When $\begin{array}{}{U}_{j}^{(k)}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{L}_{j}^{(\mathit{\text{k}})}\end{array}$ are made to asymptotically approach ±∞, then $\begin{array}{}{F}_{i}^{(\mathit{\text{k}})}\end{array}$ becomes linear function with respect to *ψ*. For example, when *∂f*_{i}/*∂ψ* > 0, substituting (6)~ (8) into (5), then $\begin{array}{}{F}_{i}^{(k)}\end{array}$ can be expressed as follows:
$$\begin{array}{}{\displaystyle {{F}_{i}}^{(k)}(\psi )={f}_{i}({\psi}^{(k)})+\sum _{j=1}^{n}\left(\frac{{{p}_{ij}}^{(k)}}{{{U}_{j}}^{(k)}-{\psi}_{j}}-\frac{{{p}_{ij}}^{(k)}}{{{U}_{j}}^{(k)}-{{\psi}_{j}}^{(k)}}\right).}\end{array}$$(9)

Here, expanding $\begin{array}{}{\displaystyle {p}_{ij}^{(k)},}\end{array}$ and transforming $\begin{array}{}{\displaystyle {F}_{i}^{(k)},}\end{array}$ (9) can be shown as follows:
$$\begin{array}{}{\displaystyle {{F}_{i}}^{(k)}(\psi )\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}}& ={f}_{i}({\psi}^{(k)})\\ & {\displaystyle +\sum _{j=1}^{n}\frac{\mathrm{\partial}{f}_{i}({\psi}^{(k)})}{\mathrm{\partial}\psi}\left(\frac{1-{\psi}_{j}/{{U}_{j}}^{(k)}}{1-{{\psi}_{j}}^{(k)}/{{U}_{j}}^{(k)}}\right)\left({\psi}_{j}-{{\psi}_{j}}^{(k)}\right).}\end{array}$$(10)

Then, taking the limit $\begin{array}{}{\displaystyle {U}_{j}^{(k)}\to \mathrm{\infty},{F}_{i}^{(\mathit{\text{k}})}}\end{array}$ is as follows:
$$\begin{array}{}{\displaystyle {{F}_{i}}^{(k)}(\psi )={f}_{i}({\psi}^{(k)})+\sum _{j=1}^{n}\frac{\mathrm{\partial}{f}_{i}({\psi}^{(k)})}{\mathrm{\partial}\psi}({\psi}_{j}-{{\psi}_{j}}^{(k)}),}\end{array}$$(11)

where this formulation is exactly identical to the expanded function for SLP. when *∂f*_{i}/*∂ψ* ≤ 0, taking the limit $\begin{array}{}{\displaystyle {L}_{j}^{(k)}\to -\mathrm{\infty},}\end{array}$ also (11) is obtained.

In order to suppress oscillation of convergence characteristics at the end of the search by the sequential linear programming, move-limit is imposed on the correction of the design variable $\begin{array}{}{\displaystyle \delta {\psi}_{j}^{(\mathit{\text{k}})}.}\end{array}$ In reference [4], when the number of iterations exceed an arbitrary value, move-limit is damped. However, this method is not efficient because the number of iterations until convergence differs for each problem. Therefore, in this paper, when the oscillation of the objective function is detected, move-limit is damped. Then, move-limit *ζ*^{k} can be defined as follows:
$$\begin{array}{}{\zeta}^{(0)}=\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}0.03\phantom{\rule{thinmathspace}{0ex}}h\\ {\zeta}^{(k)}=\left\{\begin{array}{}{\zeta}^{(k-1)}& \{({{f}_{0}}^{(k)}-{{f}_{0}}^{(k-1)})\phantom{\rule{thinmathspace}{0ex}}({{f}_{0}}^{(k-1)}-{{f}_{0}}^{(k-2)})\ge 0\}\\ {c}^{N}{\zeta}^{(k-1)}& \{({{f}_{0}}^{(k)}-{{f}_{0}}^{(k-1)})\phantom{\rule{thinmathspace}{0ex}}({{f}_{0}}^{(k-1)}-{{f}_{0}}^{(k-2)})<0\},\end{array}\right.\end{array}$$(12)

where *N* is total number of times the oscillation in objective function is detected, and *c* is a parameter (0 < *c* < 1). $\begin{array}{}{\displaystyle \delta {\psi}_{j}^{(k)}}\end{array}$ is defined as follows:
$$\begin{array}{}{\displaystyle -{\zeta}^{(k)}\le \delta {{\psi}_{j}}^{(k)}\le {\zeta}^{(k)}.}\end{array}$$(13)

In this paper, two methods are defined: F-SLP (*c* = 1) where *ζ* is fixed during optimization process, and R-SLP (*c* < 1) where *ζ* is relaxed when the objective function oscillates.

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