## 3 The Generalized Alternating Linearization Bundle Method  
In this section, we present our generalized alternating linearization bundle method with inexact first-order oracles for solving (1). Let \(k\) be the current iteration index, \(x^{j}\), \(j\in J^{k}\subseteq\{1,\cdots,k\}\) be given points generated by previous iterations, and the corresponding approximate values \(f_{x^{j}}/g_{x^{j}}\) be produced by the oracle (6). For notational convenience, we denote  
\[f^{j}_{x}:=f_{x^{j}},\;g^{j}_{x}:=g_{x^{j}},\;F^{j}_{x}:=F_{x^{j}},\;e^{j}_{x} :=\varepsilon_{x^{j}},\;\gamma^{j}_{x}:=\gamma_{x^{j}}.\]  
The approximate linearizations of \(f\) at \(x^{j}\) are given by  
\[f_{j}(\cdot):=f^{j}_{x}+\langle g^{j}_{x^{j}},-x^{j}\rangle,\;\;j\in J^{k}.\]  
From the second relation in (6), we have  
\[f(\cdot)\geq f(x^{j})+\langle g^{j}_{x^{j}},-x^{j}\rangle-\eta(\gamma^{j}_{x} )=f_{j}(\cdot),\]  
which implies that \(f_{j}\) is a lower approximation to \(f\). Next, it is natural to define the polyhedral inexact cutting-plane model of \(f\) by  
\[\tilde{f}_{k}(\cdot):=\max_{j\in J_{k}}\left\{f_{j}(\cdot)\right\}, \tag{8}\]  
which is obviously a lower polyhedral model for \(f\), that is, \(\tilde{f}_{k}(\cdot)\leq f(\cdot)\). Let \(\hat{x}^{k}\) (called _stability center_) be the \"best\" point obtained so far, which satisfies that \(\hat{x}^{k}=x^{k(l)}\) for some \(k(l)\leq k\). Frequently, it holds that \(\tilde{f}^{k(l)}_{x}=\min_{j=1,\cdots,k}f^{j}_{x}\). Thus, from (7), we have  
\[f(\hat{x}^{k})\in[f^{k}_{x},f^{k}_{x}+\varepsilon^{k(l)}_{x}],\;\;\text{ whenever}\;\;F^{k}_{x}\leq\gamma^{k}_{x}. \tag{9}\]  
By applying the bundle idea to the \"complex\" function \(f\), and keeping the simple function \(h\) unchanged, similar to traditional proximal bundle methods (see, e.g., Reference [28]), we may solve the following subproblem to obtain a new iterate \(x^{k+1}\):  
\[x^{k+1}:=\arg\min\;\tilde{f}_{k}(\cdot)+h(\cdot)+\frac{1}{2t_{k}}\|\cdot-\hat{ x}^{k}\|^{2}, \tag{10}\]where \(t_{k}>0\) is a proximal parameter. However, subproblem (10) is generally not easy to solve, so by making use of the alternating linearization idea of [PERSON] (2000), we solve two easier subproblems instead of (10). These two subproblems are interrelated: one is to find the proximal point of the polyhedron model \(\tilde{f}\) plus the linearization of \(h\), aiming at generating an aggregate linear model of \(f\) for use in the second subproblem; the other is to find the proximal point of the aggregate linear model of \(f\) plus \(h\), aiming at obtaining a new trial point. Now, we are ready to present the details of our algorithm, which generalizes the work of [PERSON] (2000). We note that the choice of the model function \(\tilde{f}_{k}\) in the algorithm may be different from the form of (8), since the subgradient aggregation strategy ([PERSON] and [PERSON], 2010) is used to compress the bundle. The algorithm generates three sequences of iterates as follows: \(\{y^{k}\}\), the sequence of intermediate points, at which the aggregate linear models of \(f\) are generated; \(\{x^{k}\}\), the sequence of trial points; \(\{\hat{x}^{k}\}\), the sequence of stability centers. We make some comments about Algorithm 1 as follows. **Step 0** (Initialization). Select an initial point \(x^{1}\in\mathbb{R}^{n}\), constants \(\kappa\in(0,1)\), \(t_{\min}>0\), and an initial stepsize \(t_{1}\geq t_{\min}\). Call the oracle (6) at \(x^{1}\) to compute the approximate values \(f_{x}^{1}\) and \(g_{x}^{1}\). Choose an initial error bound \(e_{x}^{1}\geq 0\) and a descent target \(\gamma_{x}^{1}=+\infty\). Set \(\hat{x}^{1}:=x^{1}\), \(f_{x}^{1}:=f_{x}^{1}\), \(F_{x}^{1}:=f_{x}^{1}+h(\hat{x}^{1})\), \(f_{0}:=f_{1}\), and \(\tilde{h}_{0}(\cdot):=h(x^{1})+\langle p_{h}^{0},-x^{1}\rangle\) with \(p_{h}^{0}\in\partial h(x^{1})\). Let \(i_{t}^{1}:=0\), \(l:=1\), \(k(l):=1\) and \(k:=1\). **Step 1** (Model selection).