Paula Amaral

A reformulation-linearization-convexification algorithm for optimal correction of an inconsistent system of linear constraints

In this paper, an algorithm is introduced to find an optimal solution for an optimization problem that arises in total least squares with inequality constraints, and in the correction of infeasible linear systems of inequalities. The stated problem is a nonconvex program with a special structure that allows the use of a reformulation-linearization-convexification technique for its solution. A branch-and-bound method for finding a global optimum for this problem is introduced based on this technique. Some computational experiments are included to highlight the efficacy of the proposed methodology. Inconsistent systems play a major role on the reformulation of models and are a consequence of lack of communication between decision makers. The problem of finding an optimal correction for some measure is of crucial importance in this context. The use of the Frobenius norm as a measure seems to be quite natural and leads to a nonconvex fractional programming problem. Despite being a difficult global optimization, it is possible to process it by using a branch-and-bound algorithm incorporating a local nonlinear programming method.

A Framework for Optimal Correction of Inconsistent Linear Constraints

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.

Global optimization simplex bisection revisited based on considerations by reiner horst

In this paper, the use of non-optimality spheres in a simplicial branch and bound (B&B) algorithm is investigated. In this context, some considerations regarding the use of bisection on the longest edge in relation with ideas of Reiner Horst are reminded. Three arguments highlight the merits of bisection of simplicial subsets in B&B schemes.

Compromise ratio with weighting functions in a Tabu Search multi-criteria approach to examination timetabling

University examination scheduling is a difficult and heavily administrative task, particularly when the number of students and courses is high. Changes in educational paradigms, an increase in the number of students, the aggregation of schools, more flexible curricula, among others, are responsible for an increase in the difficulty of the problem. As a consequence, there is a continuous demand for new and more efficient approaches. Optimisation and Constraint Programming communities have devoted considerable attention to this difficult problem. Just the definition of a satisfactory, not to mention optimal, timetabling may be complex. In fact, to characterise a timetabling solution, a single criteria may not be enough, since what may be considered good for one group of students may be regarded inappropriate for other students, or teachers. In this paper, four criteria were used to characterise the spreading of the exams over the examination period. A set of constraints regarding the non-overlapping of exams with students in common was considered. A multi-objective optimisation program was used to handle the four criteria and a Tabu Search was implemented to find a good feasible solution for this problem. Two new features to increase the automation of the algorithm were proposed. First, it uses a Fuzzy Inference Ruled Based System to choose the tabu tenure of the elements in the tabu list. Secondly, a modified version of the Compromise Ratio (CR) is proposed, where the usual fixed weights are replaced by weighting functions to rank the neighbourhood solutions in each iteration. Sufficient conditions which guarantee the monotonicity of the weighting functions are presented. HighlightsAutomation is one of the main motivations of the work presented in this paper.Implementation of automatic tuning strategy for the tabu tenure in the TS.A multi-criteria problem is addressed in every iteration for selecting the new solution.Combining Compromise Ratio and weighting functions avoiding continuous human interference.New results for sufficient conditions for the monotonicity of the combined operator.Possible generalizations to other problems.Modifications introduced in the Compromise Ratio can be applied to general multi-attribute problems.