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Linear complementarity, linear and nonlinear programming
01 Jan 1988-
About: The article was published on 1988-01-01 and is currently open access. It has received 1012 citations till now. The article focuses on the topics: Mixed complementarity problem & Complementarity theory.
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01 Jan 2004
TL;DR: In this paper, the search direction is obtained by solving a strict convex programming at each iterative point, which is essentially different from traditional SQP method and the global convergence of the method is proved under mild conditions.
Abstract: In this paper, LCP is converted to an equivalent nonsmooth nonlinear equation system H(x, y) = 0 by using the famous NCP function–Fischer-Burmeister function. Note that some equations in H(x, y) = 0 are nonsmooth and nonlinear hence difficult to solve while the others are linear hence easy to solve. Then we further convert the nonlinear equation system H(x, y) = 0 to an optimization problem with linear equality constraints. After that we study the conditions under which the K–T points of the optimization problem are the solutions of the original LCP and propose a method to solve the optimization problem. In this algorithm, the search direction is obtained by solving a strict convex programming at each iterative point. However, our algorithm is essentially different from traditional SQP method. The global convergence of the method is proved under mild conditions. In addition, we can prove that the algorithm is convergent superlinearly under the conditions: M is P0 matrix and the limit point is a strict complementarity solution of LCP. Preliminary numerical experiments are reported with this method.
3 citations
01 Nov 2008
TL;DR: This paper provides a brief tutorial on complementarity problems, reviews the application literature on these problems, provides a working example in the context of networked infrastructure, and concludes with an agenda for future research.
Abstract: Complementarity problems are a class of optimization problems which widely pervade infrastructure systems Complementarity problems capture the concept of network and system equilibrium without the use of a mono-objective function As a result these problems are useful for modeling infrastructure systems subject to use by actors with varied interests and objectives Complementarity problems have been applied to both technical as well as economic systems As such complementarity problems are particularly well-suited for modeling mixed or socio-technical systems This paper provides a brief tutorial on complementarity problems, reviews the application literature on these problems, provides a working example in the context of networked infrastructure, and concludes with an agenda for future research
3 citations
Cites background from "Linear complementarity, linear and ..."
...The author is aware of only three monographs [2] [3] [4] and three surveys [5] [6] [7]....
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TL;DR: In this paper, a predictor-corrector noninterior method for LCP is presented, in which the predictor step is generated by the Levenberg-Marquadt method.
Abstract: In this paper a new predictor-corrector noninterior method for LCP is presented, in which the predictor step is generated by the Levenberg-Marquadt method, which is new in the predictor-corrector-type methods, and the corrector step is generated as in [3]. The method has the following merits: (i) any cluster point of the iteration sequence is a solution of the P0 LCP; (ii) if the generalized Jacobian is nonsingular at a solution point, then the whole sequence converges to the (unique) solution of the P0 LCP superlinearly; (iii) for the P0 LCP, if an accumulation point of the iteration sequence satisfies the strict complementary condition, then the whole sequence converges to this accumulation point superlinearly. Preliminary numerical experiments are reported to show the efficiency of the algorithm.
3 citations
TL;DR: A systematic study on different graph measures in order to identify optimal organization for maximal biodiversity (defined as structural stability) and concludes that the strength in cooperation parameters are the core fact, i.e., cooperation is the real fact optimizing biodiversity among other possible structural configurations.
Abstract: Dynamical systems on graphs allow to describe multiple phenomena from different areas of Science. In particular, many complex systems in Ecology are studied by this approach. In this paper we analize the mathematical framework for the study of the structural stability of each stationary point, feasible or not, introducing a generalization for this concept, defined as Global Structural Stability. This approach would fit with the proper mathematical concept of structural stability, in which we find a full description of the complex dynamics on the phase space due to nonlinear dynamics. This fact can be analyzed as an informational field grounded in a global attractor whose structure can be completely characterized. These attractors are stable under perturbation and suppose the minimal structurally stable sets. We also study in detail, mathematically and computationally, the zones characterizing different levels of biodiversity in bipartite graphs describing mutualistic antagonistic systems of population dynamics. In particular, we investigate the dependence of the region of maximal biodiversity of a system on its connectivity matrix. On the other hand, as the network topology does not completely determine the robustness of the dynamics of a complex network, we study the correlation between structural stability and several graph measures. A systematic study on synthetic and biological graphs is presented, including 10 mutualistic networks of plants and seed-dispersal and 1000 random synthetic networks. We compare the role of centrality measures and modularity, concluding the importance of just cooperation strength among nodes when describing areas of maximal biodiversity. Indeed, we show that cooperation parameters are the central role for biodiversity while other measures act as secondary supporting functions.
3 citations
01 Jan 2007
TL;DR: The Linear Complementarity Problem (LCP) as discussed by the authors is a generalization of the problem of finding all solutions to a linear programming problem, and it has been used in many applications including linear programming, quadratic programming, two-person non-zero sum games and evolutionary games.
Abstract: We define the Linear Complementarity Problem (LCP) and outline its applications including those to Linear Programming (LP), Quadratic Programming (QP), Two person Non-Zero Sum Games and Evolutionary Games. Then we briefly discuss previous methods of solution emphasising the problem of finding all solutions. A new algorithm is then presented, and illustrated by a numerical example, which finds all solutions. It works by successive transformations of variables in order to eliminate the equations in the model.
3 citations