Finite-memory-free proximal bundle method with serious and null steps

Let $C\subseteq {\mathbb{R}}^n$ be nonempty, closed, and convex; let $f:C\to \mathbb{R}$ be convex; fix $t>0$, a serious-step fraction $m_s\in (0,1)$, a center $x_0\in C$, and an integer outer budget $N\ge 1$. A valid oracle call at $y\in C$ returns $(f(y),g_y)$ with $g_y\in \mathrm{\partial }f(y)$; otherwise it fails. Store every cut $m_y(z)=f(y)+⟨g_y,z-y⟩$ from every successful call, with no memory cap, and set $M_k$ to the maximum of the stored cuts. If no cut exists, call the oracle at the current center. If this initialization call fails, return status oracle-failure with the current center, zero completed trials, and no candidate or model residual; otherwise store its cut. Find the unique minimizer $z_k=\arg {\min }_{z\in C}(M_k(z)+\frac{\mathrm{\parallel }z-x_k{\mathrm{\parallel }}_2^2}{2t}),P_k=f(x_k)-M_k(z_k)\mathrm{.}$ If $P_k\le 0$, return status stationary-or-model-failure with $x_k,z_k,P_k$. Otherwise evaluate at $z_k$. An oracle failure returns oracle-failure. Declare a serious step exactly when $f(x_k)-f(z_k)\ge m_s{P}_k$, set $x_{k+1}=z_k$ then, and otherwise take a null step and set $x_{k+1}=x_k$. Append the valid trial cut after either step. Return budget-exhausted after $N$ trials. This finite algorithm makes no generic convergence claim.