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The structure of Bulk-Heterojunction (BHJ) materials, the main component of organic photovoltaic solar cells, is very complex, and the relationship between structure and performance is still largely an open question. Overall, there is a wide spectrum of fabrication configurations resulting in different BHJ morphologies and correspondingly different performances. Current state- of-the-art methods for assessing the performance of BHJ morphologies are either based on global quantification of morphological features or simply on visual inspection of the morphology based on experimental imaging. This makes finding optimal BHJ structures very challenging. Moreover, finding the optimal fabrication parameters to get an optimal structure is still an open question. In this paper, we propose a visual analysis framework to help answer these questions through comparative visualization and parameter space exploration for local morphology features. With our approach, we enable scientists to explore multivariate correlations between local features and performance indicators of BHJ morphologies. Our framework is built on shape-based clustering of local cubical regions of the morphology that we call patches. This enables correlating the features of clusters with intuition-based performance indicators computed from geometrical and topological features of charge paths.
Constraint Programs and Mixed Integer Programs are closely related optimization problems originating from different scientific areas. Today's state-of-the-art algorithms of both fields have several strategies in common, in particular the branch-and-bound process to recursively divide the problem into smaller sub problems. On the other hand, the main techniques to process each sub problem are different, and it was observed that they have complementary strenghts. We propose a programming framework {\sffamily SCIP} that integrates techniques from both fields in order to exploit the strenghts of both, Constraint Programming and Mixed Integer Programming. In contrast to other proposals of recent years to combine both fields, {\sffamily SCIP} does not focus on easy implementation and rapid prototyping, but is tailored towards expert users in need of full, in-depth control and high performance.
This thesis introduces the novel paradigm of "constraint integer programming" (CIP), which integrates constraint programming (CP) and mixed integer programming (MIP) modeling and solving techniques. It is supplemented by the software SCIP, which is a solver and framework for constraint integer programming that also features SAT solving techniques. SCIP is freely available in source code for academic and non-commercial purposes. Our constraint integer programming approach is a generalization of MIP that allows for the inclusion of arbitrary constraints, as long as they turn into linear constraints on the continuous variables after all integer variables have been fixed. The constraints, may they be linear or more complex, are treated by any combination of CP and MIP techniques: the propagation of the domains by constraint specific algorithms, the generation of a linear relaxation and its solving by LP methods, and the strengthening of the LP by cutting plane separation. The current version of SCIP comes with all of the necessary components to solve mixed integer programs. In the thesis, we cover most of these ingredients and present extensive computational results to compare different variants for the individual building blocks of a MIP solver. We focus on the algorithms and their impact on the overall performance of the solver. In addition to mixed integer programming, the thesis deals with chip design verification, which is an important topic of electronic design automation. Chip manufacturers have to make sure that the logic design of a circuit conforms to the specification of the chip. Otherwise, the chip would show an erroneous behavior that may cause failures in the device where it is employed. An important subproblem of chip design verification is the property checking problem, which is to verify whether a circuit satisfies a specified property. We show how this problem can be modeled as constraint integer program and provide a number of problem-specific algorithms that exploit the structure of the individual constraints and the circuit as a whole. Another set of extensive computational benchmarks compares our CIP approach to the current state-of-the-art SAT methodology and documents the success of our method.
This thesis introduces the novel paradigm of constraint integer programming (CIP), which integrates constraint programming (CP) and mixed integer programming (MIP) modeling and solving techniques. It is supplemented by the software SCIP, which is a solver and framework for constraint integer programming that also features SAT solving techniques. SCIP is freely available in source code for academic and non-commercial purposes. Our constraint integer programming approach is a generalization of MIP that allows for the inclusion of arbitrary constraints, as long as they turn into linear constraints on the continuous variables after all integer variables have been fixed. The constraints, may they be linear or more complex, are treated by any combination of CP and MIP techniques: the propagation of the domains by constraint specific algorithms, the generation of a linear relaxation and its solving by LP methods, and the strengthening of the LP by cutting plane separation. The current version of SCIP comes with all of the necessary components to solve mixed integer programs. In the thesis, we cover most of these ingredients and present extensive computational results to compare different variants for the individual building blocks of a MIP solver. We focus on the algorithms and their impact on the overall performance of the solver. In addition to mixed integer programming, the thesis deals with chip design verification, which is an important topic of electronic design automation. Chip manufacturers have to make sure that the logic design of a circuit conforms to the specification of the chip. Otherwise, the chip would show an erroneous behavior that may cause failures in the device where it is employed. An important subproblem of chip design verification is the property checking problem, which is to verify whether a circuit satisfies a specified property. We show how this problem can be modeled as constraint integer program and provide a number of problem-specific algorithms that exploit the structure of the individual constraints and the circuit as a whole. Another set of extensive computational benchmarks compares our CIP approach to the current state-of-the-art SAT methodology and documents the success of our method.
