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An Algorithm Based on Semidefinite Programming for Finding Minimax
Takustr. 7 Zuse Institute Berlin 14195 Berlin Germany BELMIRO P.M. DUARTE,GUILLAUME SAGNOL,WENG KEE WONG An algorithm based on Semidefinite Programming for finding minimax optimal designs ZIB Report 18-01 (December 2017) Zuse Institute Berlin Takustr. 7 14195 Berlin Germany Telephone: +49 30-84185-0 Telefax: +49 30-84185-125 E-mail: [email protected] URL: http://www.zib.de ZIB-Report (Print) ISSN 1438-0064 ZIB-Report (Internet) ISSN 2192-7782 An algorithm based on Semidefinite Programming for finding minimax optimal designs Belmiro P.M. Duarte a,b, Guillaume Sagnolc, Weng Kee Wongd aPolytechnic Institute of Coimbra, ISEC, Department of Chemical and Biological Engineering, Portugal. bCIEPQPF, Department of Chemical Engineering, University of Coimbra, Portugal. cTechnische Universität Berlin, Institut für Mathematik, Germany. dDepartment of Biostatistics, Fielding School of Public Health, UCLA, U.S.A. Abstract An algorithm based on a delayed constraint generation method for solving semi- infinite programs for constructing minimax optimal designs for nonlinear models is proposed. The outer optimization level of the minimax optimization problem is solved using a semidefinite programming based approach that requires the de- sign space be discretized. A nonlinear programming solver is then used to solve the inner program to determine the combination of the parameters that yields the worst-case value of the design criterion. The proposed algorithm is applied to find minimax optimal designs for the logistic model, the flexible 4-parameter Hill homoscedastic model and the general nth order consecutive reaction model, and shows that it (i) produces designs that compare well with minimax D−optimal de- signs obtained from semi-infinite programming method in the literature; (ii) can be applied to semidefinite representable optimality criteria, that include the com- mon A−; E−; G−; I− and D-optimality criteria; (iii) can tackle design problems with arbitrary linear constraints on the weights; and (iv) is fast and relatively easy to use. -
Solving Mixed Integer Linear and Nonlinear Problems Using the SCIP Optimization Suite
Takustraße 7 Konrad-Zuse-Zentrum D-14195 Berlin-Dahlem fur¨ Informationstechnik Berlin Germany TIMO BERTHOLD GERALD GAMRATH AMBROS M. GLEIXNER STEFAN HEINZ THORSTEN KOCH YUJI SHINANO Solving mixed integer linear and nonlinear problems using the SCIP Optimization Suite Supported by the DFG Research Center MATHEON Mathematics for key technologies in Berlin. ZIB-Report 12-27 (July 2012) Herausgegeben vom Konrad-Zuse-Zentrum f¨urInformationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Telefon: 030-84185-0 Telefax: 030-84185-125 e-mail: [email protected] URL: http://www.zib.de ZIB-Report (Print) ISSN 1438-0064 ZIB-Report (Internet) ISSN 2192-7782 Solving mixed integer linear and nonlinear problems using the SCIP Optimization Suite∗ Timo Berthold Gerald Gamrath Ambros M. Gleixner Stefan Heinz Thorsten Koch Yuji Shinano Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany, fberthold,gamrath,gleixner,heinz,koch,[email protected] July 31, 2012 Abstract This paper introduces the SCIP Optimization Suite and discusses the ca- pabilities of its three components: the modeling language Zimpl, the linear programming solver SoPlex, and the constraint integer programming frame- work SCIP. We explain how these can be used in concert to model and solve challenging mixed integer linear and nonlinear optimization problems. SCIP is currently one of the fastest non-commercial MIP and MINLP solvers. We demonstrate the usage of Zimpl, SCIP, and SoPlex by selected examples, give an overview of available interfaces, and outline plans for future development. ∗A Japanese translation of this paper will be published in the Proceedings of the 24th RAMP Symposium held at Tohoku University, Miyagi, Japan, 27{28 September 2012, see http://orsj.or. -
Linear Underestimators for Bivariate Functions with a Fixed Convexity Behavior
