First order rejection tests for multiple-objective optimization

The cluster problem in constrained global optimization

“…The above scaling has also been empirically observed by Goldsztejn et al [9], who reason "• • • removes the tangency between the feasible set and the objective level set, and therefore should prevent the cluster effect. "…”

A standard branch-and-bound approach for nonlinear semi-infinite problems

“…We first propose in Section 4.1 an experiment inspired from [29] focusing on the clustering effect around optima of SIPs. Next in Section 4.2.2, we propose a general comparison to the state-of-the-art alternative from [19], which is to our knowledge the best general SIP solving algorithm to date 5 .…”

“…This leads to a more powerful pruning since contracting according to the single constraint f (x) ≤ f * actually encodes the original rejection process using only upper and lower bounds. Other contractors specific to NLPs are: Optimality conditions [40,50,29] that discard subdomains that cannot contain any local optima according to first or second order optimality criteria; And monotonicity tests [40,33] which reduce a domain to one of its bounds if the objective function is proved to monotonically decrease/increase along the corresponding dimension.…”

“…The generalized gradient enclosure (17) readily allows using first order rejection tests [29] to reject decision variable domains that are proved not to contain any local minimizer, and hence no global one. This is illustrated by the following example.…”

“…This is the same problem as in Section 5.2 of [29], where the original nonsmooth constraints g(x) := i=n i=1 x 2 i + |x 1 + 1| − n ≤ 0 is now replaced by its equivalent semi-infinite expression (55).…”