Conflict analysis for infeasible subproblems is one of the key ingredients in modern SAT solvers to cope with large real-world instances. In contrast, it is common practice for today's mixed integer programming solvers to just discard infeasible subproblems and the information they reveal. In this paper we try to remedy this situation by generalizing the SAT infeasibility analysis to mixed integer programming. We present heuristics for branch-and-cut solvers to generate valid inequalities from the current infeasible subproblem and the associated branching information. SAT techniques can then be used to strengthen the resulting cuts. We performed computational experiments which show the potential of our method: On feasible MIP instances, the number of required branching nodes was reduced by 50\% in the geometric mean. However, the total solving time increased by 15\%. on infeasible MIPs arising in the context of chip verification, the number of nodes was reduced by 90\%, thereby reducing the solving time by 60\%.
The Feasibility Pump of Fischetti, Glover, Lodi, and Bertacco has proved to be a very successful heuristic for finding feasible solutions of mixed integer programs. The quality of the solutions in terms of the objective value, however, tends to be poor. This paper proposes a slight modification of the algorithm in order to find better solutions. Extensive computational results show the success of this variant: in 89 out of 121 MIP instances the modified version produces improved solutions in comparison to the original Feasibility Pump.
Hybrid Branching
(2009)
This article introduces constraint integer programming (CIP), which is a novel way to combine constraint programming (CP) and mixed integer programming (MIP) methodologies. CIP is a generalization of MIP that supports the notion of general constraints as in CP. This approach is supported by the CIP framework SCIP, which also integrates techniques for solving satisfiability problems. SCIP is available in source code and free for noncommercial use. We demonstrate the usefulness of CIP on three tasks. First, we apply the constraint integer programming approach to pure mixed integer programs. Computational experiments show that SCIP is almost competitive to current state-of-the-art commercial MIP solvers. Second, we demonstrate how to use CIP techniques to compute the number of optimal solutions of integer programs. Third, we employ the CIP framework to solve chip design verification problems, which involve some highly nonlinear constraint types that are very hard to handle by pure MIP solvers. The CIP approach is very effective here: it can apply the full sophisticated MIP machinery to the linear part of the problem, while dealing with the nonlinear constraints by employing constraint programming techniques.
Primal heuristics are an important component of state-of-the-art codes for
mixed integer programming. In this paper, we focus on primal heuristics
that only employ computationally inexpensive procedures such as rounding
and logical deductions (propagation). We give an overview of eight
different approaches. To assess the impact of these primal heuristics on
the ability to find feasible solutions, in particular early during search,
we introduce a new performance measure, the primal integral. Computational
experiments evaluate this and other measures on MIPLIB~2010 benchmark
instances.
This article introduces constraint integer programming (CIP), which is a novel way to combine constraint programming (CP) and mixed integer programming (MIP) methodologies. CIP is a generalization of MIP that supports the notion of general constraints as in CP. This approach is supported by the CIP framework SCIP, which also integrates techniques from SAT solving. SCIP is available in source code and free for non-commercial use. We demonstrate the usefulness of CIP on two tasks. First, we apply the constraint integer programming approach to pure mixed integer programs. Computational experiments show that SCIP is almost competitive to current state-of-the-art commercial MIP solvers. Second, we employ the CIP framework to solve chip design verification problems, which involve some highly non-linear constraint types that are very hard to handle by pure MIP solvers. The CIP approach is very effective here: it can apply the full sophisticated MIP machinery to the linear part of the problem, while dealing with the non-linear constraints by employing constraint programming techniques.