Takustraße 7 Konrad-Zuse-Zentrum D-14195 Berlin-Dahlem fur¨ Informationstechnik Berlin Germany MARTIN BALLERSTEINy AND DENNIS MICHAELSz AND STEFAN VIGERSKE Linear Underestimators for bivariate functions with a fixed convexity behavior This work is part of the Collaborative Research Centre “Integrated Chemical Processes in Liquid Multiphase Systems” (CRC/Transregio 63 “InPROMPT”) funded by the German Research Foundation (DFG). Main parts of this work has been finished while the second author was at the Institute for Operations Research at ETH Zurich and financially supported by DFG through CRC/Transregio 63. The first and second author thank the DFG for its financial support. Thethirdauthor was supported by the DFG Research Center MATHEON Mathematics for key technologies and the Research Campus MODAL Mathematical Optimization and Data Analysis Laboratories in Berlin. y Eidgenossische¨ Technische, Hochschule Zurich,¨ Institut fur¨ Operations Research, Ramistrasse¨ 101, 8092 Zurich (Switzerland) z Technische Universitat¨ Dortmund, Fakultat¨ fur¨ Mathematik, M/518, Vogelpothsweg 87, 44227 Dortmund (Germany) ZIB-Report 13-02 (revised) (February 2015) Herausgegeben vom Konrad-Zuse-Zentrum fur¨ Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Telefon: 030-84185-0 Telefax: 030-84185-125 e-mail: [email protected] URL: http://www.zib.de ZIB-Report (Print) ISSN 1438-0064 ZIB-Report (Internet) ISSN 2192-7782 Technical Report Linear Underestimators for bivariate functions with a fixed convexity behavior∗ A documentation for the SCIP constraint handler cons bivariate Martin Ballersteiny Dennis Michaelsz Stefan Vigerskex February 23, 2015 y Eidgenossische¨ Technische Hochschule Zurich¨ Institut fur¨ Operations Research Ramistrasse¨ 101, 8092 Zurich (Switzerland) z Technische Universitat¨ Dortmund Fakultat¨ fur¨ Mathematik, M/518 Vogelpothsweg 87, 44227 Dortmund (Germany) x Zuse Institute Berlin Takustr. -
Case Management Task Assignment Using Optaplanner
Masaryk University Faculty of Informatics Case Management Task Assignment Using OptaPlanner Master’s Thesis Bc. Marián Macik Brno, Fall 2016 Replace this page with a copy of the official signed thesis assignment anda copy of the Statement of an Author. Declaration Hereby I declare that this paper is my original authorial work, which I have worked out on my own. All sources, references, and literature used or excerpted during elaboration of this work are properly cited and listed in complete reference to the due source. Bc. Marián Macik Advisor: Mgr. Marek Grác, Ph.D. i Acknowledgement Here I would like to thank my family, friends and colleagues for their support during the work on this thesis. Moreover, I thank my consul- tant, Mgr. Ivo Bek, for guiding me during my work and for helping me overcome issues experienced when writing this thesis. I would also like to thank my advisor, Mgr. Marek Grác, Ph.D., for his help regarding the text of the thesis and for his advice in initial stages of the work. Finally, I would especially like to thank Maciej Swiderski for his help with the configuration of jBPM engine and Geoffrey De Smet for his help with OptaPlanner. ii Abstract The aim of the thesis is to analyse, design and implement a module for automated task assignment by integrating jBPM engine and Op- taPlanner. The thesis describes case management, its difference from business process management and their notations. After that, jBPM engine together with OptaPlanner are explained. In the second half of the thesis, the actual implementation and prototype application are presented, including the performance tests in different OptaPlanner configurations and scenarios. -
Nonlinear Programming Techniques for Operative Planning in Large Drinking Water Networks Jens Burgschweiger1, Bernd Gnädig2 and Marc C
14 The Open Applied Mathematics Journal, 2009, 3, 14-28 Open Access Nonlinear Programming Techniques for Operative Planning in Large Drinking Water Networks Jens Burgschweiger1, Bernd Gnädig2 and Marc C. Steinbach3,* 1Berliner Wasserbetriebe, Abt. NA-G/W, 10864 Berlin, Germany 2Düsseldorf, Germany 3Leibniz Universität Hannover, IfAM, Welfengarten 1, 30167 Hannover, Germany Abstract: Mathematical decision support for operative planning in water supply systems is highly desirable; it leads, however, to very difficult optimization problems. We propose a nonlinear programming approach that yields practically satisfactory operating schedules in acceptable computing time even for large networks. Based on a carefully