On the regularization of mixed complementarity problems

A variational inequality problem (VIP) satisfying a constraint qualification can be reduced to a mixed complementarity problem (MOP). Monotonicity of the VIP implies that the MOP is also monotone. Introducing regularizing perturbations, a sequence of strictly monotone mixed complementarity problems is generated. It is shown that, if the original problem is solvable, the sequence of computable inexact solutions of the strictly monotone MCP's is bounded and every accumulation point is a solution. Under an additional condition on the precision used for solving each subproblem, the sequence converges to the minimum norm solution of the MCP.     

Mimicry of proofs with computers: the case of Linear Algebra

In common teaching practice the habit of connecting mathematics classroom activities with reality is still substantially delegated to wor(l)d problems. During recent decades, a growing body of empirical research has documented that the practice of word problem solving in school mathematics does not match this idea of mathematical modelling and mathematization. If we wish to construct ‘real problems arising from real experiences of the child’ following the spirit of these new suggestions, we have to make changes. On the one hand we have to replace the type of activity in which we delegate the process of creating an interplay between reality and mathematics by substituting the word problems with an activity of realistic mathematical modelling, i.e. of both real-world based and quantitatively constrained sense-making; and, on the other hand, to ask for a change in teacher beliefs; furthermore, a directed effort to change the classroom socio-math norms will be needed. This paper discusses some classroom activities that takes these factors into account.     

Augmented Lagrangian Objective Penalty Function

Augmented Lagrangian function is one of the most important tools used in solving some constrained optimization problems. In this article, we study an augmented Lagrangian objective penalty function and a modified augmented Lagrangian objective penalty function for inequality constrained optimization problems. First, we prove the dual properties of the augmented Lagrangian objective penalty function, which are at least as good as the traditional Lagrangian function's. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker condition. This is especially so when the Karush-Kuhn-Tucker condition holds for convex programming of its saddle point existence. Second, we prove the dual properties of the modified augmented Lagrangian objective penalty function. For a global optimal solution, when the exactness of the modified augmented Lagrangian objective penalty function holds, its saddle point exists. The sufficient and necessary stability conditions used to determine whether the modified augmented Lagrangian objective penalty function is exact for a global solution is proved. Based on the modified augmented Lagrangian objective penalty function, an algorithm is developed to find a global solution to an inequality constrained optimization problem, and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the modified augmented Lagrangian objective penalty function is proved for a local solution. An algorithm is presented in finding a local solution, with its convergence proved under some conditions.     

Spectral condition numbers for defective elements of linear operators in hilbert spaces

In this paper we generalise matrix eigenelements results on condition numbers to linear bounded operators in complex Hilbert infinite dimensional spaces. Suitable nornls are introduced in order to interpret these results geometrically.     

Approximate minima via the ekeland variational principle 1

In this paper several consequences derived by an equivalent version of the Ekeland variational principle are presented. In particular, a notion of an approximate minimum (for a function defined in a complete metric space), which improves an earlier one given by Attouch and Riahi, is discussed. This is obtained by considering truncations of the metric function     

Inequality constrained ridge regression estimator

We carry out the idea of inequality constrained least squares (ICLS) estimation of Liew (1976) to the inequality constrained ridge regression (ICRR) estimation. We propose ICRR estimator by reducing the primal–dual relation to the fundamental problem of Dantzig and Cottle, 1967, Cottle and Dantzig, 1974 with Lemke (1962) algorithm. Furthermore, we conduct a Monte Carlo experiment.     

Operations research and optimization (ORO)

We have recently proposed a structured algorithm for solving constrained nonlinear least-squares problems and established its local two-step Q-superlinear convergence rate. The approach is based on an earlier adaptive structured scheme due to Mahdavi-Amiri and Bartels of the exact penalty method. The structured adaptation makes use of the ideas of Nocedal and Overton for handling quasi-Newton updates of projected Hessians and adapts a structuring scheme due to Engels and Martinez. For robustness, we have employed a specific nonsmooth line search strategy, taking account of the least-squares objective. Numerical results also confirm the practical relevance of our special considerations for the inherent structure of the least squares. Here, we establish global convergence of the proposed algorithm using a weaker condition than the one used by the exact penalty method of Coleman and Conn for general nonlinear programs.     

Criticism of generally accepted fundamentals and methodologies of traffic and transportation theory: A brief review

It is explained why the set of the fundamental empirical features of traffic breakdown (a transition from free flow to congested traffic) should be the empirical basis for any traffic and transportation theory that can be reliably used for control and optimization in traffic networks. It is shown that the generally accepted fundamentals and methodologies of the traffic and transportation theory are not consistent with the set of the fundamental empirical features of traffic breakdown at a highway bottleneck. To these fundamentals and methodologies of the traffic and transportation theory belong (i) Lighthill–Whitham–Richards (LWR) theory, (ii) the General Motors (GM) model class (for example, Herman, Gazis et al. GM model, Gipps’s model, Payne’s model, Newell’s optimal velocity (OV) model, Wiedemann’s model, Bando et al. OV model, Treiber’s IDM, Krauß’s model), (iii) the understanding of highway capacity as a particular (fixed or stochastic) value, and (iv) principles for traffic and transportation network optimization and control (for example, Wardrop’s user equilibrium (UE) and system optimum (SO) principles). Alternatively to these generally accepted fundamentals and methodologies of the traffic and transportation theory, we discuss the three-phase traffic theory as the basis for traffic flow modeling as well as briefly consider the network breakdown minimization (BM) principle for the optimization of traffic and transportation networks with road bottlenecks.     

基于能量最小化的肾脏计算断层扫描图像分割方法

随着成像技术的不断发展, 医学图像处理在计算机辅助诊断和病变管理中的重要作用日渐突出, 而计算断层扫描序列图像的肾脏组织分割是其中的关键步骤. 本文结合肾脏序列图像的连续性特征, 提出了一种基于活动轮廓和图割方法的能量最小化分割模型来自动分割肾脏组织. 根据相邻切片图像的形状差异与层间距之间的关系, 计算出序列图像中适合图割优化能量函数的最优范围. 能量函数采用测地活动轮廓模型和Chan-Vese模型的综合形式, 兼顾了目标的边界和区域信息. 随后, 利用图割方法优化离散化的能量函数, 驱使活动轮廓逐渐向目标边界靠近, 直至收敛为止. 对30组腹腔序列图像进行了算法测试, 实验表明基于能量最小化的分割方法能够有效地提取出序列图像中的肾脏组织, 其分割结果的平均Dice系数达到了93.7%. 关键词： 计算层析 肾脏分割 能量最小化 连续性