John W. Chinneck

Linear programming with interval coefficients

In order to solve a linear programme, the model coefficients must be fixed at specific values, which implies that the coefficients are perfectly accurate. In practice, however, the coefficients are generally estimates. The only way to deal with uncertain coefficients is to test the sensitivity of the model to changes in their values, either singly or in very small groups. We propose a new approach in which some or all of the coefficients of the LP are specified as intervals. We then find the best optimum and the worst optimum for the model, and the point settings of the interval coefficients that yield these two extremes. This provides the range of the optimised objective function, and the coefficient settings give some insight into the likelihood of these extremes.

Locating Minimal Infeasible Constraint Sets in Linear Programs

With ongoing advances in hardware and software, the bottleneck in linear programming is no longer a model solution, it is the correct formulation of large models in the first place. During initial formulation (or modification), a very large model may prove infeasible, but it is often difficult to determine how to correct it. We present a formulation aid which analyzes infeasible LP s and identifies minimal sets of inconsistent constraints from among the perhaps very large set of constraints defining the problem. This information helps to focus the search for a diagnosis of the problem, speeding the repair of the model. We present a series of filtering routines and a final integrated algorithm which guarantees the identification of at least one minimal set of inconsistent constraints. This guarantee is a significant advantage over previous methods. The algorithms are simple, relatively efficient, and easily incorporated into standard LP solvers. Preliminary computational results are reported. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Performance model driven QoS guarantees and optimization in clouds

This paper presents a method for achieving optimization in clouds by using performance models in the development, deployment and operations of the applications running in the cloud. We show the architecture of the cloud, the services offered by the cloud to support optimization and the methodology used by developers to enable runtime optimization of the clouds. An optimization algorithm is presented which accommodates different goals, different scopes and timescales of optimization actions, and different control algorithms. The optimization here maximizes profits in the cloud constrained by QoS and SLAs across a large variety of workloads.

Fast Heuristics for the Maximum Feasible Subsystem Problem

Given an infeasible set of linear constraints, finding the maximum cardinality feasible subsystem is known as themaximum feasible subsystem problem. This problem is known to be NP-hard, but has many practical applications. This paper presents improved heuristics for solving the maximum feasible subsystem problem that are significantly faster than the original, but still highly accurate.

Finding a Useful Subset of Constraints for Analysis in an Infeasible Linear Program

Infeasibility is often encountered during the process of initial model formulation or reformulation, and it can be extremely difficult to diagnose the cause, especially in large linear programs. While explanation of the error is the domain of humans or artificially intelligent assistants, algorithmic assistance is available to isolate the infeasibility to a subset of the constraints, which helps speed the diagnosis. The isolation should be infeasible, of course, and should not contain any constraints which do not contribute to the infeasibility. Algorithms for finding such irreducible inconsistent systems (IISs) of constraints have been proposed, implemented, and tested in recent years. Experience with IISs shows that a further property of the isolation is highly desirable for easing diagnosis: the isolation should contain as few model rows as possible. This article addresses the problem of finding IISs having few rows in infeasible linear programs. Theory is developed, then implemented and tested on a range of problems using a modified version of MINOS 5.4 called MINOS(IIS).

An effective polynomial-time heuristic for the minimum-cardinality IIS set-covering problem

The state-of-the-art methods for analyzing infeasible linear programs concentrate on isolating Irreducible Infeasible Sets (IISs) of constraints. However, when there are numerous infeasibilities in the model, it is also useful to identify a minimum-cardinality set of constraints which, if removed from the LP, renders it feasible. This set of constraintscovers the IISs. This paper presents a heuristic algorithm for finding a small-cardinality set of constraints which covers the IISs in an infeasible LP. Empirical tests show that it finds a true minimum-cardinality cover over all of the examples in a test set, though this performance cannot be guaranteed in general.

Active-constraint variable ordering for faster feasibility of mixed integer linear programs

The selection of the branching variable can greatly affect the speed of the branch and bound solution of a mixed-integer or integer linear program. Traditional approaches to branching variable selection rely on estimating the effect of the candidate variables on the objective function. We present a new approach that relies on estimating the impact of the candidate variables on the active constraints in the current LP relaxation. We apply this method to the problem of finding the first feasible solution as quickly as possible. Empirical experiments demonstrate a significant improvement compared to a state-of-the art commercial MIP solver.

Analyzing Infeasible Mixed-Integer and Integer Linear Programs

Algorithms and computer-based tools for analyzing infeasible linear and nonlinear programs have been developed in recent years, but few such tools exist for infeasible mixed-integer or integer linear programs. One approach that has proven especially useful for infeasible linear programs is the isolation of an Irreducible Infeasible Set of constraints (IIS), a subset of the constraints defining the overall linear program that is itself infeasible, but for which any proper subset is feasible. Isolating an IIS from the larger model speeds the diagnosis and repair of the model by focussing the analytic effort. This paper describes and tests algorithms for finding small infeasible sets in infeasible mixed-integer and integer linear programs; where possible these small sets are IISs.

MINOS(IIS): infeasibility analysis using MINOS

Abstract An irreducibly inconsistent system of constraints (IIS) is a minimal set of infeasible constraints. MINOS(IIS) is a modified version of MINOS 5.3 which analyzes an infeasible LP model and reports the constraints which constitute an IIS. This localization of the problem speeds the diagnosis and subsequent repair of the model considerably. The paper concentrates on the practical implementation of the filtering algorithms used for IIS localization in MINOS(IIS). New extensions to the algorithm which permit external guidance of the IIS localization are also presented.

The Constraint Consensus Method for Finding Approximately Feasible Points in Nonlinear Programs

This paper develops a method for moving quickly and cheaply from an arbitrary initial point at an extreme distance from the feasible region to a point that is relatively near the feasible region of a nonlinearly constrained model. The method is a variant of a projection algorithm that is shown to be robust, even in the presence of nonconvex constraints and infeasibility. Empirical results are presented.