designed model supporting gradient-based optimization algorithms, this approach employs a special initialization strategy for convergence acceleration, special minimum up and down time constraints together with pump aggregation to handle switching decisions, and several network reduction techniques for further speed-up. Results for selected application scenarios at Berliner Wasserbetriebe demonstrate the success of the approach. Keywords: Water supply, large-scale nonlinear programming, convergence acceleration, discrete decisions, network reduction. INTRODUCTION • experiments with various optimization models and methods [36, 37], Stringent requirements on cost effectiveness and environmental compatibility generate an increased demand • a first nonlinear programming (NLP) model for model-based decision support tools for designing and developed under GAMS [38], operating municipal water supply systems. This paper deals • numerical results for a substantially reduced with the minimum cost operation of drinking water network graph using (under GAMS) the SQP codes networks. Operative planning in water networks is difficult: CONOPT, SNOPT, and the augmented Lagrangian a sound mathematical model leads to nonlinear mixed- code MINOS. -
Julia: a Modern Language for Modern ML
Julia: A modern language for modern ML Dr. Viral Shah and Dr. Simon Byrne www.juliacomputing.com What we do: Modernize Technical Computing Today’s technical computing landscape: • Develop new learning algorithms • Run them in parallel on large datasets • Leverage accelerators like GPUs, Xeon Phis • Embed into intelligent products “Business as usual” will simply not do! General Micro-benchmarks: Julia performs almost as fast as C • 10X faster than Python • 100X faster than R & MATLAB Performance benchmark relative to C. A value of 1 means as fast as C. Lower values are better. A real application: Gillespie simulations in systems biology 745x faster than R • Gillespie simulations are used in the field of drug discovery. • Also used for simulations of epidemiological models to study disease propagation • Julia package (Gillespie.jl) is the state of the art in Gillespie simulations • https://github.com/openjournals/joss- papers/blob/master/joss.00042/10.21105.joss.00042.pdf Implementation Time per simulation (ms) R (GillespieSSA) 894.25 R (handcoded) 1087.94 Rcpp (handcoded) 1.31 Julia (Gillespie.jl) 3.99 Julia (Gillespie.jl, passing object) 1.78 Julia (handcoded) 1.2 Those who convert ideas to products fastest will win Computer Quants develop Scientists prepare algorithms The last 25 years for production (Python, R, SAS, DEPLOY (C++, C#, Java) Matlab) Quants and Computer Compress the Scientists DEPLOY innovation cycle collaborate on one platform - JULIA with Julia Julia offers competitive advantages to its users Julia is poised to become one of the Thank you for Julia. Yo u ' v e k i n d l ed leading tools deployed by developers serious excitement. -
Treball (1.484Mb)
Treball Final de Màster MÀSTER EN ENGINYERIA INFORMÀTICA Escola Politècnica Superior Universitat de Lleida Mòdul d’Optimització per a Recursos del Transport Adrià Vall-llaura Salas Tutors: Antonio Llubes, Josep Lluís Lérida Data: Juny 2017 Pròleg Aquest projecte s’ha desenvolupat per donar solució a un problema de l’ordre del dia d’una empresa de transports. Es basa en el disseny i implementació d’un model matemàtic que ha de permetre optimitzar i automatitzar el sistema de planificació de viatges de l’empresa. Per tal de poder implementar l’algoritme s’han hagut de crear diversos mòduls que extreuen les dades del sistema ERP, les tracten, les envien a un servei web (REST) i aquest retorna un emparellament òptim entre els vehicles de l’empresa i les ordres dels clients. La primera fase del projecte, la teòrica, ha estat llarga en comparació amb les altres. En aquesta fase s’ha estudiat l’estat de l’art en la matèria i s’han repassat molts dels models més importants relacionats amb el transport per comprendre’n les seves particularitats. Amb els conceptes ben estudiats, s’ha procedit a desenvolupar un nou model matemàtic adaptat a les necessitats de la lògica de negoci de l’empresa de transports objecte d’aquest treball. Posteriorment s’ha passat a la fase d’implementació dels mòduls. En aquesta fase m’he trobat amb diferents limitacions tecnològiques degudes a l’antiguitat de l’ERP i a l’ús del sistema operatiu Windows. També han sorgit diferents problemes de rendiment que m’han fet redissenyar l’extracció de dades de l’ERP, el càlcul de distàncies i el mòdul d’optimització. -
