Numerical Nonlinear Optimization with WORHP · Numerical Nonlinear Optimization with WORHP Christof...

Centre for Industrial Mathematics Centre for Industrial Mathematics

Numerical Nonlinear Optimization

with WORHP

Christof Büskens

London, 8.9.2011

Optimierung &

Optimale Steuerung

Centre for Industrial Mathematics

Outline

Nonlinear Optimization

WORHP

Concept

Ideas

Features

Results

Optimization &

Optimal Control


WG Optimisation and Optimal Control (2004)

16 + 4 scientists + additional staff

Industrial & scientific research


Working Areas

Parametric

Sensitivity

Analysis

Modeling &

Simulation

Real Time

Optimization &

Optimal Control

Identification

Optimization

Optimal

Control

Feedback

Control

Mathematics

Theory

Scientific Computation


Sparse NLP Solver

WORHP

We Optimize Really Huge Problems

> 1.000.000.000 variables

> 2.000.000.000 constraints


WORHP, just another NLP solver?

<1997

SNOPT

2000

IPFILTER

2001

KNITRO

2002

IPOPT

2010

WORHP

History of sparse NLP solvers:

Can WORHP be competative?


User, Market & Scientific Requirements In contrast to most established and „grown”

NLP solvers, WORHP has undergone

extensive design on the drawing board before

its implementation was started, making use of

-user requirements,

-current architectures,

-computational standards and compilers

to construct a modern NLP solver for large-

scale nonlinear optimisation.

224 Solvers

28 for nonlinear constraints

19 dense & sparse

Programming Language

C / C++

3Fortran / C

9Fortran

16

Commercialization / License Type

Company; 9

GPL type; 3

Direct; 14

Distribution by Country

AT; 2

DK; 1

CH; 1

PT; 1

BE; 1

FI; 1

RU; 1

GR; 1

BR; 1

DE

5

UK

5

AU; 1

US

13

EU/ESA

16

Algorithms

12

8

5

1 1 1 1

Sequential Quadratic Programming

Primal-Dual Interior Point

Generalized Reduced Gradient

Successive Quadratic Programming

Benders Decomposition

Levenberg Marquardt

Coordinate Search

SQP

IP

GR

Optimization Classes

4330

36

4

47

2

1

1014

11

6

20

LP-problem, mixed integer, stochastic

QP-problem, mixed integer

Semidefinite and 2nd-order cone prog.

Geometric programming

Non Linear programming

Minimization of nonsmooth functions

Semi-infinite programming

Mixed integer nonlinear programming

Network constraints

Special/constraint solvers

Dynamic programming

Control problems


The WORHP Team in Bremen

Prof. Dr.

Christof Büskens

Project leader Bremen

Dipl. math.

Tim Nikolayzik

NLP-Correction

Methods

Dennis Wassel, MPhil

“Sentinel” - Project,

associated with NLP

SADCO Project

Sonja Rauski

Hessian-Approximations


n: large m: large nonlinear

iterative solution



iteration number

iterative solution



line search or filter search direction merit function

step size



1st derivatives

2nd derivatives or approx. (BFGS)

globalisation

relaxation

sparse structure

General Idea/Problems


globalisation

relaxation

linear algebra

General Idea/Problems


WORHP (SQP/IP)

computational action

mathematical action


Finite Differences

are expensive

are inexact

are only interesting for „black-box“ problems

Really?


Group Approach by Graph Coloring

Assuming the Jacobian to have the following structure:

Usual approach:

Group approach:

(Extension for second derivatives by pair groups)

Numerical example: (Rayleigh optimal control problem)

Usual: 100.000 evaluations

Group: 6 evaluations (NP-hard)


Complex Numerical Differentiation [Martins, Kroo, Alonso]

Classical Approach:

Consider

Cauchy Riemann„s equation:

Hence:

No Cancellation Error!


SBFGS (Sparse-BFGS)

Consider

SBFGS considers this sparsity structure

SBFGS performs a BFGS update on the three blocks

Problem: Intersections!

Solution: Convexity Shifts


SBFGS

Theorem: (Superlinear Convergence) [Kalmbach, B.]

Let and appropriate functions,

s.t. and

for all and proper, symmetric

and positive definite start matrices.

Then converges superlinearly towards .

Proof: Segmentation, rang 2M update, convexity shift of kernel,

+[Griewank, Toint]: Local Convergence Analysis for Partitioned Quasi-Newton Updates, 1982.


Interfaces Fortran

Traditional

Basic-Feature

Full-Feature

C/C++

Traditional

Basic-Feature

Full-Feature

AMPL

MATLAB/SIMULINK

Plattforms:

Linux/Unix

Windows

Mac OS


CUTEr 920 problems (academical & real life)

Small and dense

Large and sparse

Implemented in AMPL

Solvers used for validation:

SNOPT KNITRO IPOPT WORHP

[<1997] [2001] [2002] [2010]


CUTEr(920)

SNOPT

7.2-8

Problems solved 827

Optimal level 810

Acceptable level 17

Not solved 93

Percentage 89,89%

Time 79569 s

KNITRO

7.0.0

887

882

5

33

96,41%

32792 s

IPOPT

3.9.2

877

869

8

43

95,33%

27056 s

WORHP

1.0

918

911

7

2

99,79%

5060 s


CUTEr


COPS 3.0 68 problems (applications)

Sparse

Midsized and Large

Implemented in AMPL

Solvers used for validation:

SNOPT KNITRO IPOPT WORHP


COPS 3.0(68)

SNOPT

7.2-8

Problems solved 63

Optimal level 60

Acceptable level 3

Not solved 5

Percentage 94,12%

Time 8858 s

KNITRO

7.0.0

64

64

0

4

94,12%

6352 s

IPOPT

3.9.2

68

68

0

0

100%

5682 s

WORHP

1.0

68

67

1

0

100%

1463 s


COPS 3.0


New large-scale NLP solver WORHP

> 1,000,000,000 variables

> 2,000,000,000 constraints

Derivatives and SBFGS

Several interfaces

Robust

Real life application

Conclusion


Thank you!

Please visit: www.worhp.de

Numerical Nonlinear Optimization with WORHP · Numerical Nonlinear Optimization with WORHP Christof...

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