TACO: a toolkit for AMPL control optimization

We describe a set of extensions to the AMPL modeling language to conveniently model mixed-integer optimal control problems for ODE or DAE dynamic processes. These extensions are realized as AMPL user functions and suffixes and do not require intrusive changes to the AMPL language standard or implementation itself. We describe and provide TACO, a Toolkit for AMPL Control Optimization that reads AMPL stub.nl files and detects the structure of the optimal control problem. This toolkit is designed to facilitate the coupling of existing optimal control software packages to AMPL. We discuss requirements, capabilities, and the current implementation. Using the example of the multiple shooting code for optimal control MUSCOD-II, a direct and simultaneous method for DAE-constrained optimal control, we demonstrate how the problem information provided by the TACO toolkit is interfaced to the solver. In addition, we show how the MS-MINTOC algorithm for mixed-integer optimal control can be used to efficiently solve mixed-integer optimal control problems modeled in AMPL. We use the AMPL extensions to model three control problem examples and we discuss how those extensions affect the representation of optimal control problems. Solutions to these problems are obtained by using MUSCOD-II and MS-MINTOC inside the AMPL environment. A collection of further AMPL control models is provided on the web site http://mintoc.de. MUSCOD-II and MS-MINTOC have been made available on the NEOS Server for Optimization, using the TACO toolkit to enable input of AMPL models.     

DrAmpl: a meta solver for optimization problem analysis

Optimization problems modeled in the AMPL modeling language (Fourer et al., in AMPL: a modeling language for mathematical programming, 2002) may be examined by a set of tools found in the AMPL Solver Library (Gay, in Hooking your solver to AMPL, 1997). DrAmpl is a meta solver which, by use of the AMPL Solver Library, dissects such optimization problems, obtains statistics on their data, is able to symbolically prove or numerically disprove convexity of the functions involved and provides aid in the decision for an appropriate solver. A problem is associated with a number of relevant solvers available on the NEOS Server for Optimization (Czyzyk et al., in IEEE J Comput Sci Eng 5:68–75, 1998) by means of a relational database. We describe the need for such a tool, the design of DrAmpl and some of its consequences, and keep in mind that a similar tool could be developed for other algebraic modeling languages.     

Modeling languages and Condor: metacomputing for optimization

A generic framework is postulated for utilizing the computational resources provided by a metacomputer to concurrently solve a large number of optimization problems generated by a modeling language. An example of the framework using the Condor resource manager and the AMPL and GAMS modeling languages is provided. A mixed integer programming formulation of a feature selection problem from machine learning is used to test the mechanism developed. Due to this application’s computational requirements, the ability to perform optimizations in parallel is necessary in order to obtain results within a reasonable amount of time. Details about the simple and easy to use tool and implementation are presented so that other modelers with applications generating many independent mathematical programs can take advantage of it to significantly reduce solution times. Received: October 28, 1998 / Accepted: December 01, 1999?Published online June 8, 2000     

Rigidity and gluing for Morse and Novikov complexes

We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold (M,) with c₁|_2(M)=[]|_2(M)=0. The rigidity results for these complexes show that the complex of a fixed generic function/hamiltonian is a retract of the Morse (respectively Novikov or Floer) complex of any other sufficiently C⁰ close generic function/hamiltonian. The gluing result is a type of Mayer-Vietoris formula for the Morse complex. It is used to express algebraically the Novikov complex up to isomorphism in terms of the Morse complex of a fundamental domain. Morse cobordisms are used to compare various Morse-type complexes without the need of bifurcation theory.     

Nonlinear and Geometric Programming – Current Status

Great strides have been made in nonlinear programming (NLP) in the last 5 years. In smooth NLP, there are now several reliable and efficient codes capable of solving large problems. Most of these implement GRG or SQP methods, and new software using interior point algorithms is under development. NLP software is now much easier to use, as it is interfaced with many modeling systems, including MSC/NASTRAN, and ANSYS for structural problems, GAMS and AMPL for general optimization, Matlab and Mathcad for general mathematical problems, and the widely used Microsoft Excel spreadsheet. For mixed integer problems, branch and bound and outer approximation codes are now available and are coupled to some of the above modeling systems, while search methods like Tabu Search and Genetic algorithms permit combinatorial, nonsmooth, and nonconvex problems to be attacked.     

