A primal—dual affine-scaling potential-reduction algorithm for linear programming

We propose a potential-reduction algorithm which always uses the primal—dual affine-scaling direction as a search direction. We choose a step size at each iteration of the algorithm such that the potential function does not increase, so that we can take a longer step size than the minimizing point of the potential function. We show that the algorithm is polynomial-time bounded. We also propose a low-complexity algorithm, in which the centering direction is used whenever an iterate is far from the path of centers.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.     

On 1-rotational Sλ(2, 3, ν) designs

In this paper, the necessary and sufficient condition for the existence of a 1-rotational S_λ (2, 3, ν) design is obtained, and it is shown that an S_λ (2, 3, ν) design, if it exists, can be always constructed cyclically.     

Adaptive basis expansion via \ell _1 trend filtering

We propose a new approach for nonlinear regression modeling by employing basis expansion for the case where the underlying regression function has inhomogeneous smoothness. In this case, conventional nonlinear regression models tend to be over- or underfitting, where the function is more or less smoother, respectively. First, the underlying regression function is roughly approximated with a locally linear function using an \(\ell _1\) penalized method, where this procedure is executed by extending an algorithm for the fused lasso signal approximator. We then extend the fused lasso signal approximator and develop an algorithm. Next, the residuals between the locally linear function and the data are used to adaptively prepare the basis functions. Finally, we construct a nonlinear regression model with these basis functions along with the technique of a regularization method. To select the optimal values of the tuning parameters for the regularization method, we provide an explicit form of the generalized information criterion. The validity of our proposed method is then demonstrated through several numerical examples.     

An analysis of the solution set to a homotopy equation between polynomials with real coefficients

We investigate the structure of the solution setS to a homotopy equationH(Z,t)=0 between two polynomialsF andG with real coefficients in one complex variableZ. The mapH is represented asH(x+iy, t)=h ₁(x, y, t)+ih ₂(x, y, t), whereh ₁ andh ₂ are polynomials from ℝ² × [0,1] into ℝ and i is the imaginary unit. Since all the coefficients ofF andG are real, there is a polynomialh ₃ such thath ₂(x, y, t)=yh₃(x, y, t). Then the solution setS is divided into two sets {(x, t)∶h ₁(x, 0, t)=0} and {(x+iy, t)∶h ₁(x, y, t)=0,h ₃(x, y, t)=0}. Using this division, we make the structure ofS clear. Finally we briefly explain the structure of the solution set to a homotopy equation between polynomial systems with real coefficients in several variables.     

Algorithmic Folding Complexity

How do we most quickly fold a paper strip (modeled as a line) to obtain a desired mountain-valley pattern of equidistant creases (viewed as a binary string)? Define the folding complexity of a mountain-valley string as the minimum number of simple folds required to construct it. We first show that the folding complexity of a length-n uniform string (all mountains or all valleys), and hence of a length-n pleat (alternating mountain/valley), is O(lg² n). We also show that a lower bound of the complexity of the problems is ??(lg² n/lg lg n). Next we show that almost all mountain-valley patterns require ??(n/lg n) folds, which means that the uniform and pleat foldings are relatively easy problems. We also give a general algorithm for folding an arbitrary sequence of length n in O(n/lg n) folds, meeting the lower bound up to a constant factor.     

Global and polynomial-time convergence of an infeasible-interior-point algorithm using inexact computation

Received April 10, 1996 / Revised version received April 30, 1998 Published online August 18, 1998     

Enhancing the superconducting transition temperature of CeRh 1-x IrxIn5 due to the strong-coupling effects of antiferromagnetic spin fluctuations: an 115In nuclear quadrupole resonance study

We report on systematic evolutions of antiferromagnetic (AFM) spin fluctuations and unconventional superconductivity (SC) in heavy-fermion (HF) compounds CeRh(1-x)Ir(x)In(5) via an (115)In nuclear-quadrupole-resonance experiment. The nuclear spin-lattice relaxation rate 1/T(1) has revealed the marked development of AFM spin fluctuations as approaching an AFM ordered state. Concomitantly, the superconducting transition temperature T(c) and the energy gap Delta0 increase drastically from T(c)= 0.4K and 2Delta0/k(B)T(c)=5 in CeIrIn(5) up to T(c) =1.2K and 2Delta0/k(B)T(c) =8.3 in CeRh(0.3)Ir(0.7)In5 , respectively. The present work suggests that the AFM spin fluctuations in close proximity to the AFM quantum critical point are indeed responsible for the strong-coupling unconventional SC in HF compounds.