A smoothing method for mathematical programs with equilibrium constraints

Received May 3, 1996 / Revised version received November 19, 1997 Published online January 20, 1999     

Drinfeld algebra deformations, homotopy comodules and the associahedra

The aim of this work is to construct a cohomology theory controlling the deformations of a general Drinfel'd algebra and thus finish the program which began in [13], [14]. The task is accomplished in three steps. The first step, which was taken in the aforementioned articles, is the construction of a modified cobar complex adapted to a non-coassociative comultiplication. The following two steps each involves a new, highly non-trivial, construction. The first construction, essentially combinatorial, defines a differential graded Lie algebra structure on the simplicial chain complex of the associahedra. The second construction, of a more algebraic nature, is the definition of a map of differential graded Lie algebras from the complex defined above to the algebra of derivations on the bar resolution. Using the existence of this map and the acyclicity of the associahedra we can define a so-called homotopy comodule structure (Definition 3.3 below) on the bar resolution of a general Drinfel'd algebra. This in turn allows us to define the desired cohomology theory in terms of a complex which consists, roughly speaking, of the bimodule and bicomodule maps from the bar resolution to the modified cobar resolution. The complex is bigraded but not a bicomplex as in the Gerstenhaber-Schack theory for bialgebra deformations. The new components of the coboundary operator are defined via the constructions mentioned above. The results of the paper were announced in [12].

     

A (2+ϵ)-approximation for precedence constrained single machine scheduling with release dates and total weighted completion time objective

We give a $(2+?)$-approximation algorithm for minimizing total weighted completion time on a single machine under release time and precedence constraints. This settles a recent conjecture on the approximability of this scheduling problem (Skutella, 2016).     

On the necessary and sufficient condition for the extended Wedderburn–Guttman theorem

Let A be a u by v matrix, and let M and N be u by p and v by q matrices, where p may not be equal to q or rank(M^′AN)<min(p,q). Recently, Galantai [A. Galantai, A note on the generalized rank reduction, Acta Math. Hungarica 116 (2007) 239–246] presented what he claimed to be the necessary and sufficient condition for rank(A-AN(M^′AN)^-M^′A)=rank(A)-rank(AN(M^′AN)^-M^′A) to hold. This rank subtractivity formula along with the condition under which it holds is called the extended Wedderburn–Guttman theorem. In this paper, we show that some of Galantai’s assertions are incorrect.     

Melt Behaviour of Poly(p-xylylene) (PPX) Coated Gel-spun UHMW-PE Fibers As Revealed by Temperature Modulated DSC

The method of temperature modulated DSC has been applied to obtain additional information about the effect of constraints on the melting behaviour of gel-spun ultra high molecular mass polyethylene (UHMW-PE) fibers coated with a high temperature stable poly(p-xylylene) (PPX) polymer. The underlying signal, corresponding to the normal DSC signal, reveals two endothermic peaks for the coated PE fibers. A shift in the underlying and magnitude signal from 142 to 145°C at 0.1 K min^–1 , a relative small magnitude signal, together with a vanishing step-like change in the phase signal with increasing PPX coating layer thickness characterize the constraints in terms of a hindrance of the melting of the unconstrained orthorhombic crystal fraction. The time constant of the melting process can be estimated as larger than the reciprocal angular frequency 1/ω=5 s of the modulation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Modified sub-population teaching-learning-based optimization for design of truss structures with natural frequency constraints

In this paper, a modified sub-population teaching-learning-based optimization (MS-TLBO) algorithm is proposed to improve the exploration and exploitation capacities by including the concept of number of teachers, adaptive teaching factor, learning through tutorial, and self-motivated learning in the basic TLBO algorithm. The multiple frequency responses to the structural optimization problems are challenging due to its search space, which is implicit, nonconvex, nonlinear, and often leading to divergence. The viability and efficiency of the proposed method are tested by five structural benchmark problems of shape and size optimization with multiple natural frequency constraints on the planar and space trusses. The results reveal that MS-TLBO is more effective as compared to the original TLBO and other state-of-the-art algorithms.     

A comparison between a collocation and weak implementation of the rigid‐body motion constraint on a particle surface

In this work, we implemented and compared two different methods to impose the rigid‐body motion constraint on a solid particle moving inside a fluid. We consider a fictitious domain method to easily manage the particle motion. As the solid as well as the fluid inertia are neglected, the particle can be discretized through its boundary only. The rigid‐body motion is imposed via Lagrange multipliers on the boundary. In the first method, such constraints are imposed in discrete points on the boundary (collocation), whereas in the second the constraint is imposed in a weak way on elements dividing the particle surface. Two test problems, that is, a spherical and an ellipsoidal particle in a sheared Newtonian fluid, are chosen to compare the methods. In both cases, the analysis is carried out in 2D as well as in 3D. The results show that for the collocation method an optimal number of collocation points exist leading to the smallest error. However, small variations in the optimal value can generate large deviations. In the weak implementation, the error is only mildly affected by the number of elements used to discretize the particle boundary and by the Lagrange multiplier's interpolation space. A further analysis is carried out to study the effect of an approximated integration of weak constraints. A comparison between the two methods showed that the same accuracy can be achieved by using less constraints if the weak discretization is used. Finally, the rigid‐body motion imposed via weak constraints leads to better conditioned linear systems. Copyright © 2009 John Wiley & Sons, Ltd.     

正交约束优化问题的一阶算法

带有正交约束的矩阵优化问题在材料计算、统计及数据分析等领域中有着广泛的应用.由于正交约束的可行域是Stiefel流形,一直以来流形上的优化方法是求解这一问题的主要方法.近年来,随着实际应用问题所要求的变量规模的扩大,传统的流形优化方法在计算上的劣势显现出来,而一些迭代简单、收敛快的新算法逐渐被提出.通过收缩方法、非收缩可行方法、不可行方法三个类别分别来介绍求解带有正交约束的矩阵优化问题的最新算法.通过分析这些方法的主要特性,以及应用问题的要求,对这类问题算法设计的研究进行了展望.     

一类具有资源约束和优先加工顺序约束极小化加权总完工时间调度优化问题研究

本文针对工件间具有链状优先约束和relocation资源约束的极小化加权总完工时间调度优化问题展开研究.针对这一NP难问题,利用relocation约束的性质和贪婪算法的思想,设计了一个多项式近似算法,并证明了当链不可中断,每个链具有相同工件数和工件间具有相同加工时间时,2为该算法的紧界.     

非线性互补约束规划问题的一个新的QP-free算法

本文研究了非线性互补约束均衡问题.利用互补函数以及光滑近似法,把非线性互补约束均衡问题转化为一个光滑非线性规划问题,得到了超线性收敛速度,数值实验结果表明本文提出的算法是可行的.