/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

On robust estimation of the location parameter

Introduction to linear and nonlinear programming

‡We describe the use of singular value decomposition in transforming genome-wide expression data from genes 3 arrays space to reduced diagonalized ‘‘eigengenes’’ 3 ‘‘eigenarrays’’ space, where the eigengenes (or eigenarrays) are unique orthonormal superpositions of the genes (or arrays). Normalizing the data by filtering out the eigengenes (and eigenarrays) that are inferred to represent noise or experimental artifacts enables meaningful comparison of the expression of different genes across different arrays in different experiments. Sorting the data according to the eigengenes and eigenarrays gives a global picture of the dynamics of gene expression, in which individual genes and arrays appear to be classified into groups of similar regulation and function, or similar cellular state and biological phenotype, respectively. After normalization and sorting, the significant eigengenes and eigenarrays can be associated with observed genome-wide effects of regulators, or with measured samples, in which these regulators are overactive or underactive, respectively.

Singular Value Decomposition for Genome-Wide Expression Data Processing and Modeling

Convex Analysis: (pms-28)

The nearest correlation matrix problem is to find a correlation matrix which is closest to a given symmetric matrix in the Frobenius norm. The well-studied dual approach is to reformulate this problem as an unconstrained continuously differentiable convex optimization problem. Gradient methods and quasi-Newton methods such as BFGS have been used directly to obtain globally convergent methods. Since the objective function in the dual approach is not twice continuously differentiable, these methods converge at best linearly. In this paper, we investigate a Newton-type method for the nearest correlation matrix problem. Based on recent developments on strongly semismooth matrix valued functions, we prove the quadratic convergence of the proposed Newton method. Numerical experiments confirm the fast convergence and the high efficiency of the method.

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A Quadratically Convergent Newton Method for Computing the Nearest Correlation Matrix

For any function f from $\mathbb R$ to $\mathbb R$, one can define a corresponding function on the space of n × n (block-diagonal) real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix-valued function inherits from f the properties of continuity, (local) Lipschitz continuity, directional differentiability, Frechet differentiability, continuous differentiability, as well as ($\rho$-order) semismoothness. Our analysis uses results from nonsmooth analysis as well as perturbation theory for the spectral decomposition of symmetric matrices. We also apply our results to the semidefinite complementarity problem, addressing some basic issues in the analysis of smoothing/semismooth Newton methods for solving this problem.

Analysis of Nonsmooth Symmetric-Matrix-Valued Functions with Applications to Semidefinite Complementarity Problems

We consider a simply constrained optimization reformulation of the Karush- Kuhn-Tucker conditions arising from variational inequalities. Based on this reformulation, we present a new Newton-type method for the solution of variational inequalities. The main properties of this method are: (a) it is well-defined for an arbitrary variational inequality problem, (b) it is globally convergent at least to a stationary point of the constrained refor- mulation, (c) it is locally superlinearly/quadratically convergent under a certain regularity condition, (d) all iterates remain feasible with respect to the constrained optimization refor- mulation, and (e) it has to solve just one linear system of equations at each iteration. Some preliminary numerical results indicate that this method is quite promising.

A QP-free constrained Newton-type method for variational inequality problems

In this paper, we reformulate the extended vertical linear complementarity problem (EVLCP(m,q)) as a nonsmooth equation H(t,x)=0, where $H: \mbox{\smallBbb R}^{n+1} \to \mbox{\smallBbb R}^{n+1}$, $t \in \mbox{\smallBbb R}$ is a parameter variable, and $x \in \mbox{\smallBbb R}$ is the original variable. H is continuously differentiable except at such points (t,x) with t=0. Furthermore H is strongly semismooth. The reformulation of EVLCP(m, q) as a nonsmooth equation is based on the so-called aggregation (smoothing) function. As a result, a Newton-type method is proposed which generates a sequence {wk=(tk,xk)} with all tk >0. We prove that every accumulation point of this sequence is a solution of EVLCP(M, q) under the assumption of row ${\cal W}_0$-property. If row ${\cal W}$-property holds at the solution point, then the convergence rate is quadratic. Promising numerical results are also presented.

A Smoothing Newton Method for Extended Vertical Linear Complementarity Problems

Higham (2002, IMA J. Numer. Anal., 22, 329–343) considered two types of nearest correlation matrix problems, namely the W-weighted case and the H-weighted case. While the W-weighted case has since been well studied to make several Lagrangian dual-based efficient numerical methods available, the H-weighted case remains numerically challenging. The difficulty of extending those methods from the W-weighted case to the H-weighted case lies in the fact that an analytic formula for the metric projection onto the positive semidefinite cone under the H-weight, unlike the case under the W-weight, is not available. In this paper we introduce an augmented Lagrangian dual-based approach that avoids the explicit computation of the metric projection under the H-weight. This method solves a sequence of unconstrained convex optimization problems, each of which can be efficiently solved by an inexact semismooth Newton method combined with the conjugate gradient method. Numerical experiments demonstrate that the augmented Lagrangian dual approach is not only fast but also robust.

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Houduo Qi

Papers

A Quadratically Convergent Newton Method for Computing the Nearest Correlation Matrix

Analysis of Nonsmooth Symmetric-Matrix-Valued Functions with Applications to Semidefinite Complementarity Problems

A QP-free constrained Newton-type method for variational inequality problems

A Smoothing Newton Method for Extended Vertical Linear Complementarity Problems

An augmented Lagrangian dual approach for the H-weighted nearest correlation matrix problem