## 3 Custom Interior-Point Method for Solving FLPRC
### Starting-Point Using Flexible Bay Structure  
In this section, we present a technique to create the starting-point for further speeding-up the computation. While our proposed primal-dual interior point method has enough computation effectiveness in itself, there is a potential advantage in utilizing a 'good' starting point from a practical perspective. In the context of using interior point method, a starting point should be strictly feasible. Therefore, we define a 'good' starting point as points that are _close to optimality_, yet is _sufficiently far from the boundary of the feasible solution_. We describe a technique to create such starting point in this section. Our method is based on a relaxed _Flexible-Bay Structure_ (FBS), in which the departments are located in parallel bays with the varying width associated the total area of the departments in that bay, using two part representation: permutation of the departments and breakpoints for the bays. Figure 1 presents an example illustrating the bay assignment and the obtained layout from the horizontal and vertical relative-positioning graphs. In our proposal, the permutation and breakpoints are specified using the relative-positioning \(\mathcal{H}_{ij},\mathcal{V}_{ij}\), and the shapes of department \((w_{i},h_{i})\) and the positioning of departments \((x_{i},y_{i})\) are determined through the relaxed-FBS decoding scheme. However, those obtained from the FBS procedure is near optimality and feasible, but not strictly feasible in general, since each pair of adjacent departments has no separation between them. Therefore, an additional small clearance \(\rho\) (say, \(10^{-5}\)) is used to make them strictly feasible. The outline of starting-point generation using the relaxed-FBS is described as Algorithm 1. ```
given\(\mathcal{H}_{ij}\), \(\mathcal{V}_{ij}\), \(\rho\) assign department \(\forall i\) to the bay \(k\) with the horizontal precedence \(\sum_{i=1}^{N}\mathcal{H}_{ij}=k\). sort department \(\forall i\) in the bay \(\forall k\) in the order of vertical precedence \(\sum_{i=1}^{N}\mathcal{V}_{ij}=k\) get bay area \(B_{k}\) as : \(B_{k}=\sum_{i\in k}S_{i}\) get bay width \(W^{B}_{k}\) as : \(W^{B}_{k}=\max_{\forall i\in k}(H/B_{k},w^{min}_{i})\) get department widths and heights as \(w_{i}=\min(W^{B}_{k},w^{max}_{i})\), \(h_{i}=S_{i}/w_{i}\) get department coordinates as \(w_{i}=\min(W^{B}_{k},w^{max}_{i})\), \(h_{i}=S_{i}/w_{i}\) \(x_{i}=w_{i}+\rho\), \(\forall i\) with \(\sum_{i=1}^{N}\mathcal{H}_{ij}=0\) (root node) \(y_{i}=h_{i}+\rho\), \(\forall i\) with \(\sum_{i=1}^{N}\mathcal{V}_{ij}=0\) (root node) \(x_{j}=x_{i}+(w_{i}+w_{j})+\rho\), \(\forall(i,j)\) with \(\mathcal{H}_{ij}=1\) \(y_{j}=x_{i}+(h_{i}+h_{j})+\rho\), \(\forall(i,j)\) with \(\mathcal{V}_{ij}=1\), where \(\rho\in\mathbb{R}\) is an additional clearance between departments. Figure 1: Example illustrating the bay assignment (shown at center) and the obtained layout (shown at right) from the horizontal and vertical relative-positioning graphs (shown at left). ### Overall Algorithm

We summarize the overall algorithm is as Algorithm 2. ```
given \(t_{0}\), \(\mathbf{A}\), \(\mathbf{b}\), \(\boldsymbol{c}\), \(\boldsymbol{\epsilon}\) find initial points \((\mathbf{v}_{0},\lambda_{0})\) by warm-start algorithm. repeat 1-4: 1. Find primal and dual search directions \([\Delta\mathbf{v}_{k}^{T},\Delta\lambda_{k}^{T}]^{T}\) by [PERSON] equation. 2. Find step size \(\gamma_{k}\in\mathbb{R}\) by back-tracking 3. Update primal and dual variables \([\mathbf{v}_{k+1}^{T},\lambda_{k+1}^{T}]:=[\mathbf{v}_{k}^{T},\lambda_{k}^{T}]+ \gamma_{k}[\Delta\mathbf{v}_{k}^{T},\Delta\lambda_{k}^{T}]^{T}\) 4. Update \(t_{k}\to t_{k+1}\) until\(||R_{t}(\mathbf{v}_{k},\lambda_{k})||\leq\epsilon\)
```

**Algorithm 2**Overall Algorithm

Theoretically, the convergence proof for the primal-dual interior-point algorithm is given in several books and papers such as [PERSON] and [PERSON] (2004) [44]. In practice, it is numerically difficult to converge when \(\epsilon\) is very small.