Relative komplexe quotienten

Let XS be a holomorphic map, and let RX×SX be an equivalence relation. The restriction of R to the fibre ^–1(S) is denoted by R_s. The quotient X/R is called a relative complex quotient, if the quotient map XX/R is holomorphic over S. Two cases are studied: (C) All fibres of are locally R_s-separable (relative Cartan quotient); (R) All fibres of are holomorphically convex, and R_s is given by tke holomorphic functions on ^–1 (s) (relative Remmert quotient).     

Implementation aspects of interactive multiobjective optimization for modeling environments: the case of GAMS-NIMBUS

Interactive multiobjective optimization methods have provided promising results in the literature but still their implementations are rare. Here we introduce a core structure of interactive methods to enable their convenient implementation. We also demonstrate how this core structure can be applied when implementing an interactive method using a modeling environment. Many modeling environments contain tools for single objective optimization but not for interactive multiobjective optimization. Furthermore, as a concrete example, we present GAMS-NIMBUS Tool which is an implementation of the classification-based NIMBUS method for the GAMS modeling environment. So far, interactive methods have not been available in the GAMS environment, but with the GAMS-NIMBUS Tool we open up the possibility of solving multiobjective optimization problems modeled in the GAMS modeling environment. Finally, we give some examples of the benefits of applying an interactive method by using the GAMS-NIMBUS Tool for solving multiobjective optimization problems modeled in the GAMS environment.     

Differentiability of the minimal average action as a function of the rotation number

Let be a finite composition of exact twist diffeomorphisms. For any real number , letA() denote the minimal average action of -invariant measures with angular rotation number . We prove thatA() is differentiable at every irrational number and that for generic it is not differentiable at rational , thus verifying conjectures of S. Aubry. Moreover, we show that these results are valid for a variational principleh which satisfies the condition which we have called elsewhere (H). As a consequence, we generalize a result due to Bangert concerning geodesics on a two dimensional torus with an arbitrary, but sufficiently smooth metric.supported by NSF grant no. DMS-8806067.01 and a Guggenheim Fellowship.     

Some mildly wild circles in Sn arising from algebraic K-Theory

We study wild embeddings of S ¹ in S ⁿ which are tame in a sense introduced by Quinn. We show that if is a finitely presented group with H ₁()=H ₂()=0, then any finiteness obstruction K ₀() can be realized on the complement of such an embedded S ¹. We also realize trivially symmetric K _–1() obstructions on the complements of such embeddings. For trivially symmetric , the embeddings constructed are shown to be isotopy homogeneous.     

Generalized Cheeger sets related to landslides

We study the maximization problem, among all subsets X of a given domain , of the quotient of the integral in X of a given function f by the integral on the boundary of X of another function g. This is a generalization of the well-known Cheeger problem corresponding to constant functions f,g. The non-constant case is motivated by applications to landslides modeling where the the supremum given by a variational blocking problem appears as a safety coefficient. We prove that this coefficient is equal to the supremum of the shape optimization problem formerly mentioned. For constant data, this amounts to studying the first eigenvalue of the 1-laplacian operator.We prove existence of optimal sets, and give some differential characterization of their internal boundary. We study their symmetry properties using the Steiner symmetrization. In dimension two, we give explicit solutions for data depending only on one variable.

---

Résumé. Nous étudions le probléme de maximisation, parmi les ensembles X inclus dans un domaine fixé , du quotient de lintégrale dune fonction donnée f dans X par lintégrale dune autre fonction g sur le bord de X. Il sagit donc dune généralisation du célébre probléme de Cheeger (correspondant au cas f, g, constants). Le cas non-constant est motivé par des applications aux glissements de terrain, oú le supremum donné par un probléme variationnel de blocage, apparaít comme un coefficient de sreté. Nous démontrons que ce coefficient est égal á loptimum du probléme doptimisation de formes mentionné précédemment. Dans le cas de données constantes, cela revient á étudier la premiére valeur propre de lopérateur 1-laplacien.Nous démontrons lexistence densembles optimaux, et donnons une caractérisation différentielle de leur bord intérieur. Nous étudions leur symétrie á laide de la symétrisation de Steiner. En dimension deux, nous exhibons des solutions explicites dans le cas oú les données ne dépendent que dune variable.

---

---

Received: 18 June 2004, Accepted: 12 July 2004, Published online: 10 December 2004Mathematics Subject Classification (2000): 49J40, 49Q10     

Constant milnor number implies constant multiplicity for quasihomogeneous singularities

We show that for certain topologically trivial deformations of an isolated hypersurface singularity the multiplicity does not change. This applies to all -constant first order deformations and to all -constant deformations of a quasihomogeneous singularity.Partially supported by the Deutsche Forschungsgemeinschaft and the National Science Foundation     

Phase retrieval for probability measures on abelian groups

IfX is a locally compact abelian group, a probability measure onX and its Fourier transform, the mapping | | is obviously not injective. The aim of this article is to find conditions under which the identification of given | | is possible up to a shift and a central symmetry.Research partially supported by the Swiss National Science Foundation.     

