WORHP, pronounced like the English word "warp

Warp

Warp, warped or warping may refer to:- Books and comics :* WaRP Graphics, an alternative comics publisher* Warp , comic book series published by First Comics based on the play Warp!...

", also referred to as eNLP (European NLP

Nonlinear programming

In mathematics, nonlinear programming is the process of solving a system of equalities and inequalities, collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function are...

 solver) by ESA, is a mathematical software library for solving continuous large scale nonlinear optimization problems numerically. The acronym WORHP is sometimes spelled out as "We Optimize Really Huge Problems", its primary intended application. WORHP is a hybrid Fortran and C

C (programming language)

C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....

 implementation and can be used from C/C++

C++

C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...

 and Fortran programs using three different interfaces of varying complexity and flexibility, and the modelling environments MATLAB

MATLAB

MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...

 and AMPL

AMPL

AMPL, an acronym for "A Mathematical Programming Language", is an algebraic modeling language for describing and solving high-complexity problems for large-scale mathematical computation AMPL, an acronym for "A Mathematical Programming Language", is an algebraic modeling language for describing and...

.

Problem formulation

WORHP is designed to solve problems of the form

subject to

with sufficiently smooth functions (objective) and (constraints) that may be nonlinear, and need not necessarily be convex. Even problems with large dimensions and can be solved efficiently, if the problem is sufficiently sparse.
Cases where objective and constraints cannot be evaluated separately, or where constraints can be evaluated element-wise can be exploited by WORHP to increase the computational efficiency.

Derivatives

WORHP requires the first derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 (Gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

) of and of (Jacobian) and second derivatives (Hessian matrix

Hessian matrix

In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...

) of the Lagrange function; in a modelling environment like AMPL, these are provided by automatic differentiation

Automatic differentiation

In mathematics and computer algebra, automatic differentiation , sometimes alternatively called algorithmic differentiation, is a set of techniques to numerically evaluate the derivative of a function specified by a computer program...

 methods, but need to be provided by the caller in other environments. First and second derivatives can be approximated by WORHP using finite differences. To reduce the otherwise prohibitively high number of necessary function evaluations in large scale sparse

Sparse matrix

In the subfield of numerical analysis, a sparse matrix is a matrix populated primarily with zeros . The term itself was coined by Harry M. Markowitz....

 problems, graph colouring theory is used to group first and second partial derivatives. Second derivatives may also be approximated using variations of the classic BFGS method

BFGS method

In numerical optimization, the Broyden–Fletcher–Goldfarb–Shanno method is a method for solving nonlinear optimization problems ....

, including block-diagonal or sparse BFGS matrices.

Development

Development of WORHP started in 2006 with funding from DLR and was continued under the eNLP label after 2008 with support by ESA / ESTEC

European Space Research and Technology Centre

The European Space Research and Technology Centre is the European Space Agency's main technology development and test centre for spacecraft and space technology. It is situated in Noordwijk, South Holland, in the western Netherlands....

 together with the Interior-Point

Interior point method

Interior point methods are a certain class of algorithms to solve linear and nonlinear convex optimization problems.The interior point method was invented by John von Neumann...

 solver ipfilter
(whose inclusion in eNLP was discontinued after 2010) to develop a European NLP solver for use in trajectory optimisation, mission analysis and aerospace applications in general.

The development of WORHP is led by the Steinbeis-Forschungszentrum Optimierung, Steuerung und Regelung and scientists of the Optimization and Optimal Control Group at the University of Bremen

University of Bremen

The University of Bremen is a university of approximately 23,500 people from 126 countries that are studying, teaching, researching, and working in Bremen, Germany...

, and at the Bundeswehr University of Munich.

Applications

Thus far WORHP has mostly been used in academic environments, such as research projects and final-year theses in various German universities.

WORHP has also been integrated into trajectory analysis tools such as LOTNAV
and ASTOS

Astos

ASTOS is a Trajectory optimization and simulation tool for launch and re-entry missions, orbit transfers, design optimization and for re-entry safety assessments. It solves Aerospace problems with a data driven interface and automatic initial guesses...

, and is currently evaluated at ESOC

European Space Operations Centre

The European Space Operations Centre is responsible for controlling ESA satellites and space probes. The centre is located in Darmstadt, Germany. It is Mission Control for most of the space projects of the ESA. Since its creation in 1967, the centre has operated 60 European space missions...

. It is also used as local optimisation tool in ESA's PRESTIGE
project awarded to University of Bremen and Politecnico di Milano on Multidisciplinary design optimization

Multidisciplinary design optimization

Multi-disciplinary design optimization is a field of engineering that uses optimization methods to solve design problems incorporating a number of disciplines. As defined by Prof. Carlo Poloni, MDO is "the art of finding the best compromise"...

.