{+}} {\bar{Φ}}_{e}' ({\bar{x}}_{e}) a s f^{' +} (γ c_{0 e}) < {\bar{Φ}}_{e}^{'} (γ c_{0 e}),

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MULTICOMMODITY NETWORK FLOWS WITH NONCONVEX ARC COSTS

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[1] *Corresponding author.