线性互补问题与多目标优化

本文研究了求解线性互补问题的一类新方法:把线性互补问题转化为多目标优化问题,利用多目标优化有效解的定义,给出了零有效解的概念;进而获得多目标优化问题的零有效解就是线性互补问题的最优解.最后给出了有解、无解线性互补问题,并分别把这些问题转化为多目标优化,采用极大极小方法求解转化后的多目标优化问题.数值实验结果表明了该方法的正确性和有效性,完善了文献[19]的数值结果.     

Ellipsoids that contain all the solutions of a positive semi-definite linear complementarity problem

This paper deals with the LCP (linear complementarity problem) with a positive semi-definite matrix. Assuming that a strictly positive feasible solution of the LCP is available, we propose ellipsoids each of which contains all the solutions of the LCP. We use such an ellipsoid for computing a lower bound and an upper bound for each coordinate of the solutions of the LCP. We can apply the lower bound to test whether a given variable is positive over the solution set of the LCP. That is, if the lower bound is positive, we know that the variable is positive over the solution set of the LCP; hence, by the complementarity condition, its complement is zero. In this case we can eliminate the variable and its complement from the LCP. We also show how we efficiently combine the ellipsoid method for computing bounds for the solution set with the path-following algorithm proposed by the authors for the LCP. If the LCP has a unique non-degenerate solution, the lower bound and the upper bound for the solution, computed at each iteration of the path-following algorithm, both converge to the solution of the LCP.Supported by Grant-in-Aids for General Scientific Research (63490010) of The Ministry of Education, Science and Culture.Supported by Grant-in-Aids for Young Scientists (63730014) and for General Scientific Research (63490010) of The Ministry of Education, Science and Culture.     

Features of thermomechanical properties of radiation-modified blends of high-density polyethylene with liquid-crystalline copolyester

The present paper surveys the investigation results of gamma-irradiated blends of high-density polyethylene (PE) with thermotropic liquid-crystalline polymer (LCP). The LCP used was a liquid crystalline copolyester of 40% poly(ethyleneterephthalate) with 60% p-(hydroxybenzoic acid). The LCP content in the blends was 0,5, and 10 wt.%. The constituents were blended with the use of a single-screw extruder. The samples were prepared by compression molding and irradiated by a Co⁶⁰ -radiation source in an inert atmosphere (argon) to relatively low absorbed doses (up to 200 kGy; 1 Mrad=10 kGy). Morphological, thermal, and mechanical properties in a wide temperature range were investigated for the irradiated and nonirradiated samples. The effects of gamma-irradiation and LCP addition on the thermomechanical behavior of PE are discussed. It was found that the LCP addition affected significantly the stress-strain behavior of PE at temperatures above the melting point. The features of thermorelaxation properties of the PE/LCP blends previously irradiated and oriented, particularly thermorelaxation and residual shrinkage stresses at isometric heating and cooling, were also established. The results obtained testify that the LCP addition makes it possible to improve considerably the thermosetting properties of irradiated PE.Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 1, pp. 124–139, January–February, 1998.     

Linear complementarity and discounted switching controller stochastic games

The class of discounted switching controller stochastic games can be solved in one step by a linear complementarity program (LCP). Following the proof of this technical result is a discussion of a special formulation and initialization of a standard LCP pivoting algorithm which has, in numerical experiments, always terminated in a complementary solution. That the LCP algorithm as formulated always finds a complementary solution has not yet been proven, but these theoretical and experimental results have the potential to provide an alternative proof of the ordered field property for these games. Numerical experimentation with the reformulated LCP is reviewed.     

On the equivalence of linear complementarity problems

We show that the Extended Linear Complementarity Problem (ELCP) can be recast as a standard Linear Complementarity Problem (LCP) provided that the surplus variables or the feasible set of the ELCP are bounded. Since many extensions of the LCP are special cases of the ELCP, this implies that these extensions can be rewritten as an LCP as well. Our equivalence proof is constructive and leads to three possible numerical solution methods for a given ELCP: regular ELCP algorithms, mixed integer linear programming algorithms, and regular LCP algorithms.     

