Nikolaos V. Sahinidis

Optimization under uncertainty: state-of-the-art and opportunities

Abstract A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Uncertainty, for instance, governs the prices of fuels, the availability of electricity, and the demand for chemicals. A key difficulty in optimization under uncertainty is in dealing with an uncertainty space that is huge and frequently leads to very large-scale optimization models. Decision-making under uncertainty is often further complicated by the presence of integer decision variables to model logical and other discrete decisions in a multi-period or multi-stage setting. This paper reviews theory and methodology that have been developed to cope with the complexity of optimization problems under uncertainty. We discuss and contrast the classical recourse-based stochastic programming, robust stochastic programming, probabilistic (chance-constraint) programming, fuzzy programming, and stochastic dynamic programming. The advantages and shortcomings of these models are reviewed and illustrated through examples. Applications and the state-of-the-art in computations are also reviewed. Finally, we discuss several main areas for future development in this field. These include development of polynomial-time approximation schemes for multi-stage stochastic programs and the application of global optimization algorithms to two-stage and chance-constraint formulations.

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.

Global optimization of nonconvex NLPs and MINLPs with applications in process design

Abstract This paper presents an algorithm for finding global solutions of nonconvex nonlinear programs (NLPs) and mixed-integer nonlinear programs (MINLPs). The approach is based on the solution of a sequence of convex underestimating subproblems generated by evolutionary subdivision of the search region. The key components of the algorithm are new optimality-based and feasibility-based range reduction tests. The former use known feasible solutions and perturbation results to exclude inferior parts of the search region from consideration, while the latter analyze constraints to obtain valid inequalities. Furthermore, the algorithm integrates these devices with an efficient local search heuristic. Computational results demonstrate that the algorithm compares very favorably to several other current approaches when applied to a large collection of global optimization and process design problems. It is typically faster, requires less storage and it produces more accurate results.

A branch-and-reduce approach to global optimization

This paper presents valid inequalities and range contraction techniques that can be used to reduce the size of the search space of global optimization problems. To demonstrate the algorithmic usefulness of these techniques, we incorporate them within the branch-and-bound framework. This results in a branch-and-reduce global optimization algorithm. A detailed discussion of the algorithm components and theoretical properties are provided. Specialized algorithms for polynomial and multiplicative programs are developed. Extensive computational results are presented for engineering design problems, standard global optimization test problems, univariate polynomial programs, linear multiplicative programs, mixed-integer nonlinear programs and concave quadratic programs. For the problems solved, the computer implementation of the proposed algorithm provides very accurate solutions in modest computational time.

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.

GPU-BLAST

Motivation: The Basic Local Alignment Search Tool (BLAST) is one of the most widely used bioinformatics tools. The widespread impact of BLAST is reflected in over 53 000 citations that this software has received in the past two decades, and the use of the word ‘blast’ as a verb referring to biological sequence comparison. Any improvement in the execution speed of BLAST would be of great importance in the practice of bioinformatics, and facilitate coping with ever increasing sizes of biomolecular databases. Results: Using a general-purpose graphics processing unit (GPU), we have developed GPU-BLAST, an accelerated version of the popular NCBI-BLAST. The implementation is based on the source code of NCBI-BLAST, thus maintaining the same input and output interface while producing identical results. In comparison to the sequential NCBI-BLAST, the speedups achieved by GPU-BLAST range mostly between 3 and 4. Availability: The source code of GPU-BLAST is freely available at http://archimedes.cheme.cmu.edu/biosoftware.html. Contact: sahinidis@cmu.edu Supplementary information: Supplementary data are available at Bioinformatics online.

Optimization model for long range planning in the chemical industry

Abstract In this paper a multiperiod MILP model is presented for the optimal selection and expansion of processes given time varying forecasts for the demands and prices of chemicals over a long range horizon. To reduce the computational expense of solving this long range planning problem, several strategies are investigated, including branch and bound, the use of integer cuts, strong cutting planes, Benders decomposition and heuristics. These procedures, which have been implemented in the program MULPLAN, are illustrated with several example problems. As is shown, the proposed model is especially useful for the study of a variety of different scenarios.

MINLP model for cyclic multiproduct scheduling on continuous parallel lines

Abstract This paper addresses the problem of cyclic multiproduct scheduling on continuous parallel production lines. This plant configuration is typically used in the manufacturing of specialty chemicals. The problem involves a combinatorial part (assignment of products to lines and their sequencing in each li and a continuous part (duration of production runs and frequency of production). To account for these two elements, a large-scale mixed integer nonline program (MINLP) is developed and an exact reformulation technique is applied to linearize it in the space of the integer variables. The Kuhn-Tucker optimality conditions are exploited in the resulting model in order to effectively apply generalized Benders decomposition. This avoids the explicit solution of extremely large nonlinear subproblems. At the same time, a computational scheme based on valid outer-approximations of the nonlinear part of the problem is proposed to strengthen the bounds of the master problem and therefore achieve fast convergence. Despite the nonlinear nature of the model, special but important cases are identified for which certain convexity conditions are satisfied and the suggested procedure guarantees the globa optimality of the solution. The proposed technique was applied to a real world problem for a polymer production plant. The corresponding MINLP containe 780 binary variables, 23,000 continuous variables and 3200 constraints.