By virtue of convexification techniques, we study best approximations to a closed set C in a Hilbert space as well as perturbation conditions relative to C and a nonlinear inequality system. Some results on equivalence of the best approximation and the basic constraint qualification are established.

Power flow optimization is generally nonlinear and non-convex, and a second-order cone relaxation has been proposed recently for convexification. We prove several sufficient conditions under which the relaxation is exact. One of these conditions seems particularly realistic and suggests guidelines on integrating distributed generations.

We consider the problem of finding the shortest curve in the plane that has unit width. This problem was first posed as the “river shore” puzzle by Ogilvy (1972) and is related to the area of on-line searching. Adhikari and Pitman (1989) proved that the optimal solution has length 2.2782 . . . We present a simpler proof, which exploits the fact that the width of a polygon does not decrease unde...

For the problem of estimating a sparse sequence of coefficients of a parametric or nonparametric generalized linear model, posterior mode estimation with a Subbotin(λ, ν) prior achieves thresholding and therefore model selection when ν ∈ [0, 1] for a class of likelihood functions. The proposed estimator also offers a continuum between the (forward/backward) best subset estimator (ν = 0), its ap...

A general class of convexification transformations is proposed to convexify the noninferior frontier of a multiobjective program. We prove that under certain assumptions the noninferior frontier could be convexified completely or partly after transformation and then weighting method can be applied to identify the noninferior solutions. Numerical experiments are given to vindicate our results.