We propose a primal-dual continuation approach for the capacitated multi-facility Weber problem (CMFWP) based on its nonlinear second-order cone program (SOCP) reformulation. The main idea of the approach is to reformulate the CMFWP as a nonlinear SOCP with a nonconvex objective function, and then introduce a logarithmic barrier term and a quadratic proximal term into the objective to construct a sequence of convexified subproblems. By this, this class of nondifferentiable and nonconvex optimization problems is converted into the solution of a sequence of nonlinear convex SOCPs. In this paper, we employ the semismooth Newton method proposed in Kanzow etal. (SIAM Journal of Optimization 20:297320, 2009) to solve the KKT system of the resulting convex SOCPs. Preliminary numerical results are reported for eighteen test instances, which indicate that the continuation approach is

In medical imaging, parameterized 3-D surface models are useful for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal processing. Here we introduce a parameterization method based on Riemann surface structure, which uses a special curvilinear net structure (conformal net) to partition the surface into a set of patches that can each be conformally mapped to a parallelogram. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable (their solutions tend to be smooth functions and the boundary conditions of the Dirichlet problem can be enforced). Conformal parameterization also helps transform partial differential equations (PDEs) that may be defined on 3-D brain surface manifolds to modified PDEs on a two-dimensional parameter domain. Since the Jacobian matrix of a conformal parameterization is

Recently there have two different effective methods proposed by Kanzow et al. in (Kanzow, 2001 [8]) and (Kanzow and Petra, 2004 [9]), respectively, which commonly use the FischerBurmeister (FB) function to recast the mixed complementarity problem (MCP) as a constrained minimization problem and a nonlinear system of equations, respectively. They all remark that their algorithms may be improved if the FB function is replaced by other NCP functions. Accordingly, in this paper, we employ the generalized FischerBurmeister (GFB) where the 2-norm in the FB function is relaxed to a general p-norm (p> 1) for the two methods and investigate how much the improvement is by changing the parameter p as well as which method is influenced more when we do so, by the performance profiles of iterations and function evaluations for the two methods with different p on MCPLIB collection.

G-equations are well-known front propagation models in combustion and are HamiltonJacobi type equations with convex but non-coercive Hamiltonians. Viscous G-equations arise from numerical discretization or modeling dissipative mechanisms. Although viscosity helps to overcome non-coercivity, we prove homogenization of an inviscid G-equation based on approximate correctors and attainability of controlled flow trajectories. We verify the attainability for two-dimensional mean zero incompressible flows, and demonstrate asymptotically and numerically that viscosity reduces the homogenized Hamiltonian in cellular flows. In the case of one-dimensional compressible flows, we found an explicit formula of homogenized Hamiltonians, as well as necessary and sufficient conditions for wave trapping (effective Hamiltonian vanishes identically). Viscosity restores coercivity and wave propagation.

Recent studies have shown that the entropy production rate for the master equation consists of two nonnegative terms: the adiabatic and non-adiabatic parts, where the non-adiabatic part is also known as the dissipation rate of a Boltzmann-Shannon relative entropy. In this paper, we provide some nonzero lower bounds for the relative entropy, the entropy production rate, and its adiabatic and non-adiabatic parts. These nonzero lower bounds not only reveal some novel dissipative properties for general nonequilibrium processes which are much stronger than the second law of thermodynamics, but also impose some new constraints on thermodynamic constitutive relations. Moreover, we also provide a mathematical application of these nonzero lower bounds by studying the long-time behavior of the master equation. Extensions to the Tsallis statistics are also discussed, including the nonzero lower bounds for the Tsallis-type relative entropy and its dissipation rate.