\section{Dealing with Boundary Optimum}\label{sec:boundary}

\revise{Here, we discuss the treatment and theoretical behavior when dealing with the boundary optimum as mentioned in \remref{rem: boundary}. First, we extend the results in \thmref{thm: width}, when not assuming the \assref{apt: exist_star} hold. We uniformly shift the constraints by a small amount ${\epsilon}_{\cFunc}$ to satisfy \assref{apt: exist_star} with the modified constraints. Formally,  $ \forall \conIdx\in\conSpace$, $\cFunc'_\conIdx(\instance) = \cFunc_\conIdx(\instance) + {\epsilon}_{\cFunc}$. Then, running \algname with these adjusted constraints, $\cFunc'_\conIdx$, instead of the original $\cFunc_\conIdx$, we have the following guarantee, which is a direct extension of \thmref{thm: width}.  We denote the $\tilde{\globalf}^* = \globalf{(\tilde{\instance}^*)}$, Here ${\tilde{\instance}^*} = \argmax_{\instance\in\searchSpace, \forall \conIdx\in\conSpace, \cFunc'_\conIdx(\instance) > 0}{\globalf(\instance)}$. }

\begin{cor}\label{cor: width}
Under the aforementioned assumptions and modifications, with a constant $\beta=2\log(\frac{2(\conNum+1)\vert \discreteSet \vert T}{\delta})$ and the acquisition function from $\algoref{alg:main}$, there exists an $\epsilon_\globalf \leq \epsilon_\cFunc$, such that after at most $T \geq \frac{\beta \widehat{\maxInfo_T} C_1}{\epsilon_\globalf^2}$ iterations, we have $\Pr{\vert CI_{\tilde{\globalf}^*, T}\vert \leq \epsilon_\globalf, \tilde{\globalf}^* \in CI_{\tilde{\globalf}^*, T }} \geq 1 - \delta$.
    Here, $C_1 = 8/\log(1+\sigma^{-2})$.
\end{cor}

\revise{This \corref{cor: width} allows us to depict the width of the global optimum defined in the enlarged feasible region similarly. Since the optimum is defined in the enlarged area, it could be an upper bound of the global optimum defined in the original feasible region, including the feasible region boundaries. That is} 
$$
\tilde{\globalf}^* \geq \argmax_{\instance\in\searchSpace, \forall \conIdx\in\conSpace, \cFunc_\conIdx(\instance) \geq 0}{\globalf(\instance)}
$$
\revise{This allows us to extend further the \corref{cor: cum-regret} that depicts the partial cumulative regret after sufficient iterations and the upper bound of the violations.}

\begin{cor}\label{cor: cum-regret-boundary}
    Under the aforementioned assumptions and modifications, when $ \forall \conIdx\in\conSpace$, $\cFunc'_\conIdx(\instance) > 0$, $\instance \neq \instance^*$, $\exists \epsilon_\cFunc > \epsilon'_\cFunc \geq \epsilon_\globalf > 0$, $\globalf^* - \globalf(\instance) \leq 2\epsilon_\globalf$, it holds that $\forall \conIdx\in\conSpace$, $\cFunc'_\conIdx(\instance) \geq \epsilon'_{\cFunc}$. We use $\beta=2\log(\frac{2(\conNum+1)\vert \discreteSet \vert T}{\delta})$ and the acquisition function from $\algoref{alg:main}$. After at most $t' \geq \frac{\beta \widehat{\maxInfo_{t'}} C_1}{{\epsilon'}_{\globalf}^2}$ iterations, we have, $\Pr{\sum_{t=t'}^{T}\reward(\instance^*) - \reward(\instance_t) \leq {\sqrt{(T-t')\beta\maxInfo_{T}{C_1}}}, \forall \instance \in \discreteSet, \cFunc_\conIdx(\instance_t) \geq -{\epsilon}_{\cFunc}} > 1-\delta$. Here, $C_1 = 8/\log(1+\sigma^{-2})$ and $t' \leq T$.
\end{cor}

\begin{proof}
    First, we have $\forall \instance \in \discreteROI$,  $2\beta^{1/2}\sigma_{f, t-1}(\instance) \leq \alpha_t \leq \epsilon_{\globalf} $.