Vrpsolver: a Generic Exact Solver for Vehicle Routing and Related Problems
VRPSolver: a Generic Exact Solver for Vehicle Routing and Related Problems Eduardo Uchoa Departamento de Engenharia de Produ¸c~ao Universidade Federal Fluminense Niter´oi- RJ, Brasil SPOC20 { School on Advanced BCP Tools { Paris A Generic Exact Solver for VRP Vehicle Routing Problem (VRP) One of the most widely studied in Combinatorial Optimization: +7,500 works published only in 2018 (Google Scholar), mostly heuristics Direct application in the real systems that distribute goods and provide services. Optimized routes can: save a lot of money reduce the environmental impacts of transportation SPOC20 { School on Advanced BCP Tools { Paris A Generic Exact Solver for VRP Example: In 2017, Amazon spent USD 21.7B in shipping, 14.2% of its net sales. As customers demand quicker service, this percentage is growing! SPOC20 { School on Advanced BCP Tools { Paris A Generic Exact Solver for VRP Vehicle Routing Problem (VRP) Reflecting the variety of real transportation systems, VRP literature is spread into hundreds of variants. For example, there are variants that consider: Vehicle capacities, Time windows, Heterogeneous fleets, Multiple depots, Split delivery, pickup and delivery, backhauling, Arc routing (Ex: garbage collection), etc, etc. Articles describing new variants appear every week SPOC20 { School on Advanced BCP Tools { Paris A Generic Exact Solver for VRP Outline of the presentation Part I - Advances on Exact CVRP algorithms Review of the advances in the last 15 years Part II - From CVRP to other classic VRP variants Part III - A Generic -
OPTIMAL CONTROL of NONHOLONOMIC MECHANICAL SYSTEMS By
OPTIMAL CONTROL OF NONHOLONOMIC MECHANICAL SYSTEMS by Stuart Marcus Rogers A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics Department of Mathematical and Statistical Sciences University of Alberta ⃝c Stuart Marcus Rogers, 2017 Abstract This thesis investigates the optimal control of two nonholonomic mechanical systems, Suslov's problem and the rolling ball. Suslov's problem is a nonholonomic variation of the classical rotating free rigid body problem, in which the body angular velocity Ω(t) must always be orthogonal to a prescribed, time-varying body frame vector ξ(t), i.e. hΩ(t); ξ(t)i = 0. The motion of the rigid body in Suslov's problem is actuated via ξ(t), while the motion of the rolling ball is actuated via internal point masses that move along rails fixed within the ball. First, by applying Lagrange-d'Alembert's principle with Euler-Poincar´e'smethod, the uncontrolled equations of motion are derived. Then, by applying Pontryagin's minimum principle, the controlled equations of motion are derived, a solution of which obeys the uncontrolled equations of motion, satisfies prescribed initial and final conditions, and minimizes a prescribed performance index. Finally, the controlled equations of motion are solved numerically by a continuation method, starting from an initial solution obtained analytically (in the case of Suslov's problem) or via a direct method (in the case of the rolling ball). ii Preface This thesis contains material that has appeared in a pair of papers, one on Suslov's problem and the other on rolling ball robots, co-authored with my supervisor, Vakhtang Putkaradze. -
AMPL Academic Price List These Prices Apply to Purchases by Degree-Awarding Institutions for Use in Noncommercial Teaching and Research Activities
