Numerical Nonlinear Optimization with WORHP
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1 Numerical Nonlinear Optimization with WORHP Christof Büskens Optimierung & Optimale Steuerung London,
2 Optimization & Optimal Control Nonlinear Optimization WORHP Concept Ideas Features Results Outline
3 WG Optimisation and Optimal Control (2004) scientists + additional staff Industrial & scientific research
4 Working Areas Feedback Control Modeling & Simulation Mathematics Real Time Optimization & Theory Optimal Control Scientific Computation Identification Parametric Sensitivity Analysis Optimal Control Optimization
5 Sparse NLP Solver WORHP We Optimize Really Huge Problems > variables > constraints
6 <1997 SNOPT Centre for WORHP, just another NLP solver? History of sparse NLP solvers: 2000 IPFILTER 2001 KNITRO 2002 IPOPT Can WORHP be competative? 2010 WORHP
7 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 largescale nonlinear optimisation. Direct; Solvers 28 for nonlinear constraints 19 dense & sparse Commercialization / License Type Company; 9 Distribution by Country Programming Language C / C++ 3 Fortran / C 9 Fortran 16 5 Algorithms GR SQP IP 8 12 Sequential Quadratic Programming Primal-Dual Interior Point Generalized Reduced Gradient Successive Quadratic Programming Benders Decomposition Levenberg Marquardt Coordinate Search Optimization Classes GPL type; RU; 3 1 BR; 1 AU; 1 US 13 EU/ESA 16 LP-problem, mixed integer, stochastic QP-problem, mixed integer Semidefinite and 2nd-order cone prog. Geometric programming Non Linear programming Minimization of nonsmooth functions UK 5 DE 5 GR; 1 AT; 2 DK; 1 CH; 1 PT; 1 BE; 1 FI; Semi-infinite programming Mixed integer nonlinear programming Network constraints Special/constraint solvers
8 The WORHP Team in Bremen Prof. Dr. Dipl. math. SADCO Project Christof Büskens Tim Nikolayzik Sonja Rauski Dennis Wassel, MPhil Project leader Bremen NLP-Correction Methods Hessian-Approximations Sentinel - Project, associated with NLP
9 Nonlinear Optimization n: large nonlinear m: large iterative solution
10 Nonlinear Optimization iteration number iterative solution
11 Nonlinear Optimization step size line search or filter merit function search direction
12 General Idea/Problems 2 nd derivatives or approx. (BFGS) relaxation 1 st derivatives globalisation sparse structure
13 General Idea/Problems relaxation globalisation linear algebra
14 WORHP (SQP/IP) mathematical action computational action
15 Finite Differences are expensive are inexact are only interesting for black-box problems Really?
16 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: evaluations Group: 6 evaluations (NP-hard)
17 Complex Numerical Differentiation [Martins, Kroo, Alonso] Classical Approach: Consider Cauchy Riemann s equation: Hence: No Cancellation Error!
18 SBFGS (Sparse-BFGS) Consider SBFGS considers this sparsity structure SBFGS performs a BFGS update on the three blocks Problem: Intersections! Solution: Convexity Shifts
19 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.
20 Fortran Traditional Interfaces Basic-Feature Full-Feature C/C++ Traditional Plattforms: Basic-Feature Linux/Unix Full-Feature Windows AMPL MATLAB/SIMULINK Mac OS
21 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]
22 CUTEr(920) SNOPT KNITRO IPOPT WORHP 1.0 Problems solved Optimal level Acceptable level Not solved Percentage 89,89% 96,41% 95,33% 99,79% Time s s s 5060 s
23 CUTEr
24 CUTEr
25 CUTEr
26 COPS problems (applications) Sparse Midsized and Large Implemented in AMPL Solvers used for validation: SNOPT KNITRO IPOPT WORHP
27 COPS 3.0(68) SNOPT KNITRO IPOPT WORHP 1.0 Problems solved Optimal level Acceptable level Not solved Percentage 94,12% 94,12% 100% 100% Time 8858 s 6352 s 5682 s 1463 s
28 COPS 3.0
29 Conclusion 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
30 Thank you! Please visit:
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