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2

You can find an answer to this in "Dynamic Planar Range Maxima Queries", by Brodal and Tsakalidis. They support insert and remove in $O(\log n)$ time. That paper points to several previous papers that addressed the question as well. In particular, I expect the simplest solution can be found in sections 8 and 9 of "Maintenance of Configurations in the Plane",...

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First, while the Gale-Shapely algorithm guarantees that at least one stable matching exists, it need not be unique$^*$. However, note that when the male-optimal and female-optimal matchings are the same, the stable matching is indeed unique, as shown in this answer. As the male-optimal and female-optimal matchings coincide, both can be considered the '...

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A multi-objective optimization problem can be formulated as \begin{align} \min\limits_{\phi \in \Phi} & \quad [f_1(\phi), f_2(\phi), \ldots, f_m(\phi)]^T \label{eq:mo-objectives} \\ s.t. & \quad g_j(\phi) \leqslant 0,~j = 1, \ldots, J \label{eq:mo-constraints} \end{align} Without loss of generality, assume that all objective functions are ...

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