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In this paper we explore the construction of arbitrarily tight  α  BB relaxations of   C  2   general non-linear non-convex functions. We illustrate the theoretical challenges of building such relaxations by deriving conditions under which it is possible for an  α  BB underestimator to provide exact bounds. We subsequently propose a methodology to build  α  BB underestimators which may be arbitrarily tight (i.e., the maximum separation distance between the original function and its underestimator is arbitrarily close to 0) in some domains that do not include the global solution (defined in the text as "sub-optimal"), assuming exact eigenvalue calculations are possible. This is achieved using a transformation of the original function into a  μ  - subenergy function and the derivation of  α  BB underestimators for the new function. We prove that this transformation results in a number of desirable bounding properties in certain domains. These theoretical results are validated in computational test cases where approximations of the tightest possible  μ  -subenergy underestimators, derived using sampling, are compared to similarly derived approximations of the tightest possible classical  α  BB underestimators. Our tests show that  μ  -subenergy underestimators produce much tighter bounds, and succeed in fathoming nodes which are impossible to fathom using classical  α  BB.

Arbitrarily tight $$\alpha $$ α BB underestimators of general non-linear functions over sub-optimal domains