Optimization of Critical Infrastructure with Fluids

Many of the world's most critical infrastructure systems control the motion of fluids. Despite their importance, the design, operation, and restoration of these infrastructures are sometimes carried out suboptimally. One reason for this is the intractability of optimization problems involving fluids, which are often constrained by partial differential equations or nonconvex physics. To address these challenges, this dissertation focuses on developing new mathematical programming and algorithmic techniques for optimization problems involving difficult nonlinear constraints that model a fluid's behavior. These new contributions bring many important problems within the realm of tractability. The first focus of this dissertation is on surface water systems. Specifically, we introduce the Optimal Flood Mitigation Problem, which optimizes the positioning of structural measures to protect critical assets with respect to a predefined flood scenario. Two solution approaches are then developed. The first leverages mathematical programming but does not tractably scale to realistic scenarios. The second uses a physics-inspired metaheuristic, which is found to compute good quality solutions for realistic scenarios. The second focus is on potable water distribution systems. Two foundational problems are considered. The first is the optimal water network design problem, for which we derive a novel convex reformulation, then develop an algorithm found to be more effective than the current state of the art on select instances. The second is the optimal pump scheduling (or Optimal Water Flow) problem, for which we develop a mathematical programming relaxation and various algorithmic techniques to improve convergence. The final focus is on natural gas pipeline systems. Two novel problems are considered. The first is the Maximal Load Delivery (MLD) problem for gas pipelines, which aims at finding a feasible steady-state operating point that maximizes load delivery for a severely damaged gas network. The second is the joint gas-power MLD problem, which couples damaged gas and power networks at gas-fired generators. In both problems, convex relaxations of nonconvex dynamical constraints are developed to increase tractability.PHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169849/1/tasseff_1.pd

Recursive McCormick Linearization of Multilinear Programs

Linear programming (LP) relaxations are widely employed in exact solution methods for multilinear programs (MLP). One example is the family of Recursive McCormick Linearization (RML) strategies, where bilinear products are substituted for artificial variables, which deliver a relaxation of the original problem when introduced together with concave and convex envelopes. In this article, we introduce the first systematic approach for identifying RMLs, in which we focus on the identification of linear relaxation with a small number of artificial variables and with strong LP bounds. We present a novel mechanism for representing all the possible RMLs, which we use to design an exact mixed-integer programming (MIP) formulation for the identification of minimum-size RMLs; we show that this problem is NP-hard in general, whereas a special case is fixed-parameter tractable. Moreover, we explore structural properties of our formulation to derive an exact MIP model that identifies RMLs of a given size with the best possible relaxation bound is optimal. Our numerical results on a collection of benchmarks indicate that our algorithms outperform the RML strategy implemented in state-of-the-art global optimization solvers.Comment: 22 pages, 11 figures, Under Revie

On the interplay of Mixed Integer Linear, Mixed Integer Nonlinear and Constraint Programming

In this thesis we study selected topics in the field of Mixed Integer Programming (MIP), in particular Mixed Integer Linear and Nonlinear Programming (MI(N)LP). We set a focus on the influences of Constraint Programming (CP). First, we analyze Mathematical Programming approaches to water network optimization, a set of challenging optimization problems frequently modeled as non-convex MINLPs. We give detailed descriptions of many variants and survey solution approaches from the literature. We are particularly interested in MILP approximations and present a respective computational study for water network design problems. We analyze this approach by algorithmic considerations and highlight the importance of certain convex substructures in these non-convex MINLPs. We further derive valid inequalities for water network design problems exploiting these substructures. Then, we treat Mathematical Programming problems with indicator constraints, recalling their most popular reformulation techniques in MIP, leading to either big-M constraints or disjunctive programming techniques. The latter give rise to reformulations in higher-dimensional spaces, and we review special cases from the literature that allow to describe the projection on the original space of variables explicitly. We theoretically extend the respective results in two directions and conduct computational experiments. We then present an algorithm for MILPs with indicator constraints that incorporates elements of CP into MIP techniques, including computational results for the JobShopScheduling problem. Finally, we introduce an extension of the class of MILPs so that linear expressions are allowed to have non-contiguous domains. Inspired by CP, this permits to model holes in the domains of variables as a special case. For such problems, we extend the theory of split cuts and show two ways of separating them, namely as intersection and lift-and-project cuts, and present computational results. We further experiment with an exact algorithm for such problems, applied to the Traveling Salesman Problem with multiple time windows

