What Could a Million Cores Do to Solve Integer Programs?
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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. -
How to Use Your Favorite MIP Solver: Modeling, Solving, Cannibalizing
How to use your favorite MIP Solver: modeling, solving, cannibalizing Andrea Lodi University of Bologna, Italy [email protected] January-February, 2012 @ Universit¨atWien A. Lodi, How to use your favorite MIP Solver Setting • We consider a general Mixed Integer Program in the form: T maxfc x : Ax ≤ b; x ≥ 0; xj 2 Z; 8j 2 Ig (1) where matrix A does not have a special structure. A. Lodi, How to use your favorite MIP Solver 1 Setting • We consider a general Mixed Integer Program in the form: T maxfc x : Ax ≤ b; x ≥ 0; xj 2 Z; 8j 2 Ig (1) where matrix A does not have a special structure. • Thus, the problem is solved through branch-and-bound and the bounds are computed by iteratively solving the LP relaxations through a general-purpose LP solver. A. Lodi, How to use your favorite MIP Solver 1 Setting • We consider a general Mixed Integer Program in the form: T maxfc x : Ax ≤ b; x ≥ 0; xj 2 Z; 8j 2 Ig (1) where matrix A does not have a special structure. • Thus, the problem is solved through branch-and-bound and the bounds are computed by iteratively solving the LP relaxations through a general-purpose LP solver. • The course basically covers the MIP but we will try to discuss when possible how crucial is the LP component (the engine), and how much the whole framework is built on top the capability of effectively solving LPs. • Roughly speaking, using the LP computation as a tool, MIP solvers integrate the branch-and-bound and the cutting plane algorithms through variations of the general branch-and-cut scheme [Padberg & Rinaldi 1987] developed in the context of the Traveling Salesman Problem (TSP). -
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. -
Nonlinear Integer Programming ∗
Nonlinear Integer Programming ∗ Raymond Hemmecke, Matthias Koppe,¨ Jon Lee and Robert Weismantel Abstract. Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Nu- merous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considera- tions and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research. Raymond Hemmecke Otto-von-Guericke-Universitat¨ Magdeburg, FMA/IMO, Universitatsplatz¨ 2, 39106 Magdeburg, Germany, e-mail: [email protected] Matthias Koppe¨ University of California, Davis, Dept. of Mathematics, One Shields Avenue, Davis, CA, 95616, USA, e-mail: [email protected] Jon Lee IBM T.J. -
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. -
Functional Description of MINTO, a Mixed Integer Optimizer
Functional description of MINTO, a mixed integer optimizer Citation for published version (APA): Savelsbergh, M. W. P., Sigismondi, G. C., & Nemhauser, G. L. (1991). Functional description of MINTO, a mixed integer optimizer. (1st version ed.) (Memorandum COSOR; Vol. 9117). Technische Universiteit Eindhoven. Document status and date: Published: 01/01/1991 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. -
Process Optimization
Process Optimization Mathematical Programming and Optimization of Multi-Plant Operations and Process Design Ralph W. Pike Director, Minerals Processing Research Institute Horton Professor of Chemical Engineering Louisiana State University Department of Chemical Engineering, Lamar University, April, 10, 2007 Process Optimization • Typical Industrial Problems • Mathematical Programming Software • Mathematical Basis for Optimization • Lagrange Multipliers and the Simplex Algorithm • Generalized Reduced Gradient Algorithm • On-Line Optimization • Mixed Integer Programming and the Branch and Bound Algorithm • Chemical Production Complex Optimization New Results • Using one computer language to write and run a program in another language • Cumulative probability distribution instead of an optimal point using Monte Carlo simulation for a multi-criteria, mixed integer nonlinear programming problem • Global optimization Design vs. Operations • Optimal Design −Uses flowsheet simulators and SQP – Heuristics for a design, a superstructure, an optimal design • Optimal Operations – On-line optimization – Plant optimal scheduling – Corporate supply chain optimization Plant Problem Size Contact Alkylation Ethylene 3,200 TPD 15,000 BPD 200 million lb/yr Units 14 76 ~200 Streams 35 110 ~4,000 Constraints Equality 761 1,579 ~400,000 Inequality 28 50 ~10,000 Variables Measured 43 125 ~300 Unmeasured 732 1,509 ~10,000 Parameters 11 64 ~100 Optimization Programming Languages • GAMS - General Algebraic Modeling System • LINDO - Widely used in business applications -
A Tutorial on Linear and Bilinear Matrix Inequalities
