Gene Regulation Network Inference With Joint Sparse Gaussian Graphical Models

Revealing biological networks is one key objective in systems biology. With microarrays, researchers now routinely measure expression profiles at the genome level under various conditions, and such data may be used to statistically infer gene regulation networks. Gaussian graphical models (GGMs) have proven useful for this purpose by modeling the Markovian dependence among genes. However, a single GGM may not be adequate to describe the potentially differing networks across various conditions, and hence it is more natural to infer multiple GGMs from such data. In this article we propose a class of nonconvex penalty functions aiming at the estimation of multiple GGMs with a flexible joint sparsity constraint. We illustrate the property of our proposed nonconvex penalty functions by simulation study. We then apply the method to a gene expression dataset from the GenCord Project, and show that our method can identify prominent pathways across different conditions. Supplementary materials for this article are available online.     

Piecewise-Convex Maximization Problems

A function F:Rⁿ R is called a piecewise convex function if it can be decomposed into , where f _j:Rⁿ R is convex for all jM={1,2...,m}. We consider subject to xD. It generalizes the well-known convex maximization problem. We briefly review global optimality conditions for convex maximization problems and carry one of them to the piecewise-convex case. Our conditions are all written in primal space so that we are able to proposea preliminary algorithm to check them.     

Convexification of a Noninferior Frontier

In a recent paper by Li (Ref. 1), a scheme was proposed to convexify an efficient frontier for a vector optimization problem by rescaling each component of the vector objective functions by its p-power. For sufficiently large p, it was shown that the transformed efficient frontier is cone-convex; hence, the usual linear scalarization (or supporting hyperplane) method can be used to find the efficient solutions. An outstanding question remains: What is the minimum value of p such that the efficient frontier can be convexified? In this note, we answer the above question by deriving some theoretical lower bounds for p.

     

非凸二次规划的粒子群算法

给出了粒子群算法中惯性权值和学习因子的一种简单改进,并将其应用到非凸二次规划的求解中,通过数值试验与现有的求解非凸二次规划问题的分支定界法进行了比较,得到了较好的结果.     

Convergence of Gradient Algorithms for Nonconvex C1+α Cost Functions*

This paper is concerned with convergence of stochastic gradient algorithms with momentum terms in the nonconvex setting. A class of stochastic momentum methods, including stochastic gradient descent, heavy ball and Nesterov’s accelerated gradient,is analyzed in a general framework under mild assumptions. Based on the convergence result of expected gradients, the authors prove the almost sure convergence by a detailed discussion of the effects of momentum and the number of upcrossings. It is worth noting that there are not additional restrictions imposed on the objective function and stepsize.Another improvement over previous results is that the existing Lipschitz condition of the gradient is relaxed into the condition of H¨older continuity. As a byproduct, the authors apply a localization procedure to extend the results to stochastic stepsizes.     

Convergence analysis of a subsampled Levenberg-Marquardt algorithm

In this work, a subsampled Levenberg-Marquardt algorithm is proposed for solving nonconvex finite-sum optimization problem. At each iteration, based on subsampled function value, gradient and simplified Hessian, a linear system is inexactly solved and the regularized parameter is updated as trust-region algorithms. Provided the sample size increases asymptotically, we prove that the generated sequence converges to a stationary point almost surely.     

Development of the Tangent Modulus from the Euler-Problem to Nonsmooth Materials

History and development of the tangent modulus from the origins to the recent nonsmooth damaging versions are presented. Load history and stability analyses of structures of nonlinear reversible or irreversible materials are based on the concept of tangent modulus. Generally, instantaneously changing tangent modulus is needed and the solution yields iteration process. In the case of inelastic problems, the switch from loading to unloading of the material behaviour results in nonsmooth material functions. Nonsmooth, generally saw-tooth like behaviour happens in composite, laminated or rock type materials, or in the interaction of concrete and the reinforcement, too. Recently, damage and localization are in the focus of structural analyses, extending the tangent modulus to the negative cases, as well. Consequently, an overview of the history and development of the tangent modulus containing the recent modifications seems to be necessary. On the other hand, the more than a century long history of the tangent modulus is a marvellous study of the parallel development of mechanics and mathematics, by following the mutual inspiring effect of them through the activity of such pioneers like P.D. Panagiotopoulos in creating Nonsmooth Mechanics.     

Parameter estimation in metabolic flux balance models for batch fermentation—Formulation & Solution using Differential Variational Inequalities (DVIs)

Recent years have witnessed a surge in research in cellular biology. There has been particular interest in the interaction between cellular metabolism and its environment. In this work we present a framework for fitting fermentation models that include this interaction. Differential equations describe the evolution of extracellular metabolites, while a Linear Program (LP) models cell metabolism, and piecewise smooth functions model the links between cell metabolism and its environment. We show that the fermentation dynamics can be described using Differential Variational Inequalities (DVIs). Discretization of the system and reformulation of the VIs using optimality conditions converts the DVI to a Mathematical Program with Complementarity Constraints (MPCC). We briefly describe an interior point algorithm for solving MPCCs. Encouraging numerical results are presented in estimating model parameters to fit model prediction and data obtained from fermentation, using cultures of Saccharomyces cerevisiae reported in the literature.