Global optimization with piecewise linear approximation

textGlobal optimization deals with the development of solution methodologies for nonlinear nonconvex optimization problems. These problems, which could arise in diverse situations ranging from optimizing hydro-power generation schedules to estimating coefficients of non-linear regression models, are difficult for traditional nonlinear solvers that iteratively search the neighborhood around a starting point. The Piecewise Linear Approximation (PLA) method that we study in this dissertation seeks to generate ‘good’ starting points, hopefully ones that lie in the basin of attraction of the globally optimal solution. In this approach, we approximate the non-linear functions in the optimization problem by piecewise linear functions defined over the vertices of a grid that partitions the domain of each nonlinear function into cells. Based on this approximation, we convert the original nonlinear program into a mixed integer program (MIP) and use the solution to this MIP as a starting point for a local nonlinear solver. In this dissertation, we validate the effectiveness of the PLA approach as a global optimization approach by applying it to a diverse set of continuous and discrete nonlinear optimization problems. Further, we develop various modeling and algorithmic strategies for enhancing the basic approach. Our computational results demonstrate that the PLA approach works well on non-convex problems and can, in some cases, provide better solutions than those provided by existing nonlinear solvers.Information, Risk, and Operations Management (IROM

A game-theoretic optimisation approach to fair customer allocation in oligopolies

Under the ever-increasing capital intensive environment that contemporary process industries face, oligopolies begin to form in mature markets where a small number of companies regulate and serve the customer base. Strategic and operational decisions are highly dependent on the firms’ customer portfolio and conventional modelling approaches neglect the rational behaviour of the decision makers, with regards to the problem of customer allocation, by assuming either static competition or a leader-follower structure. In this article, we address the fair customer allocation within oligopolies by employing the Nash bargaining approach. The overall problem is formulated as mixed integer program with linear constraints and a nonlinear objective function which is further linearised following a separable programming approach. Case studies from the industrial liquid market highlight the importance and benefits of the proposed game theoretic approach

GALINI: an extensible solver for mixed-integer quadratically-constrained problems

Many industrial relevant optimization problems can be formulated as Mixed-Integer Quadratically Constrained Problems. This class of problems are difficult to solve because of 1) the non-convex bilinear terms 2) integer variables. This thesis develops the Python library \suspect{} for detecting special structure (monotonicity and convexity) of Pyomo models. This library can be extended to provide specialized detection for complex expressions. As a motivating example, we show how the library can be used to detect the convexity of the reciprocal of the log mean temperature difference. This thesis introduces GALINI: a novel solver that is easy to extend at runtime with new 1) cutting planes, 2) primal heuristics, 3) branching strategies, 4) node selection strategies, and 5) relaxations.GALINI uses Pyomo to represent optimization problems, this decision makes it possible to integrate with the existing Pyomo ecosystem to provide, for example, building blocks for relaxations. We test the solver on two large datasets and show that the performance is comparable to existing open source solvers. Finally, we present a library to formulate pooling problems, a class of network flow problems, using Pyomo. The library provides a mechanism to automatically generate the PQ-formulation for pooling problems. Since the library keeps the knowledge of the original network, it can 1) use a mixed-integer programming primal heuristic specialized for the pooling problem to find a feasible solution, and 2) generate valid cuts for the pooling problem. We use this library to develop an extension for GALINI that uses the mixed-integer programming primal heuristic to find a feasible solution and that generates cuts at every node of the branch & cut algorithm. We test GALINI with the pooling extensions on large scale instances of the pooling problem and show that we obtain results that are comparable to or better than the best available commercial solver on dense instances.Open Acces

Modeling and Optimization of Gas Networks in Refinery

Master'sMASTER OF ENGINEERIN

Deterministic global optimization approach to bilinear process network synthesis

Master'sMASTER OF ENGINEERIN

Advances in multi-parametric mixed-integer programming and its applications

At many stages of process engineering we are confronted with data that have not yet revealed their true values. Uncertainty in the underlying mathematical model of real processes is common and poses an additional challenge on its solution. Multi-parametric programming is a powerful tool to account for the presence of uncertainty in mathematical models. It provides a complete map of the optimal solution of the perturbed problem in the parameter space. Mixed integer linear programming has widespread application in process engineering such as process design, planning and scheduling, and the control of hybrid systems. A particular difficulty arises, significantly increasing the complexity and computational effort in retrieving the optimal solution of the problem, when uncertainty is simultaneously present in the coefficients of the objective function and the constraints, yielding a general multi-parametric (mp)-MILP problem. In this thesis, we present novel solution strategies for this class of problems. A global optimization procedure for mp-MILP problems, which adapts techniques from the deterministic case to the multi-parametric framework, has been developed. One of the challenges in multi-parametric global optimization is that parametric profiles, and not scalar values as in the deterministic case, need to be compared. To overcome the computational burden to derive a globally optimal solution, two-stage methods for the approximate solution of mp-MILP problems are proposed. The first approach combines robust optimization and multi-parametric programming; whereas in the second approach suitable relaxations of bilinear terms are employed to linearize the constraints during the approximation stage. The choice of approximation techniques used in the two-stage method has impact on the conservatism of the solution estimate that is generated. Lastly, multi-parametric programming based two-stage methods are applied in pro-active short-term scheduling of batch processes when faced with varied sources of uncertainty, such of price, demand and operational level uncertainty.Open Acces