## 5 Numerical Experiments  
In this section, we aim to test the numerical efficiency of the proposed algorithm. In the fields of production and transportation, finance and insurance, power industry, and telecommunications, decision makers usually need to solve problems with uncertain information. As an effective tool to solve such problems, stochastic programming (SP) has attracted more and more attention and research on its practical instances and theories; see, for example, References [41; 42]. We consider a class of two-stage SP problems with fixed recourse, whose discretization of uncertainty into \(N\) scenarios has the form (see e.g., References [23; 43])  
\[ \begin{array}{ll}\min&f(x):=\langle c,x\rangle+\sum_{i=1}^{N}p_{i}V_{i}(x) \\ \text{s.t.}&x\in X:=\{x\in\mathbb{R}_{+}^{n}:\ Ax=b\},\end{array}  \tag{31}\]where \(x\) is the first-stage decision variable, \(c\in\mathbb{R}^{n_{1}}\), \(A\in\mathbb{R}^{m_{1}\times n_{1}}\), and \(b\in\mathbb{R}^{m_{1}}\). In addition, the recourse function is  
\[V_{i}(x):=\min_{\pi\in\mathbb{R}^{n_{2}}_{+}}\{\langle q,\pi\rangle:\ W\pi=h_{i} -T_{i}x\},\]  
where corresponding to the \(i\)th scenario \((h_{i},T_{i})\), with probability \(p_{i}>0\) for \(h_{i}\in\mathbb{R}^{m_{2}}\) and \(T_{i}\in\mathbb{R}^{m_{2}\times n_{1}}\). Here \(\pi\) is the second-stage decision variable. Clearly, by introducing the indicator function \(\delta_{X}\), problem (31) can be written as the form of (5), and then becomes the form of (1) by setting \(h(x)=\delta_{X}(x)\). The above recourse function can be written as its dual form:  
\[V_{i}(x)=\max_{y\in R^{m_{2}}}\langle h_{i}-T_{i}x,y\rangle\quad\text{s.t.}\ W^{T}y \leq q,\]  
where \(q\in\mathbb{R}^{n_{2}}\) and \(W\in\mathbb{R}^{m_{2}\times n_{2}}\). By solving these linear programming problems to return solutions with precision up to a given tolerance, one can establish an inexact oracle in the form (6), see Reference [23] for more detailed description. The instances of SP problems are downloaded from the link: [[http://pwp.gatech.edu/guanghui-lan/computer-codes/](http://pwp.gatech.edu/guanghui-lan/computer-codes/)]([http://pwp.gatech.edu/guanghui-lan/computer-codes/](http://pwp.gatech.edu/guanghui-lan/computer-codes/)). Four instances are tested, namely, SSN(50), SSN(100), 20-term(50), 20-term(100), where the integers in the brackets mean the number of scenarios \(N\). Here, the SSN instances come from the telecommunications and have been studied by [PERSON], [PERSON], and [PERSON] [44]. And the 20-term instances come from the motor freight carrier's problem and have been studied by [PERSON], [PERSON], and [PERSON] [45]. The dimensions of these instances are listed in Table 2.  
The parameters are selected as: \(\kappa=0.04\), \(t_{min}=0.1\) and \(t_{1}=1.1\). The maximum bundle size is set to be 35. All the tests were performed in MATLAB (R2014a) on a PC with Intel(R) Core(TM) i7-4790 CPU 3.60 GHz, 4 GB RAM. The quadratic programming and linear programming subproblems were solved by the MOSEK 8 toolbox for MATLAB; see [[http://www.mosek.com](http://www.mosek.com)]([http://www.mosek.com](http://www.mosek.com)). We first compare our algorithm (denoted by GALBM) with the accelerated prox-level method (APL) in Reference [43], where the tolerances of the linear programming solver of MOSEK are set by default. The results are listed in Table 3, in which the number of iterations (NI), the consumed CPU time in seconds (Time), and the returned minimum values (\(f_{*}\)) are compared. Note that we use the MATLAB commands tic and to measure the consumed CPU time. For each instance, we run 10 times and report the average CPU time. From Table 3, we see that, when similar solution quality is achieved, GALBM can significantly outperform than APL in terms of the number of iterations and CPU time. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & \(\mathbf{n_{1}}\) & \(\mathbf{m_{1}}\) & \(\mathbf{n_{2}}\) & \(\mathbf{m_{2}}\) \\ \hline SSN & 89 & 1 & 706 & 175 \\
20-term & 63 & 3 & 764 & 124 \\ \hline \hline \end{tabular}
\end{table}