Mixed integer programming has become a very powerful tool for modeling and
solving real-world planning and scheduling problems, with the breadth of
applications appearing to be almost unlimited. A critical component in
the solution of these mixed-integer programs is a set of routines commonly
referred to as presolve. Presolve can be viewed as a collection of
preprocessing techniques that reduce the size of and, more importantly,
improve the ``strength'' of the given model formulation, that is, the degree
to which the constraints of the formulation accurately describe the
underlying polyhedron of integer-feasible solutions. As our computational
results will show, presolve is a key factor in the speed with which we can
solve mixed-integer programs, and is often the difference between a model
being intractable and solvable, in some cases easily solvable. In this
paper we describe the presolve functionality in the Gurobi commercial
mixed-integer programming code.
This includes an overview, or taxonomy of the different methods that are
employed, as well as more-detailed descriptions of several of the techniques,
with some of them appearing, to our knowledge, for the first time in the
literature.
Mixed integer programming has become a very powerful tool for modeling and
solving real-world planning and scheduling problems, with the breadth of
applications appearing to be almost unlimited. A critical component in
the solution of these mixed-integer programs is a set of routines commonly
referred to as presolve. Presolve can be viewed as a collection of
preprocessing techniques that reduce the size of and, more importantly,
improve the ``strength'' of the given model formulation, that is, the degree
to which the constraints of the formulation accurately describe the
underlying polyhedron of integer-feasible solutions. As our computational
results will show, presolve is a key factor in the speed with which we can
solve mixed-integer programs, and is often the difference between a model
being intractable and solvable, in some cases easily solvable. In this
paper we describe the presolve functionality in the Gurobi commercial
mixed-integer programming code.
This includes an overview, or taxonomy of the different methods that are
employed, as well as more-detailed descriptions of several of the techniques,
with some of them appearing, to our knowledge, for the first time in the
literature.
We address the property checking problem for SoC design verification at the register transfer level (RTL) by integrating techniques from integer programming, constraint programming, and SAT solving. Specialized domain propagation and preprocessing algorithms for individual RTL operations extend a general constraint integer programming framework. Conflict clauses are learned by analyzing infeasible LPs and deductions, and by employing reverse propagation. Experimental results show that our approach outperforms SAT techniques for proving the validity of properties on circuits containing arithmetics.
Modern applications of mathematical programming must take into account a multitude of technical details, business demands, and legal requirements. Teaching the mathematical modeling of such issues and their interrelations requires real-world examples that are well beyond the toy sizes that can be tackled with the student editions of most commercial software packages. We present a new tool, which is freely available for academic use including complete source code. It consists of an algebraic modeling language and a linear mixed integer programming solver. The performance and features of the tool are in the range of current state-of-the-art commercial tools, though not in all aspects as good as the best ones. Our tool does allow the execution and analysis of large real-world instances in the classroom and can therefore enhance the teaching of problem solving issues. Teaching experience has been gathered and practical usability was tested in classes at several universities and a two week intensive block course at TU Berlin. The feedback from students and teachers has been very positive.
In the recent years there has been tremendous progress in the development of algorithms to find optimal solutions for integer programs. In many applications it is, however, desirable (or even necessary) to generate all feasible solutions. Examples arise in the areas of hardware and software verification and discrete geometry. In this paper, we investigate how to extend branch-and-cut integer programming frameworks to support the generation of all solutions. We propose a method to detect so-called unrestricted subtrees, which allows us to prune the integer program search tree and to collect several solutions simultaneously. We present computational results of this branch-and-count paradigm which show the potential of the unrestricted subtree detection.
Mixed integer programs ($MIPs$) are commonly solved with branch and bound algorithms based on linear programming. The success and the speed of the algorithm strongly depends on the strategy used to select the branching variables. Today's state-of-the-art strategy is called \emph{pseudocost branching} and uses information of previous branchings to determine the current branching. We propose a modification of \emph{pseudocost branching} which we call \emph{history branching}. This strategy has been implemented in $SIP$, a state-of-the-art $MIP$ solver. We give computational results that show the superiority of the new strategy.
