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Another important notion of the well-posedness for a minimizing problem is the well-posedness by perturbations or the extended well-posedness due to Zolezzi [13].

Then, the OLS training process is initialized by choosing a cluster as the initial batch A 0. After selecting the initial batch, the following minimizing problem is solved: text{Minimize:} {||A_{0}{x_{0}} - b_{0}||}^{2} + lambda{||x_{0}||}^{2} (1) Fig. 2 Noisy object removal.

Besides, the relations between (G TWPness for GVQEPs and that for constrained minimizing problems are exhibited.

The metric characterizations and/or sufficient criteria of the proposed well-posedness are presented, and the relations between (generalized) Tykhonov well-posedness for generalized vector quasi-equilibrium problems and that for constrained minimizing problems are discussed.

The metric characterizations and sufficient criteria of (generalized) Tykhonov well-posedness, (G TWPness for brevity, for GVQEPs are presented by applying Kuratowski noncompactness measure, and the relations between (G TWPness for GVQEPs and that for constrained minimizing problems are exhibited in Section 2.

The mathematical background of the minimizing peak problem is presented in [12] which proves that the problem of minimizing peak is NP-complete.

Hierarchical multimodal hub location problem is a cost-minimizing hub covering problem where two types of hubs and hub links, accounting for ground and air transportation, are to be established, while ensuring time-definite deliveries.

This problem is directly related to minimizing manufacturing costs and lead time.

The solution to the above minimizing problem could be given by the following theorems.

The redundancy allocation problem is formulated with the objective of minimizing design cost, when the system exhibits a multi-state reliability behavior, given system-level performance constraints.

Because μ>0, the constraint set of this dual problem is not closed so that the existence of a minimizing solution cannot be guaranteed by Weierstrass' theorem.