We study the radial-hedgehog solution on a three-dimensional (3D) spherical shell with radial boundary conditions, within the Landau-de Gennes theory for nematic liquid crystals. We prove that the radial-hedgehog solution is the unique minimizer of the Landaude Gennes energy in two separate regimes: (i) for thin shells when the temperature is below the critical nematic supercooling temperature ...

We propose a new algorithm, a reeective Newton method, for the minimization of a quadratic function of many variables subject to upper and lower bounds on some of the variables. The method applies to a general (indeenite) quadratic function, for which a local minimizer subject to bounds is required, and is particularily suitable for the large-scale problem. Our new method exhibits strong conver...

The basis pursuit denoising refers to the solution of an 12 minimization formulation which is well known as an effective method for signal denoising. In this paper we investigate an p2 formulation with p ∈ (0, 1) for denoising. Based on an analysis of the discontinuity of the global minimizer of the p2 problem with respect to regularization parameter, we propose two smoothed p2 solvers for orth...

We define a new method for global optimization, the Homotopy Optimization Method (HOM). This method differs from previous homotopy and continuation methods in that its aim is to find a minimizer for each of a set of values of the homotopy parameter, rather than to follow a path of minimizers. We define a second method, called HOPE, by allowing HOM to follow an ensemble of points obtained by per...

We develop an approach to training generative models based on unrolling a variational autoencoder into a Markov chain, and shaping the chain’s trajectories using a technique inspired by recent work in Approximate Bayesian computation. We show that the global minimizer of the resulting objective is achieved when the generative model reproduces the target distribution. To allow finer control over...

In this article, we consider the semilinear elliptic problem −ε∆u = h(|x|)(u− a(|x|))(1− u) in B1(0) with the Neumann boundary condition. The function a is a C1 function satisfying |a(x)| < 1 for x ∈ [0, 1] and a′(0) = 0. In particular we consider the case a(r) = 0 on some interval I ⊂ [0, 1]. The function h is a positive C1 function satisfying h′(0) = 0. We investigate an asymptotic profile of...

We study the long time behavior of the Hele-Shaw-Cahn-Hilliard system (HSCH) which models two phase incompressible Darcian flow in porous media with matched density but arbitrary viscosity contrast. We demonstrate that the ω-limit set of each trajectory is a single stationary solution of the system via Lojasiewicz-Simon type technique. Moreover, a rate of convergence has been established. Event...

For arbitrary real matrices F and G, the positive semi-deenite Procrustes problem is minimization of the Frr obenius norm of F ? PG with respect to positive semi-deenite symmetric P. Existing solution algorithms are based on a convex programming approach. Here an unconstrained non-convex approach is taken, namely writing P = E 0 E and optimizing with respect to E. The main result is that all lo...

The multilinear least-squares (MLLS) problem is an extension of the linear leastsquares problem. The difference is that a multilinear operator is used in place of a matrix-vector product. The MLLS is typically a large-scale problem characterized by a large number of local minimizers. It originates, for instance, from the design of filter networks. We present a global search strategy that allows...

The iteratively reweighted `1 minimization algorithm (IRL1) has been widely used for variable selection, signal reconstruction and image processing. However the convergence of the IRL1 has not been proved. In this paper, we prove that any sequence generated by the IRL1 is bounded and any accumulation point is a stationary point of the `2-`p minimization problem with 0 < p < 1. Moreover, the sta...