Branching rules revisited
(2004)
Mixed integer programs are commonly solved with linear programming based branch-and-bound algorithms. The success of the algorithm strongly depends on the strategy used to select the variable to branch on. We present a new generalization called {\sl reliability branching} of today's state-of-the-art {\sl strong branching} and {\sl pseudocost branching} strategies for linear programming based branch-and-bound algorithms. After reviewing commonly used branching strategies and performing extensive computational studies we compare different parameter settings and show the superiority of our proposed newstrategy.
MIPLIB 2003
(2005)
This paper reports on the fourth version of the Mixed Integer Programming Library. Since ({\sc miplib}) is to provide a concise set of challenging problems, it became necessary to purge instances that became too easy. We present an overview of the 27 new problems and statistical data for all 60 instances.
Branching Rules Revisited
(2005)
MIPLIB 2003
(2006)
Starting with the description of the Traveling Salesmen Problem formulation as given by van Vyve and Wolsey in the article Approximate extended formulations'', we investigate the effects of small variations onto the performance of contemporary mixed integer programming solvers. We will show that even minor changes in the formulation of the model can result in performance difference of more than a factor of 1000. As the results show it is not obvious which changes will result in performance improvements and which not.
Given a general mixed integer program (MIP), we automatically detect block structures in the constraint matrix together with the coupling by capacity constraints arising from multi-commodity-flow formulations. We identify the underlying graph and generate cutting planes based on cuts in the detected network. Our implementation adds a separator to the branch-and-cut libraries of SCIP and CPLEX. We make use of the complemented mixed integer rounding framework (cMIR) but provide a special purpose aggregation heuristic that exploits the network structure. Our separation scheme speeds-up the computation for a large set of MIPs coming from network design problems by a factor of two on average.
An error controlled finite elemente method (FEM) for solving stationary Schrödinger equations in three space dimensions is proposed. The method is based on an adaptive space discretization into tetrahedra and local polynomial basis functions of order $p=1$--$5$ defined on these tetrahedra. According to a local error estimator the triangulation is automatically adapted to the solution. Numerical results for standard problems appearing in vibrational motion and molecular structure calculations are presented and discussed. Relative precisions better than 1e-8 are obtained. For equilateral H$_3^{++}$ the adaptive FEM turns out to be superior to global basis set expansions in the literature. Our precise FEM results exclude in a definite manner the stability or metastability of equilateral H$_3^{++}$ in its groundstate.
Using the perturbational-variational Rayleigh-Ritz matrix formalism, the 1/Z-expansion for the ground state of the isoelectronic $H_2$ sequence in the range of the internuclear distance $0.2\le R \le 9.0$ is calculated. Also lower bounds of the radius of convergence, based on Kato's theory of linear operators, are given. The numerical results of the 1/Z-expansion can be compared with the exact results and do not converge in the whole R-range. This behavior is in qualitative agreement with the lower bounds for the radius of convergence and enlights some still open properties of 1/Z- expansions for this sequence in the literature. {\bf PACS:} 31.15 + q; 31.20 Di; 31.20 Tz.
The subject of this study is a multilevel Finite Element Method based on an error estimator and step by step grid refinement as an universal tool for solving time--independent Schrödinger--eigenvalue problems. Numerical results for standard problems appearing in vibrational motion and molecular electronic structure calculations are given and discussed.
This paper presents the new program package MACRON for the simulation of macromolecular kinetics including standard chemical reactions. Such problems lead to countable (possibly) infinite systems of ordinary differential equations (CODE's), which are numerically treated by the so-called discrete Galerkin method here. By a chemical compiler the required analytical preprocessing is performed, such that the complete reaction system, standard kinetics as well as macromolecular reactions, can be entered in the chemical formalism. Typical macromolecular reaction steps are chain addition, termination, chain transfer and degradation (cracking). In order to ensure efficiency and reliability, high sophisticated numerical routines are built within the package. MACRON can be used without a detailed knowledge of the used numerical methods. As an illustration the application of MACRON to some realistic problems is presented.
