A Linearly Convergent Majorized ADMM with Indefinite Proximal Terms for Convex Composite Programming and Its Applications

This paper aims to study a majorized alternating direction method of multipliers with indefinite proximal terms (iPADMM) for convex composite optimization problems. We show that the majorized iPADMM for 2-block convex optimization problems converges globally under weaker conditions than those used in the literature and exhibits a linear convergence rate under a local error bound condition. Based on these, we establish the linear rate convergence results for a symmetric Gaussian-Seidel based majorized iPADMM, which is designed for multi-block composite convex optimization problems. Moreover, we apply the majorized iPADMM to solve different types of regularized logistic regression problems. The numerical results on both synthetic and real datasets demonstrate the efficiency of the majorized iPADMM and also illustrate the effectiveness of the introduced indefinite proximal terms

Regrets of an Online Alternating Direction Method of Multipliers for Online Composite Optimization

In this paper, we investigate regrets of an online semi-proximal alternating direction method of multiplier (Online-spADMM) for solving online linearly constrained convex composite optimization problems. Under mild conditions, we establish ${\rm O}(\sqrt{N})$ objective regret and ${\rm O}(\sqrt{N})$ constraint violation regret at round $N$ when the dual step-length is taken in $(0,(1 +\sqrt{5})/2)$ and penalty parameter $\sigma$ is taken as $\sqrt{N}$. We explain that the optimal value of parameter $\sigma$ is of order ${\rm O}(\sqrt{N})$. Like the semi-proximal alternating direction method of multiplier (spADMM), Online-spADMM has the advantage to resolve the potentially non-solvability issue of the subproblems efficiently. We show the usefulness of the obtained results when applied to online quadratic optimization problem. The inequalities established for Online-spADMM are also used to develop iteration complexity of the average update of spADMM for solving linearly constrained convex composite optimization problems

An effect of large permanent charge: Decreasing flux to zero with increasing transmembrane potential to infinity

In this work, we examine effects of large permanent charges on ionic flow through ion channels based on a quasi-one dimensional Poisson-Nernst-Planck model. It turns out large positive permanent charges inhibit the flux of cation as expected, but strikingly, as the transmembrane electrochemical potential for anion increases in a particular way, the flux of anion decreases. The latter phenomenon was observed experimentally but the cause seemed to be unclear. The mechanisms for these phenomena are examined with the help of the profiles of the ionic concentrations, electric fields and electrochemical potentials. The underlying reasons for the near zero flux of cation and for the decreasing flux of anion are shown to be different over different regions of the permanent charge. Our model is oversimplified. More structural detail and more correlations between ions can and should be included. But the basic finding seems striking and important and deserving of further investigation.Comment: 27 pages, 13 figure

Smoothing SQP methods for solving degenerate nonsmooth constrained optimization problems with applications to bilevel programs

We consider a degenerate nonsmooth and nonconvex optimization problem for which the standard constraint qualification such as the generalized Mangasarian Fromovitz constraint qualification (GMFCQ) may not hold. We use smoothing functions with the gradient consistency property to approximate the nonsmooth functions and introduce a smoothing sequential quadratic programming (SQP) algorithm under the exact penalty framework. We show that any accumulation point of a selected subsequence of the iteration sequence generated by the smoothing SQP algorithm is a Clarke stationary point, provided that the sequence of multipliers and the sequence of exact penalty parameters are bounded. Furthermore, we propose a new condition called the weakly generalized Mangasarian Fromovitz constraint qualification (WGMFCQ) that is weaker than the GMFCQ. We show that the extended version of the WGMFCQ guarantees the boundedness of the sequence of multipliers and the sequence of exact penalty parameters and thus guarantees the global convergence of the smoothing SQP algorithm. We demonstrate that the WGMFCQ can be satisfied by bilevel programs for which the GMFCQ never holds. Preliminary numerical experiments show that the algorithm is efficient for solving degenerate nonsmooth optimization problem such as the simple bilevel program

Linear Rate Convergence of the Alternating Direction Method of Multipliers for Convex Composite Quadratic and Semi-Definite Programming

