A framework for optimal correction of inconsistent linear constraints Paula Amaral Dep. Mathematics UNL
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Pedro Barahona Dep. of Computer Science UNL
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July 18, 2005 Abstract The problem of inconsistency between constraints often arises in practice as the result, among others, of the complexity of real models or due to unrealistic requirements and preferences. To overcome such inconsistency two major actions may be taken: removal of constraints or changes in the coefficients of the model. This last approach, that can be generically described as “model correction” is the problem we address in this paper in the context of linear constraints over the reals. The correction of the right hand side alone, which is very close to a fuzzy constraints approach, was one of the first proposals to deal with inconsistency, as it may be mapped into a linear problem. The correction of both the matrix of coefficients and the right hand side introduces non linearity in the constraints. The degree of difficulty in solving the problem of the optimal correction depends on the objective function, whose purpose is to measure the closeness between the original and corrected model. Contrary to other norms, that provide corrections with quite rigid patterns, the optimization of the important Frobenius norm was still an open problem. We have analyzed the problem using the KKT conditions and derived necessary and sufficient conditions which enabled us to unequivocally characterize local optima, in terms of the solution of the Total Least Squares and the set of active constraints. These conditions justify a set of pruning rules, which proved, in preliminary experimental results, quite successful in a tree search procedure for determining the global minimizer.
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Introduction
Inconsistency in linear models is a problem that often occurs in practice due to a variety of causes: the complexity of the problem; conflicting goals in different groups of decision makers; lack of communication between different groups that define the constraints; different views over the problem; partial information; wrong or inaccurate estimates; over optimistic purposes; errors 1
in data; integration of different formulations and actualization of old models. If constraints often model existing real situations, they are also used as preferences regarding the “a posteriori” construction of a physical system. In the first situation, and specially in the second situation, modifications of a model can be necessary for sake of plausibility. The modifications of the model, to render it feasible, can be overcome by considering some constraints as “soft” in the sense that they can be removed altogether or, more generally, that the constraints can be corrected by appropriate changes in their coefficients. A variety of frameworks have been proposed in Constraint Programming to deal with soft constraints. In the Partial Constraint Satisfaction formalism [FrWa92], constraints are removed if they cannot be satisfied and one is interested in solutions that violate the least number of soft constraints. This is an appropriate formalism when all preferences have the same “weight” and the amount of violation is not taken into consideration, but its expressive power is quite limited. In practice, preferences are often of unequal importance and more expressive formalisms have been proposed to model these problems, namely by assigning different weights to the soft constraints. The quality of a solution is then measured by some aggregation of the weights of unsatisfied constraints, as done in hierarchical constraint solving [BoBe89]. The above formalisms do not take into account the degree in which a constraint is unsatisfied. More expressive models, namely the semi-ring and Valued-CSP [BFMR96], and Fuzzy CSP models [DuFP93] allow in different ways to assign “degrees of unsatisfaction” to tuples of values in the domain of the variables of a constraint (that do not satisfy it), so that the aggregated violation of the constraints is minimized. Many algorithms have been proposed to handle these approaches, but they are largely limited to finite domains variables. The formalisms may be adapted to mixed and continuous domains, and among these to linear constraints over the rationals/reals. Some theoretical results have been developed related to the removal of constraints. Van Loon [VLoo81] addressed the identification of Irreducible Inconsistent Systems (IIS) - a set of non-solvable relations, where each of its proper subsystems is solvable. Detection of IIS is an important tool for analyzing inconsistency and to identify constraints that can be removed in order to obtain a consistent system, and the work was later extended [WaHu92] [ChDr91]. Chakravatty [Chak94] proved that the identification of Minimum Cardinality Infeasibility Sets (MCIS) is NP-Hard. Given a set of relations C a Minimum Cardinality Infeasibility Set M is a subset of C such that C \ M is feasible and, among sets that verify this condition, M has smallest cardinality. This work is thus closely related to Partial Constraint Satisfaction in finite domains. In the context of Hierarchical Constraint Logic Programming, Holzbaur, Menezes and Barahona [HoMB96] showed how to find minimal conflict sets (equivalent to IISs) incrementally, i.e. upon the addition of a 2
constraint to a set of feasible constraints, that makes the new set infeasible. Much less work was developed, to our knowledge, regarding the correction of linear constraints. Roodman [Rood79] developed one of the first known approaches, which accounted only for changes in the right-hand side of constraints. A method based on the analysis of the final Phase I solution in the Simplex method allowed to estimate lower bounds on the amount of change in the RHS of each constraint to attain feasibility. Some additional results based on the analysis of the final solution in the Dual Simplex were also presented. Insights on how to work with multiple changes in constraints were given, in a sort of parametric approach although guided by one parameter alone. By analogy to finite domains, the degree of unsatisfaction of a system of linear constraints of the form m X
aij xj ≤ bi , i = . . . , m
j=1
may be done by adding an extra non-negative variable pi to the RHS and minimizing an aggregation of the pi ’s. In the simplest case, where the aggregation is the sum of the pi ’s, the minimization problem is mapped into straightforward linear programming. With the usual interpretation of linear constraints, changing the RHS alone aims at minimizing the amount of resources needed (the bi terms) to make all the tasks feasible. However, the tasks may become feasible not only due to an increase of the resources available, but also by an increased efficiency in their use, which require changes in the aij terms. Vatolin [Vato92] was the first to propose a study for the correction of both the coefficient matrix and the RHS, proposing a family of objective functions as the correction criteria to be minimized. He proved that for a system of equations and inequations on non negative variables, the optimal correction for the proposed cost functions could be solved by a finite number of linear programming problems. However, the number of such problems could, for some cost functions, be exponential on the number of decision variables. Moreover, the correction patterns induced by the minimization of the proposed cost functions, which did not include the Frobenius norm, were very rigid which casted some doubts on the practical application of the results obtained. This paper presents an approach to model the correction of overconstrained linear problems by minimizing all the changes in the coefficients of the linear constraints, to make them feasible. The paper is organized as follows. In section 2 we formalize our minimal discrepancy approach and present a number of variants of it, as well as the algorithms proposed in [Vato92] to solve some of these variants, analyzing their pitfalls. In section 3, we focus on the Frobenius variant and present the key result that relates 3
local minima solutions with the well known Total Least Square problem. In section 4, we refer to heuristic approaches and also to a simple method to find local optima by means of unconstrained minimization, and show how to speed up the search by integrating it with an algebraic approach. In section 5 we propose a tree-search approach to obtain global optima, as well as criteria to prune the search. We conclude with some remarks on the experimental results of the previous two sections, as well as some directions for future research.
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The correction of linear inequalities
Consider a system of linear inequalities Ax ≤ b, x ∈
