Harder, better, faster, stronger: understanding and improving the tractability of large energy system models

Energy system models based on linear programming have been growing in size with the increasing need to model renewables with high spatial and temporal detail. Larger models lead to high computational requirements. Furthermore, seemingly small changes in a model can lead to drastic differences in runtime. Here, we investigate measures to address this issue. We review the mathematical structure of a typical energy system model, and discuss issues of sparsity, degeneracy and large numerical range. We introduce and test a method to automatically scale models to improve numerical range. We test this method as well as tweaks to model formulation and solver preferences, finding that adjustments can have a substantial impact on runtime. In particular, the barrier method without crossover can be very fast, but affects the structure of the resulting optimal solution. We conclude with a range of recommendations for energy system modellers

The double pivot simplex method

The simplex method, created by George Dantzig, optimally solves a linear program by pivoting. Dantzig’s pivots move from a basic feasible solution to a different basic feasible solution by exchanging exactly one basic variable with a nonbasic variable. This paper introduces the double pivot simplex method, which can transition between basic feasible solutions using two variables instead of one. Double pivots are performed by identifying the optimal basis in a two variable linear program using a new method called the slope algorithm. The slope algorithm is fast and allows an iteration of DPSM to have the same theoretical running time as an iteration of the simplex method. Computational experiments demonstrate that DPSM decreases the average number of pivots by approximately 41% on a small set of benchmark instances

A nearly optimal randomized algorithm for explorable heap selection

Explorable heap selection is the problem of selecting the nth smallest value in a binary heap. The key values can only be accessed by traversing through the underlying infinite binary tree, and the complexity of the algorithm is measured by the total distance traveled in the tree (each edge has unit cost). This problem was originally proposed as a model to study search strategies for the branch-and-bound algorithm with storage restrictions by Karp, Saks and Widgerson (FOCS '86), who gave deterministic and randomized n⋅exp(O(logn−−−−√)) time algorithms using O(log(n)2.5) and O(logn−−−−√) space respectively. We present a new randomized algorithm with running time O(nlog(n)3) using O(logn) space, substantially improving the previous best randomized running time at the expense of slightly increased space usage. We also show an Ω(log(n)n/log(log(n))) for any algorithm that solves the problem in the same amount of space, indicating that our algorithm is nearly optimal

Row generation techniques for approximate solution of linear programming problems

Ankara : The Department of Industrial Engineering and the Institute of Engineering and Science of Bilkent University, 2010.Thesis (Master's) -- Bilkent University, 2010.Includes bibliographical references leaves 69-77.In this study, row generation techniques are applied on general linear programming problems with a very large number of constraints with respect to the problem dimension. A lower bound is obtained for the change in the objective value caused by the generation of a specific row. To achieve row selection that results in a large shift in the feasible region and the objective value at each row generation iteration, the lower bound is used in the comparison of row generation candidates. For a warm-start to the solution procedure, an effective selection of the subset of constraints that constitutes the initial LP is considered. Several strategies are discussed to form such a small subset of constraints so as to obtain an initial solution close to the feasible region of the original LP. Approximation schemes are designed and compared to make possible the termination of row generation at a solution in the proximity of an optimal solution of the input LP. The row generation algorithm presented in this study, which is enhanced with a warm-start strategy and an approximation scheme is implemented and tested for computation time and the number of rows generated. Two efficient primal simplex method variants are used for benchmarking computation times, and the row generation algorithm appears to perform better than at least one of them especially when number of constraints is large.Paç, A BurakM.S

