Constraint integer programming (CIP) is a novel paradigm which integrates constraint programming (CP), mixed integer programming (MIP), and satisfiability (SAT) modeling and solving techniques. In this paper we discuss the software framework and solver SCIP (Solving Constraint Integer Programs), which is free for academic and non-commercial use and can be downloaded in source code. This paper gives an overview of the main design concepts of SCIP and how it can be used to solve constraint integer programs. To illustrate the performance and flexibility of SCIP, we apply it to two different problem classes. First, we consider mixed integer programming and show by computational experiments that SCIP is almost competitive to specialized commercial MIP solvers, even though SCIP supports the more general constraint integer programming paradigm. We develop new ingredients that improve current MIP solving technology. As a second application, we employ SCIP to solve chip design verification problems as they arise in the logic design of integrated circuits. This application goes far beyond traditional MIP solving, as it includes several highly non-linear constraints, which can be handled nicely within the constraint integer programming framework. We show anecdotally how the different solving techniques from MIP, CP, and SAT work together inside SCIP to deal with such constraint classes. Finally, experimental results show that our approach outperforms current state-of-the-art techniques for proving the validity of properties on circuits containing arithmetic.

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SCIP: solving constraint integer programs

http://egon.cheme.cmu.edu/Papers/OaBB.pdf

An algorithmic framework for convex mixed integer nonlinear programs

Many industrial problems can be naturally formulated using mixed integer non-linear programming (MINLP) models and can be solved by spatial Branch&Bound (sBB) techniques. We study the impact of two important parts of sBB methods: bounds tightening (BT) and branching strategies. We extend a branching technique originally developed for MILP, reliability branching, to the MINLP case. Motivated by the demand for open-source solvers for real-world MINLP problems, we have developed an sBB software package named couenne (Convex Over-and Under-ENvelopes for Non-linear Estimation) and used it for extensive tests on several combinations of BT and branching techniques on a set of publicly available and real-world MINLP instances. We also compare the performance of couenne with a state-of-the-art MINLP solver.

/pdf/branching-and-bounds-tighteningtechniques-for-non-convex-5f8pe12upf.pdf

Branching and bounds tighteningtechniques for non-convex MINLP

Encompassing all the major topics students will encounter in courses on the subject, the authors teach both the underlying mathematical foundations and how these ideas are implemented in practice. They illustrate all the concepts with both worked examples and plenty of exercises, and, in addition, provide software so that students can try out numerical methods and so hone their skills in interpreting the results. As a result, this will make an ideal textbook for all those coming to the subject for the first time. Authors' note: A problem recently found with the software is due to a bug in Formula One, the third party commercial software package that was used for the development of the interface. It occurs when the date, currency, etc. format is set to a non-United States version. Please try setting your computer date/currency option to the United States option . The new version of Formula One, when ready, will be posted on Www.

Linear Programming 1: Introduction

The branch and bound principle has long been established as an effective computational tool for solving mixed integer linear programming problems. This paper investigates the computational feasibility of branch and bound methods in solving convex nonlinear integer programming problems. The efficiency of a branch and bound method often depends on the rules used for selecting the branching variables and branching nodes. Among others, the concepts of pseudo-costs and estimations are implemented in selecting these parameters. Since the efficiency of the algorithm also depends on how fast an upper bound on the objective minimum is attained, heuristic rules are developed to locate an integer feasible solution to provide an upper bound. The different criteria for selecting branching variables, branching nodes, and heuristics form a total of 27 branch and bound strategies. These strategies are computationally tested and the empirical results are presented.

Branch and Bound Experiments in Convex Nonlinear Integer Programming

This paper presents a “branch and bound” method for solving mixed integer linear programming problems. After briefly discussing the bases of the method, new concepts called pseudo-costs and estimations are introduced. Then, the heuristic rules for generating the tree, which are the main features of the method, are presented. Numerous parameters allow the user for adjusting the search strategy to a given problem.

Experiments in mixed-integer linear programming

First, this paper presents the results of experiments with algorithmic techniques for efficiently solving medium and large scale linear and mixed integer programming problems. The techniques presented here are either original or recent.

The efficient solution of large-scale linear programming problems—some algorithmic techniques and computational results

This paper discusses heuristic "branch and bound" methods for solving mixed integer linear programming problems. The research presented on here is the follow on to that recorded in [3].

After a resume of the concept of pseudo-costs and estimations, new heuristic rules for generating a tree which make use of pseudo-costs and estimations are presented. Experiments have shown that models having a low percentage of integer variables behave in a radically different way from models with a high percentage of integer variables. The new heuristic rules seem to apply generally to the first type of model.

Later, other heuristic rules are presented that are used with models having a high percentage of integer variables and with models having a special structure (models including special ordered sets.)

The rules introduced here have been implemented in the IBM Mathematical Programming System Extended/370. They are used to solve large mixed integer linear programming models.

Numerical results that permit comparisons to be made among the different rules are provided and discussed.

Gerard Ribière

Papers

Experiments in mixed-integer linear programming

The efficient solution of large-scale linear programming problems—some algorithmic techniques and computational results

Experiments in mixed-integer linear programming using pseudo-costs