A new algorithm for the linear complementarity problem allowing for an arbitrary starting point

Linear Programming

Quadratic Pareto Race

This paper presents a dynamic and visual "free search" type of a decision support system -- Quadratic Pareto Race, which enables a decision maker (DM) to freely search the efficient frontier of a multiple objective quadratic-linear programming problem by controlling the speed and direction of motion. The values of the objective functions are presented in a numeric form and as bar graphs on a display. The implementation of Quadratic Pareto Race is based on the theoretical foundations developed by Korhonen and Yu (1996). The system is implemented on a microcomputer and illustrated using a numerical example

Enumeration of PLCP-orientations of the 4-cube

The linear complementarity problem (LCP) provides a unified approach to many problems such as linear programs, convex quadratic programs, and bimatrix games. The general LCP is known to be NP-hard, but there are some promising results that suggest the possibility that the LCP with a P-matrix (PLCP) may be polynomial-time solvable. However, no polynomial-time algorithm for the PLCP has been found yet and the computational complexity of the PLCP remains open. Simple principal pivoting (SPP) algorithms, also known as Bard-type algorithms, are candidates for polynomial-time algorithms for the PLCP. In 1978, Stickney and Watson interpreted SPP algorithms as a family of algorithms that seek the sink of unique-sink orientations of $n$-cubes. They performed the enumeration of the arising orientations of the $3$-cube, hereafter called PLCP-orientations. In this paper, we present the enumeration of PLCP-orientations of the $4$-cube.The enumeration is done via construction of oriented matroids generalizing P-matrices and realizability classification of oriented matroids.Some insights obtained in the computational experiments are presented as well

Computation Reuse in Statics and Dynamics Problems for Assemblies of Rigid Bodies

The problem of determining the forces among contacting rigid bodies is fundamental to many areas of robotics, including manipulation planning, control, and dynamic simulation. For example, consider the question of how to unstack an assembly, or how to find stable regions of a rubble pile. In considering problems of this type over discrete or continuous time, we often encounter a sequence of problems with similar substructure. The primary contribution of our work is the observation that in many cases, common physical structure can be exploited to solve a sequence of related problems more efficiently than if each problem were considered in isolation. We examine three general problems concerning rigid-body assemblies: dynamic simulation, assembly planning, and assembly stability given limited knowledge of the structure\u27s geometry. To approach the dynamic simulation and assembly planning applications, we have optimized a known method for solving the system dynamics. The accelerations of and forces among contacting rigid bodies may be computed by formulating the dynamics equations and contact constraints as a complementarity problem. Dantzig\u27s algorithm, when applicable, takes n or fewer major cycles to find a solution to the linear complementarity problem corresponding to an assembly with n contacts. We show that Dantzig\u27s algorithm will find a solution in n - k or fewer major cycles if the algorithm is initialized with a solution to the dynamics problem for a subassembly with k internal contacts. Finally, we show that if we have limited knowledge of a structure\u27s geometry, we can still learn about stable regions of its surface by physically pressing on it. We present an approach for finding stable regions of planar assemblies: sample presses on the surface to identify a stable cone in wrench space, partition the space of applicable wrenches into stable and unstable regions, and map these back to the surface of the structure

A unified approach to complementarity in optimization

AbstractAn underlying general structure of complementary pivot theory is presented with applications to various problems in optimization theory. The applications include linear complementarity, fixed point theory, unconstrained and constrained convex optimization without derivatives, nonlinear complementarity, and saddle point problems

Introducing Interior-Point Methods for Introductory Operations Research Courses and/or Linear Programming Courses

In recent years the introduction and development of Interior-Point Methods has had a profound impact on optimization theory as well as practice, influencing the field of Operations Research and related areas. Development of these methods has quickly led to the design of new and efficient optimization codes particularly for Linear Programming. Consequently, there has been an increasing need to introduce theory and methods of this new area in optimization into the appropriate undergraduate and first year graduate courses such as introductory Operations Research and/or Linear Programming courses, Industrial Engineering courses and Math Modeling courses. The objective of this paper is to discuss the ways of simplifying the introduction of Interior-Point Methods for students who have various backgrounds or who are not necessarily mathematics majors

The existence of a strongly polynomial time simplex method

It is well known how to clarify whether there is a polynomial time simplex algorithm for linear programming (LP) is the most challenging open problem in optimization and discrete geometry. This paper gives a affirmative answer to this open question by the use of the parametric analysis technique that we recently proposed. We show that there is a simplex algorithm whose number of pivoting steps does not exceed the number of variables of a LP problem.Comment: 17 pages, 1 figur

An SLSPP-algorithm to compute an equilibrium in an economy with linear production technologies

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