Rigorous verification of feasibility

Abstract

This paper considers the problem of finding and verifying feasible points for constraint satisfaction problems, including those with uncertain coefficients. The main part is devoted to the problem of finding a narrow box around an approximately feasible solution for which it can be rigorously and automatically proved that it contains a feasible solution. Some examples demonstrate difficulties when attempting verification.We review some existing methods and present a new method for verifying the existence of feasible points of constraint satisfaction problems in an automatically determined narrow box. Numerical tests within GloptLab, a solver developed by the authors, show how the different methods perform. Also discussed are the search for approximately feasible points and the use of approximately feasible points within a branch and bound scheme for constraint satisfaction.

Convergent upper bounds in global minimization with nonlinear equality constraints

“…Compared to numerical results in the literature, for some of the problems our algorithm performs better than the majority of the methods analyzed in [9] (bt1, extrasim, . For two problems, hs028 and aljazzaf, our method is even the only one to successfully verify the existence of a feasible point and provide a valid upper bound for v * .…”

“…For two problems, hs028 and aljazzaf, our method is even the only one to successfully verify the existence of a feasible point and provide a valid upper bound for v * . The only two problems, for which the methods presented in [9] clearly perform better are kear2 and powell, for which the existence of a feasible point cannot be verified with our algorithm, as Assumption 1.1 is violated.…”

“…Our implementation is tested for a selection of problems from the COCONUT benchmark [40]. All selected problems are also examined by Domes and Neumaier in [9]. In the case that a test problem did not include any box constraints, these were added to the problem to make it applicable to our purposes.…”

“…In contrast, our focus is on deterministic methods. In context of rigorous upper bound determination, such approaches are introduced by Kearfott [19,20] and Domes and Neumaier [9]. They determine approximately feasible points with conventional nonlinear solvers and then construct boxes around such points, for which the existence of a feasible point can be proven via interval Newton methods.…”

Deterministic upper bounds for spatial branch-and-bound methods in global minimization with nonconvex constraints

“…In contrast, we handle continuously differentiable optimization problems, and we can generate feasible points at a much lower computational cost. Further ideas that address similar problems can be found in Domes and Neumaier (2015) and Kearfott (1998Kearfott ( , 2014.…”