The description of chain length distributions in macromolecular reaction kinetics leads to so-called countable systems of differential equations. In particular, when the appearing reaction rate coefficients depend on the chain length of the reacting macromolecules itself, an efficient numerical treatment of these systems is very difficult. Then even the evaluation of the right-hand side of the system can become prohibitively expensive with respect to computing time. In this paper we show how the discrete Galerkin method can be applied to such problems. The existing algorithm CODEX is improved by use of a multiplicative error correction scheme for time discretization and a new type of numerical preprocessing by means of a Gauss summation. Both ideas are exemplary for a wide class of approximation types and are described very briefly here. The new numerical techniques are tested on an example from soot formation, where the coagulation of molecules is modeled in terms of reaction coefficients depending on the surface of the particles and their collision frequency.
We employ the adaptive resolution approach AdResS, in its recently developed Grand Canonicallike version (GC-AdResS) [Wang et al. Phys.Rev.X 3, 011018 (2013)], to calculate the excess chemical potential, $μ^{ex}$, of various liquids and mixtures. We compare our results with those obtained from full atomistic simulations using the technique of thermodynamic integration and show a satisfactory agreement. In GC-AdResS the procedure to calculate $μ^{ex}$ corresponds to the process of standard initial equilibration of the system; this implies that, independently of the specific aim of the study, $μ^{ex}$, for each molecular species, is automatically calculated every time a GC-AdResS simulation is performed.
A great amount of material properties is strongly influenced by dislocations, the carriers of plastic deformation. It is therefore paramount to have appropriate tools to quantify dislocation substructures with regard to their features, e.g., dislocation density, Burgers vectors or line direction. While the transmission electron microscope (TEM) has been the most widely-used equipment implemented to investigate dislocations, it usually is limited to the two-dimensional (2D) observation of three-dimensional (3D) structures. We reconstruct, visualize and quantify 3D dislocation substructure models from only two TEM images (stereo pairs) and assess the results. The reconstruction is based on the manual interactive tracing of filiform objects on both images of the stereo pair. The reconstruction and quantification method are demonstrated on dark field (DF) scanning (S)TEM micrographs of dislocation substructures imaged under diffraction contrast conditions. For this purpose, thick regions (>300 nm) of TEM foils are analyzed, which are extracted from a Ni-base superalloy single crystal after high temperature creep deformation. It is shown how the method allows 3D quantification from stereo pairs in a wide range of tilt conditions, achieving line length and orientation uncertainties of 3% and 7°, respectively. Parameters that affect the quality of such reconstructions are discussed.
A great amount of material properties is strongly influenced by dislocations, the carriers of plastic deformation. It is therefore paramount to have appropriate tools to quantify dislocation substructures with regard to their features, e.g., dislocation density, Burgers vectors or line direction. While the transmission electron microscope (TEM) has been the most widely-used equipment implemented to investigate dislocations, it usually is limited to the two-dimensional (2D) observation of three-dimensional (3D) structures. We reconstruct, visualize and quantify 3D dislocation substructure models from only two TEM images (stereo-pairs) and assess the results. The reconstruction is based on the manual interactive tracing of filiform objects on both images of the stereo-pair. The reconstruction and quantification method are demonstrated on dark field (DF) scanning (S)TEM micrographs of dislocation substructures imaged under diffraction contrast conditions. For this purpose, thick regions (> 300 nm) of TEM foils are analyzed, which are extracted from a Ni-base superalloy single crystal after high temperature creep deformation. It is shown how the method allows 3D quantification from stereo-pairs in a wide range of tilt conditions, achieving line length and orientation uncertainties of 3 % and 7°, respectively. Parameters that affect the quality of such reconstructions are discussed.