Short-term supply chain management in upstream natural gas systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2009.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 253-267).Natural gas supply chain planning and optimization is important to ensure security and reliability of natural gas supply. However, it is challenging due to the distinctive features of natural gas supply chains. These features arise from the low volumetric energy density of natural gas and the significance of gas quality and pressure in supply chain operations. Contracts play a central role in the entire supply chain due to high capital cost, specificity and investment risks associated with gas infrastructure. An upstream production planning framework is crucial for supply-side optimization and scenario evaluation in the natural gas supply chain. The technical features of upstream systems imply that the most efficient mode of operation is by single entity central control of the system, while their economics favor involvement of multiple parties in ownership. To resolve this conflict, upstream systems are generally operated by a single operator on the basis of governing rules that stem from agreements between the upstream operator, multiple stakeholders and consumer facilities. These agreements govern production sharing, operational strategy and gas sales in the upstream system. A short-term operational planning framework (with a 2-12 weeks planning horizon) for upstream natural gas systems is presented that can help to maximize production infrastructure utilization and aid in its management, minimize costs and meet production targets while simultaneously satisfying governing rules. Its requirements are inspired by the Sarawak Gas Production System (SGPS), an offshore gas production system in the South China Sea, which supplies the liquefied natural gas (LNG) plant complex at Bintulu in East Malaysia. This is the first attempt to formulate a comprehensive modeling framework for an upstream gas production system that includes a production infrastructure model and a methodology to incorporate governing rules.(cont.) The model has two components: the infrastructure model is a model of the physical system, i.e., of wells, trunkline network and facilities while the contractual model is a mathematical representation of the governing rules, e.g., production-sharing contracts (PSC), customer specifications and operational rules. The model formulation and objectives are from the perspective of the upstream operator. The infrastructure model incorporates the capability to track multiple qualities of gas throughout the network and determine the optimal routing and blending of gas such that the quality specifications are satisfied at the demand nodes. Nonlinear pressure-flowrate relationships in wells and the network are included for predicting a sufficiently accurate pressure-flowrate profile thereby facilitating implementation of the production strategy on the network. Modeling of complex platform configurations with reversible lines, lines that can be shut-off in normal operation and compression facilities, further improve the realistic representation of the network. A simplified prediction of natural gas liquids (NGL) production is included to maximize NGL revenue. The contractual model represents the framework for modeling the governing rules that are central to the operation of upstream systems. Modeling of productionsharing contracts is a two-fold challenge: accounting for gas volumes and converting the logical rules as stated in the system operations manual to binary constraints. A PSC network representation is proposed to account for gas volumes as well as interactions between different PSC. PSC rules are expressed as logical expressions in terms of availability, priority and transfer Boolean-states, and converted to binary constraints. Additional logical constraints are required to model the inference and intent of the rules. Operational rules can be modeled within the same framework.(cont.) The resulting mathematical program is a mixed-integer nonlinear program (MINLP) with nonconvex functions and can be solved with the current state-of-the-art global optimization approaches, provided careful attention is paid to the model formulation.A hierarchical multi-objective approach is proposed to address multiple objectives when operating upstream systems, by optimizing a lower priority objective over the multiple optimal solutions of a program with a higher priority objective to obtain a win-win scenario. A reproducible case study that captures all the features of natural gas upstream systems is constructed to facilitate future work in algorithm development for such problems. A preliminary comparison with the existing approach indicates that substantial benefits may be possible by using the proposed approach for short-term planning. The application of a reduced-space global optimization approach to planning in upstream gas networks has also been demonstrated, which can significantly lower the number of variables in the branch-and-bound algorithm. The lower bounding problem is implemented using McCormick (convex) relaxations of computer evaluated functions and solved by implementing a nonsmooth bundle solver as a linearization tool to obtain a linear programming relaxation. The upper bounding problem is implemented using automatic differentiation and a local NLP solver. Branch-and-bound with reduction heuristics and linearization propagation is used for global optimization.This approach has been found to be competitive with current state-of-the-art global optimization algorithms for upstream planning problems.by Ajay Selot.Ph.D