An Alternating Trust Region Algorithm for Distributed Linearly Constrained Nonlinear Programs, Application to the AC Optimal Power Flow

A novel trust region method for solving linearly constrained nonlinear programs is presented. The proposed technique is amenable to a distributed implementation, as its salient ingredient is an alternating projected gradient sweep in place of the Cauchy point computation. It is proven that the algorithm yields a sequence that globally converges to a critical point. As a result of some changes to the standard trust region method, namely a proximal regularisation of the trust region subproblem, it is shown that the local convergence rate is linear with an arbitrarily small ratio. Thus, convergence is locally almost superlinear, under standard regularity assumptions. The proposed method is successfully applied to compute local solutions to alternating current optimal power flow problems in transmission and distribution networks. Moreover, the new mechanism for computing a Cauchy point compares favourably against the standard projected search as for its activity detection properties

Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States

We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, led to entirely grid-based tensor-structured 3D Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core Hamiltonian and two-electron integrals (TEI) in $O(n\log n)$ complexity using the rank-structured approximation of basis functions, electron densities and convolution integral operators all represented on 3D $n\times n\times n$ Cartesian grids. The algorithm for calculating TEI tensor in a form of the Cholesky decomposition is based on multiple factorizations using algebraic 1D ``density fitting`` scheme. The basis functions are not restricted to separable Gaussians, since the analytical integration is substituted by high-precision tensor-structured numerical quadratures. The tensor approaches to post-Hartree-Fock calculations for the MP2 energy correction and for the Bethe-Salpeter excited states, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is related to the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for finite lattice-structured systems, where one of the numerical challenges is the summation of electrostatic potentials of a large number of nuclei. The 3D grid-based tensor method for calculation of a potential sum on a $L\times L\times L$ lattice manifests the linear in $L$ computational work, $O(L)$, instead of the usual $O(L^3 \log L)$ scaling by the Ewald-type approaches

A Parametric Non-Convex Decomposition Algorithm for Real-Time and Distributed NMPC

A novel decomposition scheme to solve parametric non-convex programs as they arise in Nonlinear Model Predictive Control (NMPC) is presented. It consists of a fixed number of alternating proximal gradient steps and a dual update per time step. Hence, the proposed approach is attractive in a real-time distributed context. Assuming that the Nonlinear Program (NLP) is semi-algebraic and that its critical points are strongly regular, contraction of the sequence of primal-dual iterates is proven, implying stability of the sub-optimality error, under some mild assumptions. Moreover, it is shown that the performance of the optimality-tracking scheme can be enhanced via a continuation technique. The efficacy of the proposed decomposition method is demonstrated by solving a centralised NMPC problem to control a DC motor and a distributed NMPC program for collaborative tracking of unicycles, both within a real-time framework. Furthermore, an analysis of the sub-optimality error as a function of the sampling period is proposed given a fixed computational power.Comment: 16 pages, 9 figure

"Virus hunting" using radial distance weighted discrimination

Motivated by the challenge of using DNA-seq data to identify viruses in human blood samples, we propose a novel classification algorithm called "Radial Distance Weighted Discrimination" (or Radial DWD). This classifier is designed for binary classification, assuming one class is surrounded by the other class in very diverse radial directions, which is seen to be typical for our virus detection data. This separation of the 2 classes in multiple radial directions naturally motivates the development of Radial DWD. While classical machine learning methods such as the Support Vector Machine and linear Distance Weighted Discrimination can sometimes give reasonable answers for a given data set, their generalizability is severely compromised because of the linear separating boundary. Radial DWD addresses this challenge by using a more appropriate (in this particular case) spherical separating boundary. Simulations show that for appropriate radial contexts, this gives much better generalizability than linear methods, and also much better than conventional kernel based (nonlinear) Support Vector Machines, because the latter methods essentially use much of the information in the data for determining the shape of the separating boundary. The effectiveness of Radial DWD is demonstrated for real virus detection.Comment: Published at http://dx.doi.org/10.1214/15-AOAS869 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Partially distributed outer approximation

This paper presents a novel partially distributed outer approximation algorithm, named PaDOA, for solving a class of structured mixed integer convex programming problems to global optimality. The proposed scheme uses an iterative outer approximation method for coupled mixed integer optimization problems with separable convex objective functions, affine coupling constraints, and compact domain. PaDOA proceeds by alternating between solving large-scale structured mixed-integer linear programming problems and partially decoupled mixed-integer nonlinear programming subproblems that comprise much fewer integer variables. We establish conditions under which PaDOA converges to global minimizers after a finite number of iterations and verify these properties with an application to thermostatically controlled loads and to mixed-integer regression