Enhanced Photon Upconversion Using Erbium-Doped Nanoparticles Interacting with Silicon Metasurfaces
(2021)
The operation of a railway network as large as Deutsche Bahn's Intercity Express (ICE) hinges on a number of factors, such as the availability of personnel and the assignment of physical vehicles to a timetable schedule, a problem known as the rolling stock rotation problem (RSRP). In this paper, we consider the problem of creating an alternative timetable in the case that there is a long-term disruption, such as a strike, and the effects that this alternative timetable has on the resulting vehicle rotation plan. We define a priority measure via the Analytic Hierarchy Process (AHP) to determine the importance of each trip in the timetable and therefore which trips to cancel or retain. We then compare our results with those of a limited timetable manually designed by Deutsche Bahn (DB). We find that while our timetable results in a more expensive rotation plan, its flexibility lends itself to a number of simple improvements. Furthermore, our priority measure has the potential to be integrated into the rolling stock rotation optimization process, in particular, the Rotation Optimizer for Railways (ROTOR) software, via the cost function. Ultimately, our method provides the foundation for an automated way of creating a new timetable quickly, and potentially in conjunction with a new rotation plan, in the case of a limited scenario.
Inferring Proteolytic Processes from Mass Spectrometry Time Series Data Using Degradation Graphs
(2012)
We propose a mathematical optimization model and its solution for joint chance constrained DC Optimal Power Flow. In this application, it is particularly important that there is a high probability of transmission limits being satisfied, even in the case of uncertain or fluctuating feed-in from renewable energy sources. In critical network situations where the network risks overload, renewable energy feed-in has to be curtailed by the transmission system operator (TSO). The TSO can reduce the feed-in in discrete steps at each network node. The proposed optimization model minimizes curtailment while ensuring that there is a high probability of transmission limits being maintained. The latter is modeled via (joint) chance constraints that are computationally challenging. Thus, we propose a solution approach based on the robust safe approximation of these constraints. Hereby, probabilistic constraints are replaced by robust constraints with suitably defined uncertainty sets constructed from historical data. The ability to discretely control the power feed-in then leads to a robust optimization problem with decision-dependent uncertainties, i.e. the uncertainty sets depend on decision variables. We propose an equivalent mixed-integer linear reformulation for box uncertainties with the exact linearization of bilinear terms. Finally, we present numerical results for different test cases from the Nesta archive, as well as for a real network. We consider the discrete curtailment of solar feed-in, for which we use real-world weather and network data. The experimental tests demonstrate the effectiveness of this method and run times are very fast. Moreover, on average the calculated robust solutions only lead to a small increase in curtailment, when compared to nominal solutions.
Changes in knee shape and geometry resulting from total knee arthroplasty can affect patients in numerous important ways: pain, function, stability, range of motion, and kinematics. Quantitative data concerning these changes have not been previously available, to our knowledge, yet are essential to understand individual experiences of total knee arthroplasty and thereby improve outcomes for all patients. The limiting factor has been the challenge of accurately measuring these changes. Our study objective was to develop a conceptual framework and analysis method to investigate changes in knee shape and geometry, and prospectively apply it to a sample total knee arthroplasty population. Using clinically available computed tomography and radiography imaging systems, the three-dimensional knee shape and geometry of nine patients (eight varus and one valgus) were compared before and after total knee arthroplasty. All patients had largely good outcomes after their total knee arthroplasty. Knee shape changed both visually and numerically. On average, the distal condyles were slightly higher medially and lower laterally (range: +4.5 mm to −4.4 mm), the posterior condyles extended farther out medially but not laterally (range: +1.8 to −6.4 mm), patellofemoral distance increased throughout flexion by 1.8–3.5 mm, and patellar thickness alone increased by 2.9 mm (range: 0.7–5.2 mm). External femoral rotation differed preop and postop. Joint line distance, taking cartilage into account, changed by +0.7 to −1.5 mm on average throughout flexion. Important differences in shape and geometry were seen between pre-total knee arthroplasty and post-total knee arthroplasty knees. While this is qualitatively known, this is the first study to report it quantitatively, an important precursor to identifying the reasons for the poor outcome of some patients. Using the developed protocol and visualization techniques to compare patients with good versus poor clinical outcomes could lead to changes in implant design, implant selection, component positioning, and surgical technique. Recommendations based on this sample population are provided. Intraoperative and postoperative feedback could ultimately improve patient satisfaction.