In 1987, Northby presented an efficient lattice based search and optimization procedure to compute ground states of n-atom Lennard-Jones clusters and reported putative global minima for 13 ~< n ~< 150. In this paper, we introduce simple data structures which reduce the time complexity of the Northby algorithm for lattice search from O(n~) per move to O(n~) per move for an n-atom cluster involvi...

A minimization problem for the Ginzburg-Landau functional is considered on an annulus with thickness R in the class of functions F where only the degree (the winding number), d, on the boundary is prescribed. The existence of a critical thickness Rc which depends on the GinzburgLandau parameter δ and d is established. It is shown that if R > Rc then any function in separated form in polar coord...

In this correspondence, a constrained least-squares multichannel image restoration approach is proposed, in which no prior knowledge of the noise variance at each channel or the degree of smoothness of the original image is required. The regularization functional for each channel is determined by incorporating both within-channel and cross-channel information. It is shown that the proposed smoo...

We consider a nonlocal isoperimetric problem defined in the whole space RN , whose nonlocal part is given by a Riesz potential with exponent α ∈ (0, N − 1) . We show that critical configurations with positive second variation are local minimizers and satisfy a quantitative inequality with respect to the L1 -norm. This criterion provides the existence of a (explicitly determined) critical thresh...

Figure 2. The trajectories of CM, NAG, and SGD are shown. Although the value of the momentum is identical for both experiments, CM exhibits oscillations along the high-curvature directions, while NAG exhibits no such oscillations. The global minimizer of the objective is at (0,0). The red curve shows gradient descent with the same learning rate as NAG and CM, the blue curve shows NAG, and the g...

Given a p > 2, we prove existence of global minimizers for a p-GinzburgLandau-type energy over maps on R with degree d = 1 at infinity. For the analogous problem on the half-plane we prove existence of a global minimizer when p is close to 2. The key ingredient of our proof is the degree reduction argument that allows us to construct a map of degree d = 1 from an arbitrary map of degree d > 1 w...

We study global Mumford-Shah minimizers in R , introduced by Bonnet as blow-up limits of Mumford-Shah minimizers. We prove a new monotonicity formula for the energy of u when the singular set K is contained in a smooth enough cone. We then use this monotonicity to prove that for any reduced global minimizer (u,K) in R, if K is contained in a half-plane and touching its edge, then it is the half...

In the paper, we consider a continuation approach for the binary quadratic program (BQP) based on a class of NCP-functions. More specifically, we recast the BQP as an equivalent minimization and then seeks its global minimizer via a global continuation method. Such approach had been considered in [11] which is based on the Fischer-Burmeister function. We investigate this continuation approach a...

We propose a smoothing trust region filter algorithm for nonsmooth nonconvex least squares problems. We present convergence theorems of the proposed algorithm to a Clarke stationary point or a global minimizer of the objective function under certain conditions. Preliminary numerical experiments show the efficiency of the proposed algorithm for finding zeros of a system of polynomial equations w...