A theory of motivation and satisfaction with application to decision support

The paper deals with optimization of allocation of human resources among different activities. It is assumed that an individual is characterized by a risk averse and constant return to scale utility function of two variables: motivation to perform and reward following the performance. The individual is trying to maximize the utility by the best allocation of his time resources among the activities and by selecting the best portfolio of activities. Motivations are regarded, generally, as the product of the individual's preferences (i.e. subjective choice probabilities), productivities of time, output prices, performance and access probabilities, etc., while the rewards are profits or salaries connected with each activity. Satisfaction is defined as the maximum of utility attained for the optimum allocation and selection strategies. It is shown that for the given equitable reward rate, the optimum allocation and portfolio selection strategies can be derived explicitly and the derivation does not require the explicit knowledge of the individual's utility function.     

The Deformation Spaces of Projective Structures on 3-Dimensional Coxeter Orbifolds

A real projective structure on a 3-orbifold is given by locally modeling the orbifold by real projective geometry. We present some methodology to study Coxeter groups which are fundamental groups of 3-orbifolds with representations in and deformation spaces. There are related examples by Benoist. These examples give us nontrivial deformation spaces of projective structures.     

Detection of a signal of known shape in a multichannel system

We consider an n-channel signal detection system. Each of its channels may contain (or not contain) a signal. We assume that the signal is a function of known shape observed in the white Gaussian noise of level > 0. Let k be the number of channels containing signals. We study the asymptotically minimax variant of the detection problem as n depending on k and on the relation signal-noise in the channels. Bibliography: 9 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 88–112.This research was partially supported by the Russian Foundation for Basic Research, grants 00-01-00111, 00-15-96019, 02-01-00262, and 02-01-04001.Translated by Yu. I. Ingster and I. A. Suslina.     

On the magnification of cantor sets and their limit models

For aC ¹⁺ hyperbolic (cookie-cutter) Cantor setC we consider the limits of sequences of closed subsets ofR obtained by arbitrarily high magnifications around different points ofC. It is shown that a well defined set of limit models exists for the infinitesimal geometry, orscenery, in the Cantor set. IfCC} is a diffeomorphic copy ofC then the set of limit models of C is the same as that ofC. Furthermore every limit model is made of Cantor sets which areC ¹⁺ diffeomorphic withC (for some >0, (0,1)), but not all suchC ¹⁺ copies ofC occur in the limit models. We show the relation between this approach to the asymptotic structure of a Cantor set and Sullivan's scaling function. An alternative definition of a fractal is discussed.With 1 Figure     

Implicit elliptic differential equations

Let be a bounded domain in ⁿ (n 3) having a smooth boundary, letY be a closed, connected and locally connected subset of ^h , letf be a real-valued function defined on × ^h × ^nh ×Y, and letL be a linear, second-order elliptic operator. In this paper, the existence of strong solutionsu W ^2,p (, ^h ) W ₀ ^1,p (, ^h ) (n<p<+) to the implicit elliptic equationf(x, u, Du, Lu)=0, whereu=(u ₁,u ₂, ...,u _h ),Du=(Du ₁,Du ₂, ...,Du _h ) andLu=(Lu ₁,Lu ₂, ...,Lu _h ), is established. The abstract framework where the equation is studied is that of set-valued analysis.Dedicated to Professor G. Pulvirenti on the occasion of his sixtieth birthday     

A MIQCP formulation for B-spline constraints

This paper presents a mixed-integer quadratically constrained programming (MIQCP) formulation for B-spline constraints. The formulation can be used to obtain an exact MIQCP reformulation of any spline-constrained optimization problem problem, provided that the polynomial spline functions are continuous. This reformulation allows practitioners to use a general-purpose MIQCP solver, instead of a special-purpose spline solver, when solving B-spline constrained problems. B-splines are a powerful and widely used modeling tool, previously restricted from optimization due to lack of solver support. This contribution may encourage practitioners to use B-splines to model constraint functions. However, as the numerical study suggests, there is still a large gap between the solve times of the general-purpose solvers using the proposed formulation, and the special-purpose spline solver CENSO, the latter being significantly lower.