\delta_D^k
= \min_{k}  \Big\{ \min \{ -\bar{c}_{j^k} ~| ~j^k \in N^k
\hspace{0.1in}\mbox{and} \hspace{0.1in} \bar{c}_{j^k}<0 \}
\Big\}.$$ Considering Algorithm \[mainAlg\], our next lemma is an improvement of the one in [@km13].

Let $\x^k$ and $\x^{k+1}$ be the $k$th and $(k+1)$th iterates generated by Algorithm \[mainAlg\]. If $\x^k$ is not optimal and $\x^k \neq \x^{k+1}$, then, we have $$\c^{\Tr} \x^k - \c^{\Tr} \x^{k+1} \ge \delta_D \gamma_{\ell}.
\label{myIneq}$$ \[diffObj\]

Since $\x^k \neq \x^{k+1}$, from the derivation of (\[twoDim\]), the difference of the objective functions between $k$th and $(k+1)$th iterations is actually the solution of (\[twoDim\]), which is smaller than the special case when only one entering variable $x_{\jmath_2^k}$, which would generate the longest step among $\bar{c}_{j^k}<0$ for all $j^k \in N^k$, is selected. Let $\bar{\x}_{(\jmath_1^k,\jmath_2^k)}$ be the optimal solution of (\[twoDim\]) at iteration $k$. Therefore $$\begin{aligned}
\c^{\Tr} \x^k - \c^{\Tr} \x^{k+1} 
& = & - \bar{\c}_{(\jmath_1^k,\jmath_2^k)}^{\T}
\bar{\x}_{(\jmath_1^k,\jmath_2^k)}  
\nonumber \\
& \ge &  -\bar{c}_{\jmath_2^k}\bar{x}_{\jmath_2^k}
\nonumber \\
& \ge &  \delta_D \gamma_{\ell}.
\label{myIneq1}\end{aligned}$$ This finishes the proof. [&lt;1.5em -1.5em plus0em minus0.5em height0.75em width0.5em depth0.25em]{}

Lemma \[diffObj\] says that for Algorithm \[mainAlg\], the objective value decreases in every iteration at least by a constant $\delta_D \gamma_{\ell}$.\[constantD\]

From Lemmas \[firstL\], \[diffObj\] and Remark \[constantD\], it is easy 