Mohit Tawarmalani

A polyhedral branch-and-cut approach to global optimization

Abstract.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.

Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming

Preface. Acknowledgements. List of Figures. List of Tables. 1. Introduction. 2. Convex Extensions. 3. Project Disaggregation. 4. Relaxations of Factorable Programs. 5. Domain Reduction. 6. Node Partitioning. 7. Implementation. 8. Refrigerant Design Problem. 9. The Pooling Problem. 10. Miscellaneous Problems. 11. GAMS/BARON: A Tutorial. A: GAMS Model for Pooling Problems. Bibliography. Index. Author Index.

Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

Abstract.This work addresses the development of an efficient solution strategy for obtaining global optima of continuous, integer, and mixed-integer nonlinear programs. Towards this end, we develop novel relaxation schemes, range reduction tests, and branching strategies which we incorporate into the prototypical branch-and-bound algorithm. In the theoretical/algorithmic part of the paper, we begin by developing novel strategies for constructing linear relaxations of mixed-integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties. We then use Lagrangian/linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming. This theory leads to new range reduction schemes, including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch-and-bound tree. Finally, we incorporate these relaxation and reduction strategies in a branch-and-bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems. In the computational part of the paper, we describe our implementation discussing, wherever appropriate, the use of suitable data structures and associated algorithms. We present computational experience with benchmark separable concave quadratic programs, fractional 0–1 programs, and mixed-integer nonlinear programs from applications in synthesis of chemical processes, engineering design, just-in-time manufacturing, and molecular design.

Cloudward bound: planning for beneficial migration of enterprise applications to the cloud

In this paper, we tackle challenges in migrating enterprise services into hybrid cloud-based deployments, where enterprise operations are partly hosted on-premise and partly in the cloud. Such hybrid architectures enable enterprises to benefit from cloud-based architectures, while honoring application performance requirements, and privacy restrictions on what services may be migrated to the cloud. We make several contributions. First, we highlight the complexity inherent in enterprise applications today in terms of their multi-tiered nature, large number of application components, and interdependencies. Second, we have developed a model to explore the benefits of a hybrid migration approach. Our model takes into account enterprise-specific constraints, cost savings, and increased transaction delays and wide-area communication costs that may result from the migration. Evaluations based on real enterprise applications and Azure-based cloud deployments show the benefits of a hybrid migration approach, and the importance of planning which components to migrate. Third, we shed insight on security policies associated with enterprise applications in data centers. We articulate the importance of ensuring assurable reconfiguration of security policies as enterprise applications are migrated to the cloud. We present algorithms to achieve this goal, and demonstrate their efficacy on realistic migration scenarios.

A finite branch-and-bound algorithm for two-stage stochastic integer programs

Abstract.This paper addresses a general class of two-stage stochastic programs with integer recourse and discrete distributions. We exploit the structure of the value function of the second-stage integer problem to develop a novel global optimization algorithm. The proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination. Computational experiments on standard test problems indicate superior performance of the proposed algorithm in comparison to those in the existing literature.

Convex extensions and envelopes of lower semi-continuous functions

Abstract. 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

Semidefinite Relaxations of Fractional Programs via Novel Convexification Techniques

x/y

Multiterm polyhedral relaxations for nonconvex, quadratically constrained quadratic programs

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.

Explicit convex and concave envelopes through polyhedral subdivisions

In a recent work, we introduced the concept of convex extensions for lower semi-continuous functions and studied their properties. In this work, we present new techniques for constructing convex and concave envelopes of nonlinear functions using the theory of convex extensions. In particular, we develop the convex envelope and concave envelope of z=x/y over a hypercube. We show that the convex envelope is strictly tighter than previously known convex underestimators of x/y. We then propose a new relaxation technique for fractional programs which includes the derived envelopes. The resulting relaxation is shown to be a semidefinite program. Finally, we derive the convex envelope for a class of functions of the type f(x,y) over a hypercube under the assumption that f is concave in x and convex in y.

Global Optimization of 0-1 Hyperbolic Programs

This article addresses the generation of strong polyhedral relaxations for nonconvex, quadratically constrained quadratic programs (QCQPs). Using the convex envelope of multilinear functions as our starting point, we develop a polyhedral relaxation for QCQP, along with a cutting plane algorithm for its implementation. Our relaxations are multiterm, i.e. they are derived from the convex envelope of the sum of multiple bilinear terms of quadratic constraints, thereby providing tighter bounds than the standard termwise relaxation of the bilinear functions. Computational results demonstrate the usefulness of the proposed cutting planes.