6 research outputs found

    Heuristics with Performance Guarantees for the Minimum Number of Matches Problem in Heat Recovery Network Design

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    Heat exchanger network synthesis exploits excess heat by integrating process hot and cold streams and improves energy efficiency by reducing utility usage. Determining provably good solutions to the minimum number of matches is a bottleneck of designing a heat recovery network using the sequential method. This subproblem is an NP-hard mixed-integer linear program exhibiting combinatorial explosion in the possible hot and cold stream configurations. We explore this challenging optimization problem from a graph theoretic perspective and correlate it with other special optimization problems such as cost flow network and packing problems. In the case of a single temperature interval, we develop a new optimization formulation without problematic big-M parameters. We develop heuristic methods with performance guarantees using three approaches: (i) relaxation rounding, (ii) water filling, and (iii) greedy packing. Numerical results from a collection of 51 instances substantiate the strength of the methods

    Symmetry and degeneracy in nonconvex optimisation problems: Application to heat recovery networks

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    Many optimisation problems are formulated with nonconvexities in the objective function and the set of constraints. Nonconvex optimisation has applications in a wide range of disciplines and this thesis examines scheduling and process network design problems. Two main solution approaches are used to deal with such problems: exact and approximation algorithms. Exact algorithms guarantee to solve a problem to global optimality but may require exponential time. On the other hand, approximation algorithms can generate near-optimal solutions in reasonable time. Both sets of algorithms could benefit from insights on the special structure of optimisation problems, e.g. symmetry and degeneracy. This thesis proposes novel structures i.e. matrices and graphs, for detecting symmetry in Quadratically Constrained Quadratic Programs. In several critically important engineering applications, such as Heat Exchanger Network Synthesis (HENS), symmetry and degeneracy have not been characterised yet. This work investigates the minimum number of matches, e.g. heat exchanger units, which is the current bottleneck in designing HENS. We classify special cases with many equivalent optimal solutions and define symmetry and degeneracy. Due to the aforementioned complexities, we report via computational results that state-of-the-art approaches cannot solve the minimum number of matches problem to global optimality for moderately-sized instances. Hence this thesis develops three classes of heuristics with performance guarantees to the minimum number of matches problem. Each of these heuristics is either novel or provably the best in its class. Our work has interesting implications for solving the problem exactly, e.g. the analysis into reducing big-M parameters or the possibility of quickly generating good primal feasible solutions. Detecting special structures in optimisation problems and dealing with instances of HENS is neither trivial nor easy. This thesis provides an in-depth analysis of these problems and develops fundamental tools to efficiently solve challenging optimisation problems via both exact and approximation approaches.Open Acces

    Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs

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    Symmetry in mathematical optimization may create multiple, equivalent solutions. In nonconvex optimization, symmetry can negatively affect algorithm performance, e.g., of branch-and-bound when symmetry induces many equivalent branches. This paper develops detection methods for symmetry groups in quadratically-constrained quadratic optimization problems. Representing the optimization problem with adjacency matrices, we use graph theory to transform the adjacency matrices into binary layered graphs. We enter the binary layered graphs into the software package nauty that generates important symmetric properties of the original problem. Symmetry pattern knowledge motivates a discretization pattern that we use to reduce computation time for an approximation of the point packing problem. This paper highlights the importance of detecting and classifying symmetry and shows that knowledge of this symmetry enables quick approximation of a highly symmetric optimization problem