Structure and properties of injection molded blends of liquid crystal polymer (40 PET/60 PHB) with poly(butylene terephthalate)

Blends of a poly(butylene terephthalate) (PBT) and a liquid crystalline polymer (LCP) were prepared in a single screw extruder. The LCP used was a liquid crystalline copolyester of 40% polyethylene terephthalate with 60% p-hydroxybenzoic acid (40 PET/60 PHB). Specimens for mechanical testing were prepared by injection molding. Rheological, morphological, microstructural, and mechanical properties were investigated by capillary rheometry, polarized light microscopy, x-ray diffractometry, and a tensile tester, respectively. For the LCP content higher than 5 wt. % a significant decrease of the viscosity was noted. The morphology of the dispersed LCP phase varies between droplets and oriented fibrils. The degree of cystallinity of PBT and the crystal sizes were found to be affected by LCP addition. The tensile mechanical behavior of the PBT + LCP blends was also found to be affected by their compositions. The elasticity modulus and the parameter of elasticity anisotropy monotonically increase with the LCP content. The strain at break shows a drastic decrease for the blends containing more than 10 wt. % LCP. The long-term creep test shows that the creep compliance decreases noticeably when the LCP content increases. Institute of Polymer Mechanics of the Latvian Academy of Science, Riga, LV-1006, Latvia. Ecole Europeénne de Chimie des Polymeres et des Matériaux, Université Louis Pasteur, F-67000 Strasbourg, France. Poznan University of Technology, Laboratory of Crystallography and Polymer Physics, PL-60-965 Poznan, Poland. Published in Mekhanika Kompozitnykh Materialov, Vol. 32, No. 5, pp. 676–689, September–October, 1996.     

A Smoothing-Type Algorithm for Solving Linear Complementarity Problems with Strong Convergence Properties

In this paper, we construct an augmented system of the standard monotone linear complementarity problem (LCP), and establish the relations between the augmented system and the LCP. We present a smoothing-type algorithm for solving the augmented system. The algorithm is shown to be globally convergent without assuming any prior knowledge of feasibility/infeasibility of the problem. In particular, if the LCP has a solution, then the algorithm either generates a maximal complementary solution of the LCP or detects correctly solvability of the LCP, and in the latter case, an existing smoothing-type algorithm can be directly applied to solve the LCP without any additional assumption and it generates a maximal complementary solution of the LCP; and that if the LCP is infeasible, then the algorithm detect correctly infeasibility of the LCP. To the best of our knowledge, such properties have not appeared in the existing literature for smoothing-type algorithms. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10571134), the Natural Science Foundation of Tianjin (Grant No. 07JCYBJC05200), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.     

A FPTAS for computing a symmetric Leontief competitive economy equilibrium

In this paper, we consider a linear complementarity problem (LCP) arisen from the Nash and Arrow–Debreu competitive economy equilibria where the LCP coefficient matrix is symmetric. We prove that the decision problem, to decide whether or not there exists a complementary solution, is NP-complete. Under certain conditions, an LCP solution is guaranteed to exist and we present a fully polynomial-time approximation scheme (FPTAS) for approximating a complementary solution, although the LCP solution set can be non-convex or non-connected. Our method is based on approximating a quadratic social utility optimization problem (QP) and showing that a certain KKT point of the QP problem is an LCP solution. Then, we further show that such a KKT point can be approximated with a new improved running time complexity ${{O}((\frac{n^4}{\epsilon})\log\log(\frac{1}{\epsilon}))}$ arithmetic operation in accuracy ${\epsilon \in (0,1)}$ . We also report preliminary computational results which show that the method is highly effective. Applications in competitive market model problems with other utility functions are also presented, including global trading and dynamic spectrum management problems.     

Mechanical properties of blends of liquid crystalline copolyesters with polyprophylene

Conclusions A significant effect of the addition of LCP on the mechanical properties and their anisotropy has been established. Already, if one considers the shape of curves of the stress-strain relationship it can be seen that curves typical for semicrystalline polymers (pure polypropylene) with clearly visible yield point and significant cold drawing leading to an anisotropic stiffening are changing into curves without yielding and with a brittle failure (LC-rich blends). Generally, the tensile elasticity modulus increases with increasing LCP content for both MD and TD. The maximum value of anisotropy of elastic properties was noted for a rather low content of LCP (c = 5%). On the contrary, the stress at yield decreases with increasing LCP content. The same was observed for the strain at yield but in both cases an important increase of anisotropy has taken place. Consequently, the total elongation during drawing (strain at break) showed a drastic decrease for blends with higher LCP content (about 60–80 times). The addition of the LCP to polypropylene has led to a stiffness increase (higher elasticity modulus) but simultaneously to a considerable plasticity decrease. As a confirmation of these observations, there served also the creep test where a decrease of the creep compliance (by two times) for LC-rich blends as compared with pure PP was noted.It also should be emphasized that, generally, a smaller effect of LCP content on the elastic deformation was noted than that on the time dependent effects (nonelastic creep deformation).Published in Mekhanika Kompozitnykh Materialov, Vol. 30, No. 4, pp. 442–450, July–August, 1994.