    Then $\forall \instance \neq \tilde{\instance}^*$ and $\instance \in \discreteROI$, we have $$\Pr{\tilde{\globalf}^* - \globalf(\instance) \leq \vert CI_{\tilde{\globalf}^*, t}\vert + \UCB_{\globalf, t}(\instance) - \LCB_{\globalf, t}(\instance) \leq 2\alpha_t \leq 2\epsilon_{\globalf}} \geq 1-\delta$$ 
    Then by assumption, $\forall \instance \in \discreteROI$, $\forall \conIdx\in\conSpace$, we have probability at least $1-\delta$, $\cFunc'_\conIdx(\instance) \geq \epsilon'_{\cFunc}$. Hence we have both $\cFunc_\conIdx(\instance) \geq \epsilon_{\cFunc} - \epsilon'_{\cFunc}$  and $\instance \notin U_{\cFunc'_\conIdx, t}$. According to the algorithm, it regresses to GP-UCB by \cite{srinivas2009gaussian} between $t'$ and $T$.
    \begin{align*}
        \sum_{t=t'}^{T}(\reward(\instance^*) - \reward(\instance_t)) &\leq 
        \sum_{t=t'}^{T}(\reward(\tilde{\instance}^*) - \reward(\instance_t))^2\\
        &\leq \beta{C_1}(\maxInfo_{T} - \maxInfo_{t'})\\
        &\leq \beta{C_1}\maxInfo_{T}(1-t'/T)
    \end{align*}
     By Cauchy-Schwarz, we have
     \begin{align*}
        \sum_{t=t'}^{T}(\reward(\instance^*) - \reward(\instance_t)) &\leq \sqrt{(T-t') \sum_{t=t'}^{T}(\reward(\instance^*) - \reward(\instance_t))^2}\\
        &\leq \sqrt{\frac{(T-t')^2}{T}\beta{C_1}\maxInfo_{T}}\\
        &\leq \sqrt{(T-t')\beta{C_1}\maxInfo_{T}}
    \end{align*}
\end{proof}

\revise{Note that by enlarging the feasible region with ${\epsilon}_{\cFunc}$, we don't risk losing the feasible region to enable \algname to identify both interior and boundary optimum. We don't change the definition of $\globalf^*$. Instead, we only leverage the modified running constraints and verify the feasibility with the original constraints. For the algorithms aiming at violation tolerant objectives like CONFIG \citep{xu2023constrained}, there is no similar guarantee with straightforward modification, e.g., adding small ${\epsilon}_{\cFunc}$ to the threshold while not risking losing the feasible region. We denote the modified \algname as \algname-EPSILON. We include the corresponding comparison on the noise-free Ackley-10D-2C as exactly defined in \citet{eriksson2021scalable}, where the feasible region is less than $2.2*10^{-3}\%$ of the whole search space, and the optimum lies on the boundary of feasible region by construction. }

\begin{figure*}[t]
    \centering
        {
      \includegraphics[trim={0cm 0cm 0cm 0cm}, width=.8\textwidth]{./fig/ackley_10d_2c.png}
    }
  \caption{The figure illustrates the simple regret for Ackley-10D-2C. The results are collected from 15 independent trials. The shaded area denotes the 98\% confidence interval. We reproduce the reported performance of SCBO using the corresponding Botorch tutorial. Then, we fix the kernel choices and other hyperparameters to make a fair comparison. For \algname-EPSILON, we set $\epsilon_\cFunc=1.2$.
  }   
  \label{fig:exps:ackley_10d_2c}
\end{figure*}

{As is shown in \figref{fig:exps:ackley_10d_2c},  \algname is initially outperformed by SCBO while converging to the near-optimal area after the sufficient budget as SCBO. \algname-EPSILON archives the best convergence throughout the optimization with the proposed minor tweak. In contrast, CONFIG fails to converge to the global optimum, possibly due to its tolerance of the constraints violation, and by definition, no reward is incurred for a point outside the feasible region.}