AMPL Optimization Inc. 211 Hope Street #339 Mountain View, CA 94041, U.S.A. [email protected] — www.ampl.com +1 773-336-AMPL (-2675) AMPL Academic Price List These prices apply to purchases by degree-awarding institutions for use in noncommercial teaching and research activities. Products covered by academic prices are full-featured and have no arbitrary limits on problem size. Single Floating AMPL $400 $600 Linear/quadratic solvers: CPLEX free 1-year licenses available: see below Gurobi free 1-year licenses available: see below Xpress free 1-year licenses available: see below Nonlinear solvers: Artelys Knitro $400 $600 CONOPT $400 $600 LOQO $300 $450 MINOS $300 $450 SNOPT $320 $480 Alternative solvers: BARON $400 $600 LGO $200 $300 LINDO Global $700 $950 . Basic $400 $600 (limited to 3200 nonlinear variables) Web-based collaborative environment: QuanDec $1400 $2100 AMPL prices are for the AMPL modeling language and system, including the AMPL command-line and IDE development tools and the AMPL API programming libraries. To make use of AMPL it is necessary to also obtain at least one solver having an AMPL interface. Solvers may be obtained from us or from another source. As listed above, we offer many popular solvers for direct purchase; refer to www.ampl.com/products/solvers/solvers-we-sell/ to learn more, including problem types supported and methods used. Our prices for these solvers apply to the versions that incorporate an AMPL interface; a previously or concurrently purchased copy of the AMPL software is needed to use these versions. Programming libraries and other forms of these solvers are not included. -
MP-Opt-Model User's Manual, Version
MP-Opt-Model User's Manual Version 2.1 Ray D. Zimmerman August 25, 2020 © 2020 Power Systems Engineering Research Center (PSerc) All Rights Reserved Contents 1 Introduction6 1.1 Background................................6 1.2 License and Terms of Use........................7 1.3 Citing MP-Opt-Model..........................8 1.4 MP-Opt-Model Development.......................8 2 Getting Started9 2.1 System Requirements...........................9 2.2 Installation................................9 2.3 Sample Usage............................... 10 2.4 Documentation.............................. 14 3 MP-Opt-Model { Overview 15 4 Solver Interface Functions 16 4.1 LP/QP Solvers { qps master ...................... 16 4.1.1 QP Example............................ 19 4.2 MILP/MIQP Solvers { miqps master .................. 20 4.2.1 MILP Example.......................... 22 4.3 NLP Solvers { nlps master ....................... 22 4.3.1 NLP Example 1.......................... 25 4.3.2 NLP Example 2.......................... 26 4.4 Nonlinear Equation Solvers { nleqs master .............. 29 4.4.1 NLEQ Example 1......................... 31 4.4.2 NLEQ Example 2......................... 34 5 Optimization Model Class { opt model 38 5.1 Adding Variables............................. 38 5.1.1 Variable Subsets......................... 39 5.2 Adding Constraints............................ 40 5.2.1 Linear Constraints........................ 40 5.2.2 General Nonlinear Constraints.................. 41 5.3 Adding Costs............................... 42 5.3.1 Quadratic -
Clean Sky – Awacs Final Report
Clean Sky Joint Undertaking AWACs – Adaptation of WORHP to Avionics Constraints Clean Sky – AWACs Final Report Table of Contents 1 Executive Summary ......................................................................................................................... 2 2 Summary Description of Project Context and Objectives ............................................................... 3 3 Description of the main S&T results/foregrounds .......................................................................... 5 3.1 Analysis of Optimisation Environment .................................................................................... 5 3.1.1 Problem sparsity .............................................................................................................. 6 3.1.2 Certification aspects ........................................................................................................ 6 3.2 New solver interfaces .............................................................................................................. 6 3.3 Conservative iteration mode ................................................................................................... 6 3.4 Robustness to erroneous inputs ............................................................................................. 7 3.5 Direct multiple shooting .......................................................................................................... 7 3.6 Grid refinement ......................................................................................................................