A Progressive Hedging based branch-and-bound algorithm for mixed-integer stochastic programs

Abstract

Progressive Hedging (PH) is a well-known algorithm for solving multi-stage stochastic convex optimization problems. Most previous extensions of PH for mixed-integer stochastic programs have been implemented without convergence guarantees. In this paper, we present a new framework that shows how PH can be utilized while guaranteeing convergence to globally optimal solutions of mixed-integer stochastic convex programs. We demonstrate the effectiveness of the proposed framework through computational experiments.

Appendix A: SSLP formulation

We provide the formulation of the stochastic server location problems (SSLPs) of Ntaimo and Sen (2005), which we use in Sect. 5. The problem considers a set of locations $\mathcal {J}$, zones $\mathcal {Z}$, and clients $\mathcal {I}$. The goal is to obtain (i) a cost-efficient allocation of servers into locations specified in $\mathcal {J}$, and (ii) a cost-efficient assignment of clients to these servers under random client availability. We list the decision variables of the problem below.

$z_j$::

1 if server is located at site j, 0 otherwise

$y_{ij}^s$::

1 if client i is served by server at location j under scenario s, 0 otherwise

$y_{j0}^s$::

amount of demand overflow due to server capacity limitations.

The variable $y_{j0}^s$ must not take large values, therefore is penalized using the penalty function $\phi _j^s$. Each problem instance takes the following input parameters:

$c_j$::

cost of locating a server at location $j \in \mathcal {J}$

$q_{ij}$::

revenue from client $i \in \mathcal {I}$ when served by server at location j

$d_{ij}$::

demand of client i from server at location j

u::

server capacity

v::

upper bound on the total number of located servers

$w_z$::

minimum number of servers to be located in zone $z \in \mathcal {Z}$

$\mathcal {J}_z$::

subset of server locations that belong to zone z

$\gamma _i$::

random availability of client i

$\gamma _i^s$::

realization of $\gamma _i$ under scenario s (1 if client i is present in scenario s, 0 otherwise).

The first- and the second-stage subproblems are respectively given below.

$$\begin{aligned} \min ~&\sum _{j \in \mathcal {J}} c_j z_j + \mathbb {E}\left[ h(z, \gamma )\right] \nonumber \\ \text {s.t.} \quad ~&\sum _{j \in \mathcal {J}} z_j \le v \end{aligned}$$

(13a)

$$\begin{aligned}&\sum _{j \in \mathcal {J}_z} z_{j} \ge w_z, \quad \forall z \in \mathcal {Z} \nonumber \\&z_j \in \mathbb {B}, \quad \forall z \in \mathcal {Z}, \end{aligned}$$

(13b)

$$\begin{aligned} h(z, \gamma ^s) = \min ~&- \left( \sum _{i \in \mathcal {I}} \sum _{j \in \mathcal {J}} q_{ij}^s y_{ij}^s - \sum _{j \in \mathcal {J}} \phi _j^s\left( y_{j0}^s\right) \right) \nonumber \\ \text {s.t.} \quad ~&\sum _{i \in \mathcal {I}} d_{ij} y_{ij}^s - y^s_{j0} \le u z_j, \quad \forall i \in \mathcal {I}, \quad j \in \mathcal {J} \end{aligned}$$

(13c)

$$\begin{aligned}&\sum _{j \in \mathcal {J}} y_{ij}^s = \gamma _i^s, ~ \forall i \in \mathcal {I} \nonumber \\&y^s_{ij} \in \mathbb {B}, ~ \forall i \in \mathcal {I}, \quad j \in \mathcal {J} \nonumber \\&y^s_{j0} \ge 0, \quad \forall j \in \mathcal {J}. \end{aligned}$$

(13d)

Constraints (13a) limit the maximum number of server allocations; (13b) ensures each zone receives the minimum number of required servers; (13c) models the server capacity restrictions; (13d) ensures that each client is served by a single server.

In the original SSLP instances of Ntaimo and Sen (2005), the penalty function is assumed to have a linear form $\phi _j^s (y_{j0}^s) = q_{j0}^s y_{j0}^s$, where $q_{j0}^s$ is a large coefficient. The quadratic extension of the problem upgrades this penalty function as $\phi _j^s( y_{j0}^s ) = q_{j0}^s \big ( y_{j0}^s \big )^2$. In order to increase the significance of the new quadratic penalty function, the server capacity parameter u is adjusted to be smaller in the new instances.