Improved and Simplified Validation of Feasible Points Inequality and Equality Constrained Problems

In validated branch and bound algorithms for global optimization, upper bounds on the global optimum are obtained by evaluating the objective at an approximate optimizer; the upper bounds are then used to eliminate subregions of the search space. For constrained optimization, in general, a small region must be constructed within which existence of a feasible point can be proven, and an upper bound on the objective over that region is obtained. We had previously proposed a perturbation technique for constructing such a region. In this work, we propose a much simplified and improved technique, based on an orthogonal decomposition of the normal space to the constraints. In purely inequality constrained problems, a point, rather than a region, can be used, and, for equality and inequality constrained problems, the region lies in a smaller-dimensional subspace, giving rise to sharper upper bounds. Numerical experiments on published test sets for global optimization provide evidence of the superiority of the new approach within our GlobSol environment.