Two dimensional search algorithms for linear programming

Linear programming is one of the most important classes of optimization problems. These mathematical models have been used by academics and practitioners to solve numerous real world applications. Quickly solving linear programs impacts decision makers from both the public and private sectors. Substantial research has been performed to solve this class of problems faster, and the vast majority of the solution techniques can be categorized as one dimensional search algorithms. That is, these methods successively move from one solution to another solution by solving a one dimensional subspace linear program at each iteration. This dissertation proposes novel algorithms that move between solutions by repeatedly solving a two dimensional subspace linear program. Computational experiments demonstrate the potential of these newly developed algorithms and show an average improvement of nearly 25% in solution time when compared to the corresponding one dimensional search version. This dissertation\u27s research creates the core concept of these two dimensional search algorithms, which is a fast technique to determine an optimal basis and an optimal solution to linear programs with only two variables. This method, called the slope algorithm, compares the slope formed by the objective function with the slope formed by each constraint to determine a pair of constraints that intersect at an optimal basis and an optimal solution. The slope algorithm is implemented within a simplex framework to perform two dimensional searches. This results in the double pivot simplex method. Differently than the well-known simplex method, the double pivot simplex method simultaneously pivots up to two basic variables with two nonbasic variables at each iteration. The theoretical computational complexity of the double pivot simplex method is identical to the simplex method. Computational results show that this new algorithm reduces the number of pivots to solve benchmark instances by approximately 40% when compared to the classical implementation of the simplex method, and 20% when compared to the primal simplex implementation of CPLEX, a high performance mathematical programming solver. Solution times of some random linear programs are also improved by nearly 25% on average. This dissertation also presents a novel technique, called the ratio algorithm, to find an optimal basis and an optimal solution to linear programs with only two constraints. When the ratio algorithm is implemented within a simplex framework to perform two dimensional searches, it results in the double pivot dual simplex method. In this case, the double pivot dual simplex method behaves similarly to the dual simplex method, but two variables are exchanged at every step. Two dimensional searches are also implemented within an interior point framework. This dissertation creates a set of four two dimensional search interior point algorithms derived from primal and dual affine scaling and logarithmic barrier search directions. Each iteration of these techniques quickly solves a two dimensional subspace linear program formed by the intersection of two search directions and the feasible region of the linear program. Search directions are derived by orthogonally partitioning the objective function vector, which allows these novel methods to improve the objective function value at each step by at least as much as the corresponding one dimensional search version. Computational experiments performed on benchmark linear programs demonstrate that these two dimensional search interior point algorithms improve the average solution time by approximately 12% and the average number of iterations by 15%. In conclusion, this dissertation provides a change of paradigm in linear programming optimization algorithms. Implementing two dimensional searches within both a simplex and interior point framework typically reduces the computational time and number of iterations to solve linear programs. Furthermore, this dissertation sets the stage for future research topics in multidimensional search algorithms to solve not only linear programs but also other critical classes of optimization methods. Consequently, this dissertation\u27s research can become one of the first steps to change how commercial and open source mathematical programming software will solve optimization problems

Schnelle SVM Regularisierungspfadberechnung: Theorie und Implementierung

Im Fokus dieser Diplomarbeit steht die Implementierung und theoretische Analyse eines neuen Algorithmus zur Berechnung des gesamten Lösungspfades von allgemeinen Support Vector Machines bezüglich ihres Regularisierungsparameters. Dessen optimaler Wert wird im Ergebnis durch einen solchen Pfad deutlich leichter auffindbar. Erreicht wird dieses Ziel mit Hilfe der nicht-approximativen Criss-Cross Methode aus dem Bereich der linearen Komplementaritätsprobleme. Neben dem geometrischen Verhalten dieser Methode wird insbesondere auf deren effi ziente Initialisierung zu Beginn eines Lösungspfades eingegangen. Darüber hinaus zeigt diese Arbeit auf, dass auch Probleme der Conjoint Analyse in Support Vector Machines überführt und entsprechend gelöst werden können. Abschließend werden die theoretischen Resultate anhand von Conjoint-Analyse-Datensätzen und solchen für Support Vector Machines veranschaulich

Foundational Factorization Algorithms for the Efficient Roundoff-Error-Free Solution of Optimization Problems

LU and Cholesky factorizations play a central role in solving linear and mixed-integer programs. In many documented cases, the round-off errors accrued during the construction and implementation of these factorizations cause the misclassification of suboptimal solutions as optimal and infeasible problems as feasible and vice versa. Such erroneous outputs bring the reliability of optimization solvers into question and, therefore, it is imperative to eliminate these round off errors altogether and to do so efficiently to ensure practicality. Firstly, this work introduces two round off-error-free factorizations (REF) constructed exclusively in integer arithmetic: the REF LU and Cholesky factorizations. Additionally, it develops supplementary integer-preserving substitution algorithms, thereby providing a complete tool set for solving systems of linear equations (SLEs) exactly and efficiently. An inherent property of the REF factorization algorithms is that their entries' bit-length--- i.e., the number of bits required for expression--- is bounded polynomially. Unlike the exact rational arithmetic methods used in practice, however, the algorithms herein presented do not require any greatest common divisor operations to guarantee this pivotal property. Secondly, this work derives various useful theoretical results and details computational tests to demonstrate that the REF factorization framework is considerably superior to the rational arithmetic LU factorization approach in computational performance and storage requirements. This is significant because the latter approach is the solution validation tool of choice of state-of-the-art exact linear programming solvers due to its ability to handle both numerically difficult and intricate problems. An additional theoretical contribution and further computational tests also demonstrate the predominance of the featured framework over Q-matrices, which comprise an alternative integer-preserving approach relying on the basis adjunct matrix. Thirdly, this work develops special algorithms for updating the REF factorizations. This is necessary because applying the traditional approach to the REF factorizations is inefficient in terms of entry growth and computational effort. In fact, these inefficiencies virtually wipe out all the computational savings commonly expected of factorization updates. Hence, the current work develops REF update algorithms that differ significantly from their traditional counterparts. The featured REF updates are column/row addition, deletion, and replacement

Generating general-purpose cutting planes for mixed-integer programs

Franz WesselmannPaderborn, Univ., Diss., 201