A polyhedral branch-and-cut approach to global optimization

A variety of nonlinear, including semidefinite, relaxations have been developed in recent years for nonconvex optimization problems. Their potential can be realized only if they can be solved with sufficient speed and reliability. Unfortunately, state-of-the-art nonlinear programming codes are significantly slower and numerically unstable compared to linear programming software.In this paper, we facilitate the reliable use of nonlinear convex relaxations in global optimization via a polyhedral branch-and-cut approach. Our algorithm exploits convexity, either identified automatically or supplied through a suitable modeling language construct, in order to generate polyhedral cutting planes and relaxations for multivariate nonconvex problems. We prove that, if the convexity of a univariate or multivariate function is apparent by decomposing it into convex subexpressions, our relaxation constructor automatically exploits this convexity in a manner that is much superior to developing polyhedral outer approximators for the original function. The convexity of functional expressions that are composed to form nonconvex expressions is also automatically exploited.Root-node relaxations are computed for 87 problems from globallib and minlplib, and detailed computational results are presented for globally solving 26 of these problems with BARON 7.2, which implements the proposed techniques. The use of cutting planes for these problems reduces root-node relaxation gaps by up to 100% and expedites the solution process, often by several orders of magnitude.The research was supported in part by ExxonMobil Upstream Research Company, the National Science Foundation under awards DMII 0115166 and CTS 0124751, and the Joint NSF/NIGMS Initiative to Support Research in the Area of Mathematical Biology under NIH award GM072023.     

Convex extensions and envelopes of lower semi-continuous functions

 We define a convex extension of a lower semi-continuous function to be a convex function that is identical to the given function over a pre-specified subset of its domain. Convex extensions are not necessarily constructible or unique. We identify conditions under which a convex extension can be constructed. When multiple convex extensions exist, we characterize the tightest convex extension in a well-defined sense. Using the notion of a generating set, we establish conditions under which the tightest convex extension is the convex envelope. Then, we employ convex extensions to develop a constructive technique for deriving convex envelopes of nonlinear functions. Finally, using the theory of convex extensions we characterize the precise gaps exhibited by various underestimators of $x/y$ over a rectangle and prove that the extensions theory provides convex relaxations that are much tighter than the relaxation provided by the classical outer-linearization of bilinear terms. Received: December 2000 / Accepted: May 2002 Published online: September 5, 2002 RID="*" ID="*" The research was funded in part by a Computational Science and Engineering Fellowship to M.T., and NSF CAREER award (DMI 95-02722) and NSF/Lucent Technologies Industrial Ecology Fellowship (NSF award BES 98-73586) to N.V.S. Key words. convex hulls and envelopes – multilinear functions – disjunctive programming – global optimization     

Performance studies of compact GGAG:Ce,B thermal neutron detector coupled to Si-based photosensors

Pramana - This work concerns the modelling of the dynamics of polarisation of a centrosymmetric material subjected to an amplitude-modulated excitation (AME). The dynamic equation obtained for the...     

Uncatalyzed Michael addition of indoles: synthesis of some novel 3-alkylated indoles via a three-component reaction in solvent-free conditions

Synthesis of some novel 3-alkylated indoles via an uncatalyzed Michael addition of indoles using three components in one-pot solvent-free conditions is reported. The mechanism was established by performing the reaction in two steps. The reaction was also studied in different solvents and an important solvent effect was noticed.