Hermann Schichl

On twisted tensor products of algebras

(1995). On twisted tensor products of algebras. Communications in Algebra: Vol. 23, No. 12, pp. 4701-4735.

Interval Analysis on Directed Acyclic Graphs for Global Optimization

A directed acyclic graph (DAG) representation of optimization problems represents each variable, each operation, and each constraint in the problem formulation by a node of the DAG, with edges representing the flow of the computation. Using bounds on ranges of intermediate results, represented as weights on the nodes and a suitable mix of forward and backward evaluation, it is possible to give efficient implementations of interval evaluation and automatic differentiation. It is shown how to combine this with constraint propagation techniques to produce narrower interval derivatives and slopes than those provided by using only interval automatic differentiation preceded by constraint propagation. The implementation is based on earlier work by L.V. Kolev, (1997), Reliable Comput., 3, 83–93 on optimal slopes and by C. Bliek, (1992), Computer Methods for Design Automation, PhD Thesis, Department of Ocean Engineering, Massachusetts Institute of Technology on backward slope evaluation. Care is taken to ensure that rounding errors are treated correctly. Interval techniques are presented for computing from the DAG useful redundant constraints, in particular linear underestimators for the objective function, a constraint, or a Lagrangian. The linear underestimators can be found either by slope computations, or by recursive backward underestimation. For sufficiently sparse problems the work is proportional to the number of operations in the calculation of the objective function (resp. the Lagrangian).

Exclusion Regions for Systems of Equations

Branch and bound methods for finding all zeros of a nonlinear system of equations in a box frequently have the difficulty that subboxes containing no solution cannot be easily eliminated if there is a nearby zero outside the box. This has the effect that near each zero many small boxes are created by repeated splitting, whose processing may dominate the total work spent on the global search. This paper discusses the reasons for the occurrence of this so-called cluster effect and how to reduce the cluster effect by defining exclusion regions around each zero found that are guaranteed to contain no other zero and hence can safely be discarded. Such exclusion regions are traditionally constructed using uniqueness tests based on the Krawczyk operator or the Kantorovich theorem. These results are reviewed; moreover, refinements are proved that significantly enlarge the size of the exclusion region. Existence and uniqueness tests are also given.

GLOPT { A Program for Constrained Global Optimization

GLOPT is a Fortran77 program for global minimization of a block-separable objective function subject to bound constraints and block-separable constraints. It finds a nearly globally optimal point that is near a true local minimizer. Unless there are several local minimizers that are nearly global, we thus find a good approximation to the global minimizer.

Models and the History of Modeling

After a very fast tour through 30,000 years of modeling history, we describe the basic ingredients to models in general, and to mathematical models in particular.

Using directed acyclic graphs to coordinate propagation and search for numerical constraint satisfaction problems

The paper of H. Schichl & A. Neumaier has given the fundamentals of interval analysis on DAGs for global optimization and constraint propagation. We show in This work how constraint propagation on DAGs can be made efficient and practical by: (i) working on partial DAG representations; and (ii) enabling the flexible choice of the interval inclusion functions during propagation. We then propose a new simple algorithm, which coordinates constraint propagation and exhaustive search for solving numerical constraint satisfaction problems. The experiments carried out on different problems show that the new approach outperforms previously available propagation techniques by an order of magnitude or more in speed, while being roughly the same quality w.r.t. enclosure properties.

Smart robust voltage control for distribution networks using interval arithmetic and state machine concepts

Due to the connection of a great number of distributed generation (DG) plants, a critical voltage regulation problem in medium voltage distribution networks arises. After a synthetic survey of different strategies reported in literature to solve this problem, a voltage control concept for smart distribution networks with high penetration of distributed generation is presented. It makes use of all the possible means of intervention usually available in a distribution network and asks for an additional communication infrastructure due to its centralized structure. The proposed concept is a compromise between nearly global optimal operation and ease of implementation. Control actions are hierarchically organized by means of a state machine which constitutes the core of the decision logic of the controller. The voltage controller decisions regarding reactive and active power control of the controllable distributed plants are based on a linearised model of the distribution network. The model is valid in a broad range of operation. Uncertainties are accounted by interval arithmetic methods. Efficient operation is guaranteed by constrained optimization. The control concept has been extensively validated by simulation runs with load and generation profiles from real life operation in a distribution network located in Vorarlberg, Austria. Results are very promising so that the real implementation as a prototype will be carried out in two distribution networks in two different regions of Austria in the near future.

Global Optimization in the COCONUT Project

In this article, a solver platform for global optimization is presented, as it is developed in the COCONUT project. After a short introduction, a short description is given of the basic algorithmic concept and of all relevant components, the strategy engine, inference engines, and the remaining modules. A compact description of the search graph and its nodes and of the internal model representation using directed acyclic graphs (DAGs) completes the presentation.

Transposition Theorems and Qualification-Free Optimality Conditions

New theorems of the alternative for polynomial constraints (based on the Positivstellensatz from real algebraic geometry) and for linear constraints (generalizing the transposition theorems of Motzkin and Tucker) are proved. Based on these, two Karush-John optimality conditions—holding without any constraint qualification—are proved for single- or multiobjective constrained optimization problems. The first condition applies to polynomial optimization problems only, and gives for the first time necessary and sufficient global optimality conditions for polynomial problems. The second condition applies to smooth local optimization problems and strengthens known local conditions. If some linear or concave constraints are present, the new version reduces the number of constraints for which a constraint qualification is needed to get the Kuhn-Tucker conditions.

Verified Global Optimization for Estimating the Parameters of Nonlinear Models

Nonlinear parameter estimation is usually achieved via the minimization of some possibly non-convex cost function. Interval analysis allows one to derive algorithms for the guaranteed characterization of the set of all global minimizers of such a cost function when an explicit expression for the output of the model is available or when this output is obtained via the numerical solution of a set of ordinary differential equations. However, cost functions involved in parameter estimation are usually challenging for interval techniques, if only because of multi-occurrences of the parameters in the formal expression of the cost. This paper addresses parameter estimation via the verified global optimization of quadratic cost functions. It introduces tools for the minimization of generic cost functions. When an explicit expression of the output of the parametric model is available, significant improvements may be obtained by a new box exclusion test and by careful manipulations of the quadratic cost function. When the model is described by ODEs, some of the techniques available in the previous case may still be employed, provided that sensitivity functions of the model output with respect to the parameters are available.