In this paper, we aim to provide a comprehensive analysis on the linear rate convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex composite optimization problems. Under a certain error bound condition, we establish the global linear rate of convergence for a more general semi-proximal ADMM with the dual steplength being restricted to be in the open interval $(0, (1+\sqrt{5})/2)$. In our analysis, we assume neither the strong convexity nor the strict complementarity except an error bound condition, which holds automatically for convex composite quadratic programming. This semi-proximal ADMM, which includes the classic ADMM, not only has the advantage to resolve the potentially non-solvability issue of the subproblems in the classic ADMM but also possesses the abilities of handling multi-block convex optimization problems efficiently. We shall use convex composite quadratic programming and quadratic semi-definite programming as important applications to demonstrate the significance of the obtained results. Of its own novelty in second-order variational analysis, a complete characterization is provided on the isolated calmness for the nonlinear convex semi-definite optimization problem in terms of its second order sufficient optimality condition and the strict Robinson constraint qualification for the purpose of proving the linear rate convergence of the semi-proximal ADMM when applied to two- and multi-block convex quadratic semi-definite programming

Let the Cloud Watch Over Your IoT File Systems

Smart devices produce security-sensitive data and keep them in on-device storage for persistence. The current storage stack on smart devices, however, offers weak security guarantees: not only because the stack depends on a vulnerable commodity OS, but also because smart device deployment is known weak on security measures. To safeguard such data on smart devices, we present a novel storage stack architecture that i) protects file data in a trusted execution environment (TEE); ii) outsources file system logic and metadata out of TEE; iii) running a metadata-only file system replica in the cloud for continuously verifying the on-device file system behaviors. To realize the architecture, we build Overwatch, aTrustZone-based storage stack. Overwatch addresses unique challenges including discerning metadata at fine grains, hiding network delays, and coping with cloud disconnection. On a suite of three real-world applications, Overwatch shows moderate security overheads

A Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion Equations

In this paper, we first devise an ensemble hybridizable discontinuous Galerkin (HDG) method to efficiently simulate a group of parameterized convection diffusion PDEs. These PDEs have different coefficients, initial conditions, source terms and boundary conditions. The ensemble HDG discrete system shares a common coefficient matrix with multiple right hand side (RHS) vectors; it reduces both computational cost and storage. We have two contributions in this paper. First, we derive an optimal $L^2$ convergence rate for the ensemble solutions on a general polygonal domain, which is the first such result in the literature. Second, we obtain a superconvergent rate for the ensemble solutions after an element-by-element postprocessing under some assumptions on the domain and the coefficients of the PDEs. We present numerical experiments to confirm our theoretical results

A Regularized Semi-Smooth Newton Method With Projection Steps for Composite Convex Programs

The goal of this paper is to study approaches to bridge the gap between first-order and second-order type methods for composite convex programs. Our key observations are: i) Many well-known operator splitting methods, such as forward-backward splitting (FBS) and Douglas-Rachford splitting (DRS), actually define a fixed-point mapping; ii) The optimal solutions of the composite convex program and the solutions of a system of nonlinear equations derived from the fixed-point mapping are equivalent. Solving this kind of system of nonlinear equations enables us to develop second-order type methods. Although these nonlinear equations may be non-differentiable, they are often semi-smooth and their generalized Jacobian matrix is positive semidefinite due to monotonicity. By combining with a regularization approach and a known hyperplane projection technique, we propose an adaptive semi-smooth Newton method and establish its convergence to global optimality. Preliminary numerical results on $\ell_1$-minimization problems demonstrate that our second-order type algorithms are able to achieve superlinear or quadratic convergence.Comment: 25 pages, 4 figure

Energy-efficient population coding constrains network size of a neuronal array system

Here, we consider the open issue of how the energy efficiency of neural information transmission process in a general neuronal array constrains the network size, and how well this network size ensures the neural information being transmitted reliably in a noisy environment. By direct mathematical analysis, we have obtained general solutions proving that there exists an optimal neuronal number in the network with which the average coding energy cost (defined as energy consumption divided by mutual information) per neuron passes through a global minimum for both subthreshold and superthreshold signals. Varying with increases in background noise intensity, the optimal neuronal number decreases for subthreshold and increases for suprathreshold signals. The existence of an optimal neuronal number in an array network reveals a general rule for population coding stating that the neuronal number should be large enough to ensure reliable information transmission robust to the noisy environment but small enough to minimize energy cost.Comment: 21 pages, 4 figure

A conjugate gradient method for electronic structure calculations

In this paper, we study a conjugate gradient method for electronic structure calculations. We propose a Hessian based step size strategy, which together with three orthogonality approaches yields three algorithms for computing the ground state energy of atomic and molecular systems. Under some mild assumptions, we prove that our algorithms converge locally. It is shown by our numerical experiments that the conjugate gradient method is efficient