Solution of oligopoly market equilibrium problem using modified newton method

The paper aims to find the solution of oligopoly market equilibrium problem through system of nonlinear equations. We propose modified newton method to obtain the solution of system of nonlinear equations. We show that our proposed method has higher order of convergence. Keywords: System of nonlinear equations, modified newton method, oligopolistic market equilibrium problem.Comment: arXiv admin note: substantial text overlap with arXiv:2209.0100

An output-sensitive algorithm for multi-parametric LCPs with sufficient matrices

This paper considers the multi-parametric linear complementarity problem (pLCP) with sufficient matrices. The main result is an algorithm to find a polyhedral decomposition of the set of feasible parameters and to construct a piecewise affine function that maps each feasible parameter to a solution of the associated LCP in such a way that the function is affine over each cell of the decomposition. The algorithm is output-sensive in the sense that its time complexity is polynomial in the size of the input and linear in the size of the output, when the problem is non-degenerate. We give a lexicographic perturbation technique to resolve degeneracy as well. Unlike for the non-parametric case, the resolution turns out to be nontrivial, and in particular, it involves linear programming (LP) duality and multi-objective LP

Complementarity Problem With Nekrasov ZZ tensor

It is worth knowing that a particular tensor class belongs to $P$-tensor which ensures the compactness to solve tensor complementarity problem (TCP). In this study, we propose a new class of tensor, Nekrasov $Z$ tensor, in the context of the tensor complementarity problem. We show that the class of $P$-tensor contains the class of even ordered Nekrasov $Z$ tensors with positive diagonal elements. In this context, we propose a procedure by which a Nekrasov $Z$ tensor can be transformed into a tensor which is diagonally dominant. Keywords: Diagonally dominant tensor, Nekrasov tensors, Nonsingular $H$ tensor, Nekrasov $Z$ tensor, $P$-tensors, Tensor complementarity proble

New Relaxation Modulus Based Iterative Method for Large and Sparse Implicit Complementarity Problem

This article presents a class of new relaxation modulus-based iterative methods to process the large and sparse implicit complementarity problem (ICP). Using two positive diagonal matrices, we formulate a fixed-point equation and prove that it is equivalent to ICP. Also, we provide sufficient convergence conditions for the proposed methods when the system matrix is a $P$-matrix or an $H_+$-matrix. Keyword: Implicit complementarity problem, $H_{+}$-matrix, $P$-matrix, matrix splitting, convergenceComment: arXiv admin note: substantial text overlap with arXiv:2303.1251

Solution of nonlinear system of equations through homotopy path

The paper aims to show the equivalency between nonlinear complementarity problem and the system of nonlinear equations. We propose a homotopy method with vector parameter $\lambda$ in finding the solution of nonlinear complementarity problem through a system of nonlinear equations. We propose a smooth and bounded homotopy path to obtain solution of the system of nonlinear equations under some conditions. An oligopolistic market equilibrium problem is considered to show the effectiveness of the proposed homotopy continuation method. Keywords: Nonlinear complementarity problem, system of nonlinear equations, homotopy function with vector parameter, bounded smooth curve, oligopolistic market equilibrium.Comment: arXiv admin note: text overlap with arXiv:2209.0038

On Covering Smooth Manifolds with a Q-arrangement of Simplicies: An inductive Characterization of Q-matrices

This paper is concerned with a covering problem of smooth manifolds of dimension n − 1 by stitching 2 n n-simplices formed with 2 n-lists of points along their common (n − 1)-facets. The n-simplices are in bijective correspondence with the vertices of an n-dimensional hypercube; they could be degenerate and are allowed to overlap. We leverage the underlying inductive nature of the problem to give a (non-constructive) topological characterization. We show that for low dimensions such characterization reduces to studying the local geometry around the specific points serving to form the simplices, solving thereby the problem for n ≤ 3. This covering problem provides a geometric equivalent reformulation of a relatively old, yet unsolved, problem that originated in the optimization community: under which conditions on the n × n matrix M , does the so called linear complementarity problem given by w − M z = q, w, z ≥ 0, and w.z = 0, have a solution (w, z) for all vectors q ∈ R n. If the latter property holds, the matrix M is said to be a Q-matrix

Contact modeling as applied to the dynamic simulation of legged robots

The recent studies in robotics tend to develop legged robots to perform highly dynamic movement on rough terrain. Before implementing on robots, the reference generation and control algorithms are preferably tested in simulation and animation environments. For simulation frameworks dedicated to the test of legged locomotion, the contact modeling is of pronounced signi cance. Simulation requires a correct contact model for obtaining realistic results. Penalty based contact modeling is a popular approach that de nes contact as a spring - damper combination. This approach is simple to implement. However, penetration is observed in this model. Interpenetration of simulated objects results in less than ideal realism. In contrast to penalty based method, exact contact model de nes the constraints of contact forces and solves them by using analytical methods. In this thesis, a quadruped robot is simulated with exact contact model. The motion of system is solved by the articulated body method (ABM). This algorithm has O(n) computational complexity. The ABM is employed to avoid calculation of the inverse of matrices. The contact is handled as a linear complementarity problem and solved by using the projected Gauss Seidel algorithm. Joint and contact friction terms consisting of viscous and Coulomb friction components are implemented

Some properties of fully semimonotone Q~0-matrices

Stone [Ph.D. thesis, Dept. of Operations Research, Stanford University, Stanford, CA, 1981] proved that within the class of $Q_0$-matrices, the $U$-matrices are $P_0$-matrices and conjectured that the same must be true for fully semimonotone $(E_0^f )$ matrices. In this paper we show that this conjecture is true for matrices of order up to $4 \times 4$ and partially resolve it for higher order matrices. This is done by establishing the result that if $A$ is in $E_0^f \cap Q_0$ and if every proper principal minor of $A$ is nonnegative, then $A$ is a $P_0$-matrix. Using this key result we settle the conjecture for a number of special cases of matrices of general order. These special cases include $E_0^f$-matrices which are either symmetric or nonnegative or copositive-plus or $Z$-matrices or $E$-matrices. Also the conjecture is established for $5 \times 5$ matrices with all diagonal entries positive. While trying to settle the conjecture, we obtained a number of results on $Q_0$-matrices. The main among these are characterizations of nonnegative $Q_0$-matrices and symmetric semimonotone $Q_0$-matrices; results providing sufficient conditions under which, principal submatrices of order $(n - 1)$ of a $n \times n$$Q_0$-matrix are also in $Q_0$