Approximate Lagrange multiplier convergence for secant-function-augmented SOPICs

Let $S_i\leq 0$ be the $i$-th scalar-only path inequality constraint (SOPIC), let $\eta_i$ be its Lagrange multiplier, let $P_i\leq 1$ be the equivalent rescaling of the constraint, and let $\rho_i$ be the secant-penalty weight. The approximate multiplier is defined by

$$\tilde{\eta}_i=-\frac{\rho_i\sec\left(\frac{\pi}{2}P_i\right)}{S_i}.$$

Approximate multiplier convergence conjecture. Under the direct-adjoining and secant-function-augmentation formulations, $\tilde{\eta}_i$ approaches $\eta_i$ as $\rho_i\to 0$:

$$\tilde{\eta}_i\longrightarrow\eta_i.$$

The approximation is motivated by equating the corresponding Hamiltonians and is intended to satisfy the complementarity slackness conditions. The paper reports only empirical verification, using DIDO solutions to retrieve $\eta_i$; no rigorous equivalence or convergence proof is provided, so the conjecture remains open.