Rigorous filtering using linear relaxations

Abstract

Abstract. This paper presents rigorous filtering methods for continuous constraint satisfaction problems based on linear relaxations. Filtering or pruning stands for reducing the search space of constraint satisfaction problems. Discussed are old and new approaches for rigorously enclosing the solution set of linear systems of inequalities, as well as different methods for computing linear relaxations. This allows custom combinations of relaxation and filtering. Care is taken to ensure that all methods correctly account for rounding errors in the computations. Although most of the results apply more generally, strong emphasis is given to relaxing and filtering quadratic constraints, as implemented in the GloptLab environment, which internally exploits a quadratic structure. Demonstrative examples and tests comparing the different linear relaxation methods are also presented.

Using interval unions to solve linear systems of equations with uncertainties

“…The midpoint preconditioner takes the inverse of the midpoint of the interval hull of A and the Gauss-Jordan preconditioner that is based on the interval version of this method discussed by [6]. We also propose a mixed strategy that combines the original system and the Gauss-Jordan preconditioner to improve the efficiency and the quality of solutions, see the Algorithm 4.…”

“…In the interval case, the midpoint preconditioner is the common choice in a number of problems. Optimal linear programming preconditioners are designed by [14] in the context of the interval Newton operator and the Gauss-Jordan preconditioner is proposed by [6]. See also [10] and [15] for recent methods on optimal preconditioning.…”

“…The first one is the midpoint method: it takes the inverse of the midpoint of the hull matrix of the system A as the preconditioner. The second one is the Gauss-Jordan preconditioner which is based on the Gauss-Jordan elimination as discussed in [6].…”

An extension of the $$\alpha \hbox {BB}$$ α BB -type underestimation to linear parametric Hessian matrices

“…A generalization of this method using non-diagonal shifting appeared in Akrotirianakis et al [5] and Skjäl et al [38,39], and a different form of an understimator using exponentials instead of a quadratic function as a perturbation function was discussed in [3,4,14,15]. Other forms of convex and linear relaxations were investigated in Anstreicher [7], Domes and Neumaier [10], and Scott et al [37], for instance. The Hessian of g(x) reads…”

On the efficient Gerschgorin inclusion usage in the global optimization $$\alpha \hbox {BB}$$ α BB method

“…A global optimization method QBB based on convex underestimators and branch & bound scheme on simplices, was proposed in [28]. Other convex and linear relaxations were investigated in [7,8,26], for instance. Let us consider the classical αBB approach utilizing the form (1).…”