An efficient algorithm for a sharp approximation of universally quantified inequalities

Abstract

This paper introduces a new algorithm for solving a subclass of quantified constraint satisfaction problems (QCSP) where existential quantifiers precede universally quantified inequalities on continuous domains. This class of QCSPs has numerous applications in engineering and design. We propose here a new generic branch and prune algorithm for solving such continuous QCSPs. Standard pruning operators and solution identification operators are specialized for universally quantified inequalities. Special rules are also proposed for handling the parameters of the constraints. First experimentation show that our algorithm outperforms the state of the art methods.

Efficient handling of universally quantified inequalities

“…1], and x = [0, 15]. The solution set of this very simple CSP is the interval [9,15]. To reject values of x that do not satisfy c(x), we apply Contract f (x,0.5)≤0 (x), which reduces x to x = [4.75, 15].…”

“…Usually this inclusion is strict, meaning that bisecting the domains actually decreases the pessimism of the interval evaluations. The heuristic we propose is to define a threshold on the ratio wid( (z 1 ∪ z 2 )) widz (15) over which the parameters domains are not bisected. Such a decision relies on three interval evaluations of the function f , which turns out to be cheap w.r.t.…”

“…Therefore, the constraint ∀y ∈ y, f (x, y) ≤ 0 is equivalent to f (x, 1) ≤ 0. Then, the pruning contracts x = [0, 15] to [9,15] and the solution identification proves that [9,15] contains only solutions. On this simple example, we obtain an exact description of the solution set.…”