NUMERIC RATING SCALE ANALYSIS OF TRIGEMINAL NEURALGIA PATIENTS BEFORE AND AFTER MICROVASCULAR DECOMPRESSION

Background: Trigeminal Neuralgia is a common condition of facial pain and its significantly affect patients’ daily life. Microvascular decompression is one of the interventional pain management for trigeminal neuralgia. There is still a little data obtained on evaluation of facial pain after microvascular decompression in Indonesia. Objective: This research aimed to evaluate facial pain after microvascular decompression of trigeminal neuralgia patients in Dr. Soetomo General Hospital, PHC Hospital, and Bangil General Hospital, Indonesia. Methods: The research design was a pretest-posttest with total sampling. Data were obtained from medical records from January 2018 until June 2019. Researches used Numeric Rating Scale (NRS) as pain measurement. The data obtained were analyzed by descriptive statistical test, normality test, and paired t-test. Results: Trigeminal Neuralgia patients that has been treated with microvascular decompression have an average facial reduction from 7.33±2.29 to 1.89±3.41 with p = 0.001. This result showed that the microvascular decompression has significantly reduce facial pain in patients with trigeminal neuralgia. Conclusion: Microvascular decompression significantly reduce the facial pain of trigeminal neuralgia patients.

Characterization of the dual problem of linear matrix inequality for H-infinity output feedback control problem via facial reduction

This paper deals with the minimization of $$H_\infty $$ H ∞ output feedback control. This minimization can be formulated as a linear matrix inequality (LMI) problem via a result of Iwasaki and Skelton 1994. The strict feasibility of the dual problem of such an LMI problem is a valuable property to guarantee the existence of an optimal solution of the LMI problem. If this property fails, then the LMI problem may not have any optimal solutions. Even if one can compute parameters of controllers from a computed solution of the LMI problem, then the computed $$H_\infty $$ H ∞ norm may be very sensitive to a small change of parameters in the controller. In other words, the non-strict feasibility of the dual tells us that the considered design problem may be poorly formulated. We reveal that the strict feasibility of the dual is closely related to invariant zeros of the given generalized plant. The facial reduction is useful in analyzing the relationship. The facial reduction is an iterative algorithm to convert a non-strictly feasible problem into a strictly feasible one. We also show that facial reduction spends only one iteration for so-called regular $$H_\infty $$ H ∞ output feedback control. In particular, we can obtain a strictly feasible problem by using null vectors associated with some invariant zeros. This reduction is more straightforward than the direct application of facial reduction.

Robust principal component analysis using facial reduction

We introduce a novel approach for robust principal component analysis (RPCA) for a partially observed data matrix. The aim is to recover the data matrix as a sum of a low-rank matrix and a sparse matrix so as to eliminate erratic noise (outliers). This problem is known to be NP-hard in general. A classical approach to solving RPCA is to consider convex relaxations. One such heuristic involves the minimization of the (weighted) sum of a nuclear norm part, that promotes a low-rank component, with an $$\ell _1$$ ℓ 1 norm part, to promote a sparse component. This results in a well-structured convex problem that can be efficiently solved by modern first-order methods. However, first-order methods often yield low accuracy solutions. Moreover, the heuristic of using a norm consisting of a weighted sum of norms may lose some of the advantages that each norm had when used separately. In this paper, we propose a novel nonconvex and nonsmooth reformulation of the original NP-hard RPCA model. The new model adds a redundant semidefinite cone constraint and solves small subproblems using a PALM algorithm. Each subproblem results in an exposing vector for a facial reduction technique that is able to reduce the size significantly. This makes the problem amenable to efficient algorithms in order to obtain high-level accuracy. We include numerical results that confirm the efficacy of our approach.

Cone-constrained rational eigenvalue problems

This work deals with the eigenvalue analysis of a rational matrix-valued function subject to complementarity constraints induced by a polyhedral cone $K$. The eigenvalue problem under consideration has the general structure \[ \left(\sum_{k=0}^d \lambda^k A_k + \sum_{k =1}^m \frac{p_k(\lambda)}{q_k(\lambda)} \,B_k\right) x = y , \quad K\ni x \perp y\in K^\ast, \] where $K^\ast$ denotes the dual cone of $K$. The unconstrained version of this problem has been discussed in [Y.F. Su and Z.J. Bai. Solving rational eigenvalue problems via linearization. \emph{SIAM J. Matrix Anal. Appl.}, 32:201--216, 2011.] with special emphasis on the implementation of linearization-based methods. The cone-constrained case can be handled by combining Su and Bai's linearization approach and the so-called facial reduction technique. In essence, this technique consists in solving one unconstrained rational eigenvalue problem for each face of the polyhedral cone $K$.