Relaxations and discretizations for the pooling problem

The pooling problem is a folklore NP-hard global optimization problem that finds applications in industries such as petrochemical refining, wastewater treatment and mining. This paper assimilates the vast literature on this problem that is dispersed over different areas and gives new insights on prevalent techniques. We also present new ideas for computing dual bounds on the global optimum by solving high-dimensional linear programs. Finally, we propose discretization methods for inner approximating the feasible region and obtaining good primal bounds. Valid inequalities are derived for the discretized models, which are formulated as mixed integer linear programs. The strength of our relaxations and usefulness of our discretizations is empirically validated on random test instances. We report best known primal bounds on some of the large-scale instances

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

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