difference lies in the secondary objective function $h$, which is added by a vanishing multiplier $\lambda_k$. Second, we do averaging in order to accelerate the convergence speed of the algorithm. For this, we consider a weighted average sequence $\{\bar x_k\}$ defined as below: $$\begin{aligned}
\label{weighted alg INC}
    \bar x_{k+1} := \sum_{t=0}^{k} \psi_{t,k} x_t,  \hbox{ where } \psi_{t,k} \triangleq \frac{\gamma_t^r}{\sum_{i=0}^{k} \gamma_i^r},  
    \end{aligned}$$ in which $r<1$ is a constant, controlling the weights. Note that in Algorithm \[algorithm:IRIG\] follows from the relation by applying induction, (see e.g., Proposition 3 in [@Yousefian18]).

Convergence analysis {#sec: conv ana}
====================

In this section, [[our goal is to show that]{}]{} the generated sequence $\{\bar x_k\}$ by Algorithm \[algorithm:IRIG\] converges to the unique optimal solution of problem [[(see Theorem \[thm conv for xbar INC\])]{}]{}.

\[bound on gradient by beck\] (a) Note that from Theorem 3.16, pg. 42 of [@Beck17], Assumption \[assum:properties\] implies that there exist constants $C_f, C_h \in \mathbb{R}$ such that $\|g_{f_i}(x)\| \leq C_f$ and $\|g_{h}(x)\| \leq C_h$ for all $i=1,\cdots, m$ and $x\in X$, where $g_{f_i}(x) \in \partial f_i(x)$ and $g_{h}(x) \in \partial h(x)$.\
(b) From Theorem 3.61, pg. 71 of [@Beck17], functions $f_i$ and $h$ are Lipschitz over $X$ with parameters $C_f$ and $C_h$, respectively, 