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Continuous Constraints: Updating the Technology

Objective

Current numerical techniques cannot reliably enumerate all the solutions of a non-linear continuous Constraints Solving and Optimisation Problem (CSOP). Complete solution techniques, on the other hand, can only handle problems that are too small to be relevant for industrial applications. The goal of this project is to develop new complete solving techniques that scale up to industrial applications, that is, to push up the limit of tractability by at least one order of magnitude. We will develop new inference algorithms that efficiently exploit the properties of new abstractions and we will develop new ways of combining different techniques. The result will be new and improved complete solving techniques for non-linear continuous CSOP, a set of combination algorithms and a benchmarking suite. Current numerical techniques cannot reliably enumerate all the solutions of a non-linear continuous Constraints Solving and Optimisation Problem (CSOP). Complete solution techniques, on the other hand, can only handle problems that are too small to be relevant for industrial applications. The goal of this project is to develop new complete solving techniques that scale up to industrial applications, that is, to push up the limit of tractability by at least one order of magnitude. We will develop new inference algorithms that efficiently exploit the properties of new abstractions and we will develop new ways of combining different techniques. The result will be new and improved complete solving techniques for non-linear continuous CSOP, a set of combination algorithms and a benchmarking suite.

OBJECTIVES
Many industrial problems can be modelled as constraint satisfaction and optimisation problems (CSOP). A CSOP consists of variables that take on a value from a given set of values, of constraints that restrict the possible value combinations and, if relevant, of an optimisation criterion. A complete solver is able to compute all solutions exhaustively and to find the global optima of an optimisation problem. We will focus on problems that involve non-linear expressions and constraints on variables with continuous domains.
Existing complete solvers can only handle toy problems or rough approximations of 10 to 20 variables. Most industrially relevant problems are at least one order of magnitude larger: they may involve hundreds of variables. Our aim is to develop complete techniques that can handle non-linear continuous CSOPs that involve at least 100 variables. This will make complete solvers for continuous CSOPs appealing to industry.

DESCRIPTION OF WORK
Global optimisation is a NP-hard problem. Therefore, pushing upward the limits of tractability depends on finding the representations for which the solving techniques work with least effort or on combining techniques in order to better exploit their individual strengths and compensate for their individual weaknesses. Our approach will be dual. On the one hand, we will explore new abstractions and representations for the problems and the constraints. Indeed, Individually improving existing techniques, although needed, brings incremental gains only and we believe that it will be ultimately limited. Based on this, we aim at creating new tools, specifically designed to make inferences based on non-linear constraints and better adapted for large-scale problems. On the other hand, we will work on combining different techniques, both existing and new ones. Individual inference technique may not be effective over large operating regions and for different problem types. However, by combining them we believe that significant gains in inference power will be made. As a result there is a real potential for making good progress in the size of the problems that can be solved. Our integration framework will be Constraint Programming (CP). CP is by nature a flexible approach in which one can specify problem-solving strategies, modify the constraints that define a problem and add new constraints. In addition, CP is very expressive; it can deal with any types of constraints and it can handle heterogeneous problems. Our dual approach has good chances to achieve a quantum leap in the size of the problems that can be solved. Very little work has been done on either direction up to now. Although our main objective is at the representation and algorithmic level, we also aim at developing techniques that can be practically used and we want to make the approach appealing to industry. We will assemble a set of benchmarking and we will measure our results against these problems.

Funding Scheme

CSC - Cost-sharing contracts

Coordinator

ILOG
Address
9, Rue De Verdun
94250 Gentilly
France

Participants (6)

ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
Switzerland
Address
Ecublens
1015 Lausanne
FACULDADE CIENCIAS E TECNOLOGIA DA UNIVERSIDADE DE COIMBRA
Portugal
Address
Colegio S. Jeronimo, Largo D. Dinis
3000-141 Coimbra
TECHNISCHE UNIVERSITAET DARMSTADT
Germany
Address
Karolinenplatz 5
64289 Darmstadt
UNIVERSITAET WIEN - INSTITUT FUER MATHEMATIK
Austria
Address
Strudlhofgasse 4
1090 Wien
UNIVERSITE CATHOLIQUE DE LOUVAIN
Belgium
Address
1 Place De L'universite
1348 Louvain-la-neuve
UNIVERSITE DE NANTES
France
Address
Quai De Tourville 1
44035 Nantes