Refine
Year of publication
- 2020 (224) (remove)
Document Type
- Article (122)
- In Proceedings (35)
- ZIB-Report (30)
- Master's Thesis (9)
- Book chapter (7)
- Book (4)
- Doctoral Thesis (4)
- Other (4)
- In Collection (3)
- Bachelor's Thesis (2)
Keywords
Institute
- Modeling and Simulation of Complex Processes (71)
- Numerical Mathematics (67)
- Visual and Data-centric Computing (37)
- Visual Data Analysis (36)
- Applied Algorithmic Intelligence Methods (33)
- Computational Nano Optics (32)
- Mathematical Optimization (32)
- Mathematical Algorithmic Intelligence (30)
- Distributed Algorithms and Supercomputing (27)
- AI in Society, Science, and Technology (19)
In graphical representations of public transportation networks, there is often some degree of uncertainty in the arc values, due to delays or transfer times. This uncertainty can be expressed as a parameterized weight on the transfer arcs. Classical shortest path algorithms often have difficulty handling parameterized arc weights and a tropical geometry approach has been shown as a possible solution. The connection between the classical shortest path problem and tropical geometry is well establish: Tropically multiplying the n × n adjacency matrix of a graph with itself n − 1 times results in the so-called Kleene star, and is a matrix-form solution to the all-pairs shortest path problem. Michael Joswig and Benjamin Schröter showed in their paper The Tropical Geometry of Shortest Paths that the same method can be used to find the solution to the all-pairs shortest path problem even in the case of variable arc weights and they proposed an algorithm to solve the single-target shortest path problem in such a case. The solution takes the form of a polyhedral subdivision of the parameter space. As the number of variable arc weights grows, the time needed to execute an implementation of this algorithm grows exponentially. As the size of a public transportation network grows, the number of variable arc weights grows exponentially as well. However, it has been observed that in public transportation networks, there are usually only a few possible shortest routes. Geometrically, this means that there should be few polyhedra in the polyhedral subdivision. This algorithm is used on an example of a real-world public transportation network and an analysis of the polyhedral subdivision is made. Then a geometrical approach is used to analyze the impact of limiting the number of transfers, and thereby limiting the number of parameterized arcs used, as an estimation of the solution to the all-pairs shortest path problem
One of the fundamental steps in the optimization of public transport is line planning. It involves determining lines and assigning frequencies of service such that costs are minimized while also maximizing passenger comfort and satisfying travel demands. We formulate the problem as a mixed integer linear program that considers all circuit-like lines in a graph and allows free passenger routing. Traveler and operator costs are included in a linear scalarization in the objective. We apply said programming problem to the Parametric City, which is a graph model introduced by Fielbaum, Jara-Díaz and Gschwender that exibly represents different cities. In his dissertation, Fielbaum solved the line planning problem for various parameter choices in the Parametric City. In a first step, we therefore review his results and make comparative computations. Unlike Fielbaum we arrive at the conclusion that the optimal line plan for this model indeed depends on the demand. Consequently, we analyze the line planning problem in-depth: We find equivalent, but easier to compute formulations and provide a lower bound by LP-relaxation, which we show to be equivalent to a multi-commodity flow problem. Further, we examine what impact symmetry has on the solutions. Supported both by computational results as well as by theoretical analysis, we reach the conclusion that symmetric line plans are optimal or near-optimal in the Parametric City. Restricting the model to symmetric line plans allows for a \kappa-factor approximation algorithm for the line planning problem in the Parametric City.
Constrained second-order convex optimization algorithms are the method of choice when a high accuracy solution to a problem is needed, due to their local quadratic convergence. These algorithms require the solution of a constrained quadratic subproblem at every iteration. We present the \emph{Second-Order Conditional Gradient Sliding} (SOCGS) algorithm, which uses a projection-free algorithm to solve the constrained quadratic subproblems inexactly. When the feasible region is a polytope the algorithm converges quadratically in primal gap after a finite number of linearly convergent iterations. Once in the quadratic regime the SOCGS algorithm requires O(log(log1/ε)) first-order and Hessian oracle calls and O(log(1/ε)log(log1/ε)) linear minimization oracle calls to achieve an ε-optimal solution. This algorithm is useful when the feasible region can only be accessed efficiently through a linear optimization oracle, and computing first-order information of the function, although possible, is costly.
Network and Storage
(2020)
Natural gas is considered by many to be the most important energy source for the future. The objectives of energy commodities strategic problems can be mainly related to natural gas and deal with the definition of the “optimal” gas pipelines design which includes a number of related sub problems such as: Gas stations (compression) location and Gas storage locations, as well as compression station design and optimal operation.
Demand Side Management (DSM) is usually considered as a process of energy consumption shifting from peak hours to off-peak times. DSM does not always reduce total energy consumption, but it helps to meet energy demand and supply. For example, it balances variable generation from renewables (such as solar and wind) when energy demand differs from renewable generation.
Markov chain (MC) algorithms are ubiquitous in machine learning and statistics and many other disciplines. Typically, these algorithms can be formulated as acceptance rejection methods. In this work we present a novel estimator applicable to these methods, dubbed Markov chain importance sampling (MCIS), which efficiently makes use of rejected proposals. For the unadjusted Langevin algorithm, it provides a novel way of correcting the discretization error. Our estimator satisfies a central limit theorem and improves on error per CPU cycle, often to a large extent. As a by-product it enables estimating the normalizing constant, an important quantity in Bayesian machine learning and statistics.
The complexity in large-scale optimization can lie in both handling the objective function and handling the constraint set. In this respect, stochastic Frank-Wolfe algorithms occupy a unique position as they alleviate both computational burdens, by querying only approximate first-order information from the objective and by maintaining feasibility of the iterates without using projections. In this paper, we improve the quality of their first-order information by blending in adaptive gradients. We derive convergence rates and demonstrate the computational advantage of our method over the state-of-the-art stochastic Frank-Wolfe algorithms on both convex and nonconvex objectives. The experiments further show that our method can improve the performance of adaptive gradient algorithms for constrained optimization.
This paper studies the empirical efficacy and benefits of using projection-free first-order methods in the form of Conditional Gradients, a.k.a. Frank-Wolfe methods, for training Neural Networks with constrained parameters. We draw comparisons both to current state-of-the-art stochastic Gradient Descent methods as well as across different variants of stochastic Conditional Gradients. In particular, we show the general feasibility of training Neural Networks whose parameters are constrained by a convex feasible region using Frank-Wolfe algorithms and compare different stochastic variants. We then show that, by choosing an appropriate region, one can achieve performance exceeding that of unconstrained stochastic Gradient Descent and matching state-of-the-art results relying on L2-regularization. Lastly, we also demonstrate that, besides impacting performance, the particular choice of constraints can have a drastic impact on the learned representations.