A new algorithm for computing contact forces between solid objects with friction is presented. The algorithm allows a mix of contact points with static and dynamic friction. In contrast to previous approaches, the problem of computing contact forces is not transformed into an optimization problem. Because of this, the need for sophisticated optimization software packages is eliminated. For both systems with and without friction, the algorithm has proven to be considerably faster, simpler, and more reliable than previous approaches to the problem. In particular, implementation of the algorithm by nonspecialists in numerical programming is quite feasible.

A method for analytically calculating the forces between systems of rigid bodies in resting (non-colliding) contact is presented. The systems of bodies may either be in motion or static equilibrium and adjacent bodies may touch at multiple points. The analytic formulation of the forces between bodies in non-colliding contact can be modified to deal with colliding bodies. Accordingly, an improved method for analytically calculating the forces between systems of rigid bodies in colliding contact is also presented. Both methods can be applied to systems with arbitrary holonomic geometric constraints, such as linked figures. The analytical formulations used treat both holonomic and non-holonomic constraints in a consistent manner.

Given an undirected graph G with nonnegative edges costs and nonnegative vertex penalties, the prize collecting Steiner problem in graphs (PCSPG) seeks a tree of G with minimum weight. The weight of a tree is the sum of its edge costs plus the sum of the penalties of those vertices not spanned by the tree. In this paper we present an integer programming formulation of the PCSPG and describe an algorithm to obtain lower bounds for the problem. The algorithm is based on polyhedral cutting planes and is initiated with tests that attempt to reduce the size of the input graph. Computational experiments were carried out to evaluate the strength of the formulation through its linear programming relaxation. The algorithm found optimal solutions in 85% of 114 problems tested. Of those optimal solutions, 97% were integral, thus producing feasible upper bounds. Nine new best known upper bounds were produced for the test set. Tight lower bounds were produced in 89% of the instances. ...

We generalize the subgradient optimization method for nondifferentiable convex programming to utilize conditional subgradients. Firstly, we derive the new method and establish its convergence by generalizing convergence results for traditional subgradient optimization. Secondly, we consider a particular choice of conditional subgradients, obtained by projections, which leads to an easily implementable modification of traditional subgradient optimization schemes. To evaluate the subgradient projection method we consider its use in three applications: uncapacitated facility location, two-person zero-sum matrix games, and multicommodity network flows. Computational experiments show that the subgradient projection method performs better than traditional subgradient optimization; in some cases the difference is considerable. These results suggest that our simple modification may improve subgradient optimization schemes significantly. This finding is important as such schemes are very popula...

. In both the design and use of large-scale mathematical programming systems, a substantial portion of the e#ort has no direct relation to the variables and constraints, but is instead concerned with the description, manipulation and display of data. Established principles of database design do not apply directly to mathematical programming, however, because there are significant di#erences of organization and content between the data for an optimization model and the data for a conventional database application such as payroll or order entry. This paper derives fundamental principles of database construction for largescale mathematical programming, by use of a steel mill planning model as an example. Alternative formulations for the model---which incorporate aspects of production and network linear programming---are presented at the outset, and are shown to correspond to relational and hierarchical database schemes that have contrasting strengths and weaknesses. A particular implement...

The development of new mathematical theory and its application in software systems for the solution of hard optimization problems have a long tradition in mathematical programming. In this tradition we implemented ABACUS, an object-oriented software framework for branch-and-cut-and-price algorithms for the solution of mixed integer and combinatorial optimization problems. This paper discusses some difficulties in the implementation of branch-and-cut-and-price algorithms for combinatorial optimization problems and shows how they are managed by ABACUS.

: In this paper we consider the Project Scheduling Problem with resource constraints, where the objective is to minimize the project makespan. We present a new 0-1 linear programming formulation of the problem that requires an exponential number of variables, corresponding to all feasible subsets of activities that can be simultaneously executed without violating resource or precedence constraints. Different relaxations of the above formulation are used to derive new lower bounds, which dominate the value of the longest path on the precedence graph and are tighter than the bound proposed by Stinson et al. (1978). A tree search algorithm based on the above formulation, that uses new lower bounds and dominance criteria is also presented. Computational results indicate that the exact algorithm can solve hard instances that cannot be solved by the best algorithms reported in the literature. Correspondence to: A. Mingozzi, Dept. of Mathematics, University of Bologna, via Sacchi 3, 47023 Ces...

this paper we explore an approach which enables all of the problems listed above to be solved at a single stroke: use Lisp as the source language for the numeric and graphical code! This is not a new idea --- it was tried at MIT and UCB in the 1970's. While these experiments were modestly successful, the particular systems are obsolete. Fortunately, some of those ideas used in Maclisp [37], NIL [38] and Franz Lisp [20] were incorporated in the subsequent standardization of Common Lisp (CL) [35]. In this new setting it is appropriate to re-examine the theoretical and practical implications of writing numeric code in Lisp. The popular conceptions of Lisp's inefficiency for numerics have been based on rumor, supposition, and experience with early and (in fact) inefficient implementations. It is certainly possible to continue to write inefficient programs: As one example of the results of de-emphasizing numerics in the design, consider the situation of the basic arithmetic operators. The definitions of these functions require that they are generic, (e.g. "+" must be able to add any combination of several precisions of floats, arbitrary-precision integers, rational numbers, and complexes), The very simple way of implementing this arithmetic -- by subroutine calls -- is also very inefficient. Even with appropriate declarations to enable more specific treatment of numeric types, compilers are free to ignore declarations and such implementations naturally do not accommodate the needs of intensive number-crunching. (See the appendix for further discussion of declarations). Be this as it may, the situation with respect to Lisp has changed for the better in recent years. With the advent of ANSI standard Common Lisp, several active vendors of implementations and one active universi...

A finitely convergent non-simplex method for large scale structured linear programming problems arising in stochastic programming is presented. The method combines the ideas of the Dantzig-Wolfe decomposition principle and modern nonsmooth optimization methods. Algorithmic techniques taking advantage of properties of stochastic programs are described and numerical results for large real world problems reported. Keywords: Stochastic Programming, Decomposition. iii iv Regularized decomposition of stochastic programs: algorithmic techniques and numerical results Andrzej Ruszczy'nski 1. Introduction A large class of operations research problems lead to linear programming models of the form min c T x Mx = r; (1:1) x min x x max : In certain applications, however, some coefficients of the resource/demand vector r or some entries of the matrix M are uncertain. They can be modeled as random variables r(!) and M(!) with ! 2 \Omega\Gamma where(\Omega ; B; P ) is a probability ...

This paper considers an identical-jobs, re-entrant flow, cyclic scheduling problem in repetitive manufacturing environments. An integer programming formulation is developed with system performance measured using a weighted linear combination of cycle length and flow time. The variables in this formulation decompose into two sets: a combinatorial set, called the "cyclic precedence structure," that characterizes all precedence relations between operations in a cyclic schedule, and a set of continuous production timing variables. Given the entire precedence structure, we efficiently solve the resulting production timing subproblem by determining the optimal cycle length and operation start times. We show that under certain conditions, a single cyclic schedule with the specified precedence structure simultaneously minimizes both flow time and cycle length. The related cycle offset subproblem of mapping operations to cycles to minimize flow time while achieving a target throughput, given ...