Liqun Qi

A nonsmooth version of Newton's method

Newtons method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized Jacobian instead of the derivative. This extension includes the B-derivative version of Newtons method as a special case. Convergence theorems are proved under the condition of semismoothness. It is shown that the gradient function of the augmented Lagrangian forC2-nonlinear programming is semismooth. Thus, the extended Newtons method can be used in the augmented Lagrangian method for solving nonlinear programs.

Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations

This paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally Lipschitzian functions. For the directional derivative-based and the generalized Jacobian-based Newton methods, both the iterates and the corresponding function values are locally, superlinearly convergent. Globally, a limiting point of the iterate sequence generated by the damped, directional derivative-based Newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one, provided that the system is strongly BD-regular and semismooth at this point. In this case, the convergence is superlinear. A general attraction theorem is presented, which can be applied to two algorithms proposed by Han, Pang and Rangaraj. A hybrid method, which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent, is also presented.

Eigenvalues of a real supersymmetric tensor

In this paper, we define the symmetric hyperdeterminant, eigenvalues and E-eigenvalues of a real supersymmetric tensor. We show that eigenvalues are roots of a one-dimensional polynomial, and when the order of the tensor is even, E-eigenvalues are roots of another one-dimensional polynomial. These two one-dimensional polynomials are associated with the symmetric hyperdeterminant. We call them the characteristic polynomial and the E-characteristic polynomial of that supersymmetric tensor. Real eigenvalues (E-eigenvalues) with real eigenvectors (E-eigenvectors) are called H-eigenvalues (Z-eigenvalues). When the order of the supersymmetric tensor is even, H-eigenvalues (Z-eigenvalues) exist and the supersymmetric tensor is positive definite if and only if all of its H-eigenvalues (Z-eigenvalues) are positive. An mth-order n-dimensional supersymmetric tensor where m is even has exactly n(m-1)^n^-^1 eigenvalues, and the number of its E-eigenvalues is strictly less than n(m-1)^n^-^1 when m>=4. We show that the product of all the eigenvalues is equal to the value of the symmetric hyperdeterminant, while the sum of all the eigenvalues is equal to the sum of the diagonal elements of that supersymmetric tensor, multiplied by (m-1)^n^-^1. The n(m-1)^n^-^1 eigenvalues are distributed in n disks in C. The centers and radii of these n disks are the diagonal elements, and the sums of the absolute values of the corresponding off-diagonal elements, of that supersymmetric tensor. On the other hand, E-eigenvalues are invariant under orthogonal transformations.

A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities

Abstract.In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions, we use a single smoothing approximation function and Robinson’s normal equation to reformulate NCP and BVI as an equivalent nonsmooth equation H(u,x)=0, where H:ℜ2n→ℜ2n, u∈ℜn is a parameter variable and x∈ℜn is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {zk=(uk,xk)} such that the mapping H(·) is continuously differentiable at each zk and may be non-differentiable at the limiting point of {zk}. We prove that three most often used Gabriel-Moré smoothing functions can generate strongly semismooth functions, which play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods. We do not require any function value of F or its derivative value outside the feasible region while at each step we only solve a linear system of equations and if we choose a certain smoothing function only a reduced form needs to be solved. Preliminary numerical results show that the proposed methods for particularly chosen smoothing functions are very promising.

Nonsmooth Equations: Motivation and Algorithms

This paper reports on some recent developments in the area of solving of nonsmooth equations by generalized Newton methods. The emphasis is on three topics: motivation, characterization of superlin...

A smoothing method for mathematical programs with equilibrium constraints

The mathematical program with equilibrium constraints (MPEC) is an optimization problem with variational inequality constraints. MPEC problems include bilevel programming problems as a particular case and have a wide range of applications. MPEC problems with strongly monotone variational inequalities are considered in this paper. They are transformed into an equivalent one-level nonsmooth optimization problem. Then, a sequence of smooth, regular problems that progressively approximate the nonsmooth problem and that can be solved by standard available software for constrained optimization is introduced. It is shown that the solutions (stationary points) of the approximate problems converge to a solution (stationary point) of the original MPEC problem. Numerical results showing viability of the approach are reported.

Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities

The smoothing Newton method for solving a system of nonsmooth equations F(x) = 0, which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the kth step, the nonsmooth function F is approximated by a smooth function f(.,∈ κ ), and the derivative of f(.,∈ κ ) at x k is used as the Newton iterative matrix. The merits of smoothing methods and smoothing Newton methods are global convergence and convenience in handling. In this paper, we show that the smoothing Newton method is also superlinearly convergent if F is semismooth at the solution and f satisfies a Jacobian consistency property. We show that most common smooth functions, such as the Gabriel-More function, have this property. As an application, we show that for box constrained variational inequalities if the involved function is P- uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically).

Does diffusion kurtosis imaging lead to better neural tissue characterization? A rodent brain maturation study

Diffusion kurtosis imaging (DKI) can be used to estimate excess kurtosis, which is a dimensionless measure for the deviation of water diffusion profile from Gaussian distribution. Several recent studies have applied DKI to probe the restricted water diffusion in biological tissues. The directional analysis has also been developed to obtain the directionally specific kurtosis. However, these studies could not directly evaluate the sensitivity of DKI in detecting subtle neural tissue alterations. Brain maturation is known to involve various biological events that can affect water diffusion properties, thus providing a sensitive platform to evaluate the efficacy of DKI. In this study, in vivo DKI experiments were performed in normal Sprague-Dawley rats of 3 different ages: postnatal days 13, 31 and 120 (N=6 for each group). Regional analysis was then performed for 4 white matter (WM) and 3 gray matter (GM) structures. Diffusivity and kurtosis estimates derived from DKI were shown to be highly sensitive to the developmental changes in these chosen structures. Conventional diffusion tensor imaging (DTI) parameters were also computed using monoexponential model, yielding reduced sensitivity and directional specificity in monitoring the brain maturation changes. These results demonstrated that, by measuring directionally specific diffusivity and kurtosis, DKI offers a more comprehensive and sensitive detection of tissue microstructural changes. Such imaging advance can provide a better MR diffusion characterization of neural tissues, both WM and GM, in normal, developmental and pathological states.

Towards better MR characterization of neural tissues using directional diffusion kurtosis analysis

MR diffusion kurtosis imaging (DKI) was proposed recently to study the deviation of water diffusion from Gaussian distribution. Mean kurtosis, the directionally averaged kurtosis, has been shown to be useful in assessing pathophysiological changes, thus yielding another dimension of information to characterize water diffusion in biological tissues. In this study, orthogonal transformation of the 4th order diffusion kurtosis tensor was introduced to compute the diffusion kurtoses along the three eigenvector directions of the 2nd order diffusion tensor. Such axial (K(//)) and radial (K( upper left and right quadrants)) kurtoses measured the kurtoses along the directions parallel and perpendicular, respectively, to the principal diffusion direction. DKI experiments were performed in normal adult (N=7) and formalin-fixed rat brains (N=5). DKI estimates were documented for various white matter (WM) and gray matter (GM) tissues, and compared with the conventional diffusion tensor estimates. The results showed that kurtosis estimates revealed different information for tissue characterization. For example, K(//) and K( upper left and right quadrants) under formalin fixation condition exhibited large and moderate increases in WM while they showed little change in GM despite the overall dramatic decrease of axial and radial diffusivities in both WM and GM. These findings indicate that directional kurtosis analysis can provide additional microstructural information in characterizing neural tissues.

Finding the Largest Eigenvalue of a Nonnegative Tensor

In this paper we propose an iterative method for calculating the largest eigenvalue of an irreducible nonnegative tensor. This method is an extension of a method of Collatz (1942) for calculating the spectral radius of an irreducible nonnegative matrix. Numerical results show that our proposed method is promising. We also apply the method to studying higher-order Markov chains.