Journal of Process Control 10 (2000) 363±385 www.elsevier.com/locate/jprocont Review A tutorial on linear and bilinear matrix inequalities Jeremy G. VanAntwerp, Richard D. Braatz* Large Scale Systems Research Laboratory, Department of Chemical Engineering, University of Illinois at Urbana-Champaign, 600 South Mathews Avenue, Box C-3, Urbana, Illinois 61801-3792, USA Abstract This is a tutorial on the mathematical theory and process control applications of linear matrix inequalities (LMIs) and bilinear matrix inequalities (BMIs). Many convex inequalities common in process control applications are shown to be LMIs. Proofs are included to familiarize the reader with the mathematics of LMIs and BMIs. LMIs and BMIs are applied to several important process control applications including control structure selection, robust controller analysis and design, and optimal design of experiments. Software for solving LMI and BMI problems is reviewed. # 2000 Published by Elsevier Science Ltd. All rights reserved. 1. Introduction One of the main reasons for this is that process control engineers are generally unfamiliar with the mathematics A linear matrix inequality (LMI) is a convex con- of LMI/BMIs, and there is no introductory text avail- straint. Consequently, optimization problems with con- able to aid the control engineer in learning these vex objective functions and LMI constraints are mathematics. As of the writing of this paper, the only solvable relatively eciently with o-the-shelf software. text that covers LMIs in any depth is the research The form of an LMI is very general. Linear inequalities, monograph of Boyd and co-workers [22]. Although this convex quadratic inequalities, matrix norm inequalities, monograph is a useful roadmap for locating LMI and various constraints from control theory such as results scattered throughout the electrical engineering Lyapunov and Riccati inequalities can all be written as literature, it is not a textbook for teaching the concepts LMIs. -
Neos & Htcondor: Optimizing Your World
Optimizing Your neos & World neos: Network-Enabled Optimization System Mathematical Formulation Maximize ∑j∈Vbjyj−α∗∑(i,j)∈Adijxij Subject to exiting node on tour ∑(i,j)∈Axij−yi=0,∀i∈V Subject to entering node on tour ∑(i,j)∈Axij−yi=0,∀j∈V AMPL Model set V; set LINKS := {i in V, j in V: i <> j}; param alpha >= 0; param d{LINKS} >= 0; param b{V} >= 0; #default benefit of visiting a bar neos param c{V} >= 0; # default cost of one drink param B default 30; # default maximum budget for drinks http://www.neos-guide.org/content/bar-crawl-optimization neos & HTCondor neos Workflow Optimization Job Solver Categories bco lp sdp neos co milp sio Results cp minco slp go miocp socp Browser Web kestrel nco uco lno ndo Solver Names AlphaECP csdp LOQO PGAPack ASA ddsip LRAMBO proxy BARON DICOPT MILES PSwarm BDMLP Domino MINLP QSopt_EX Email Email BiqMac DSDP MINOS RELAX4 BLMVM feaspump MINTO SBB Solver Pool bnbs FilMINT MOSEK scip at UW-Madison Bonmin filter MUSCOD-II SD bpmpd filterMPEC NLPEC SDPA Cbc Gurobi NMTR SDPLR Clp icos nsips SDPT3 concorde Ipopt OOQP SeDuMi Custom CONDOR KNITRO PATH SNOPT CONOPT LANCELOT PATHNLP SYMPHONY Couenne L-BFGS-B PENBMI TRON XMLRPC CPLEX LINDOGlobal PENSDP XpressMP Off-Site Solvers Argonne National Lab Solver Inputs Arizona State University AMPL jpg OSIL SPARSE Kestrel C LP RELAX4 SPARSE_SDPA (AMPL or GAMS) University of Klagenfurt CPLEX MATLAB_BINARY SDPA TSP University of Minho Fortran MOSEL SDPLR ZIMPL GAMS MPS SMPS Jobs Per Month 140000 120000 100000 80000 Minho 60000 Klagenfurt Arizona State 40000 Argonne 20000 -
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. -
Domain Reduction Techniques for Global NLP and MINLP Optimization 2
Domain reduction techniques for global NLP and MINLP optimization Y. Puranik1 and Nikolaos V. Sahinidis1 1Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA Abstract Optimization solvers routinely utilize presolve techniques, including model simplification, reformulation and domain reduction techniques. Domain reduction techniques are especially important in speeding up convergence to the global optimum for challenging nonconvex nonlinear programming (NLP) and mixed- integer nonlinear programming (MINLP) optimization problems. In this work, we survey the various techniques used for domain reduction of NLP and MINLP optimization problems. We also present a computational analysis of the impact of these techniques on the performance of various widely available global solvers on a collection of 1740 test problems. Keywords: Constraint propagation; feasibility-based bounds tightening; optimality-based bounds tightening; domain reduction 1 Introduction combinatorial complexity introduced by the integer variables, nonconvexities in objective function or We consider the following mixed-integer nonlinear the feasible region lead to multiple local minima programming problem (MINLP): and provide a challenge to the optimization of such problems. min f(~x) Branch-and-bound [138] based methods can be s.t. g(~~x) ≤ 0 exploited to solve Problem 1 to global optimal- (1) ~xl ≤ ~x ≤ ~xu ity. Inspired by branch-and-bound for discrete Rn−m Zm ~x ∈ × programming problems [109], branch-and-bound was arXiv:1706.08601v1 [cs.DS] 26 Jun 2017 adapted for continuous problems by Falk and MINLP is a very general representation for opti- Soland [58]. The algorithm proceeds by bounding mization problems and includes linear programming the global optimum by a valid lower and upper (LP), mixed-integer linear programming (MIP) and bound throughout the search process. -
54 OP08 Abstracts
54 OP08 Abstracts CP1 Dept. of Mathematics Improving Ultimate Convergence of An Aug- [email protected] mented Lagrangian Method Optimization methods that employ the classical Powell- CP1 Hestenes-Rockafellar Augmented Lagrangian are useful A Second-Derivative SQP Method for Noncon- tools for solving Nonlinear Programming problems. Their vex Optimization Problems with Inequality Con- reputation decreased in the last ten years due to the com- straints parative success of Interior-Point Newtonian algorithms, which are asymptotically faster. In the present research We consider a second-derivative 1 sequential quadratic a combination of both approaches is evaluated. The idea programming trust-region method for large-scale nonlin- is to produce a competitive method, being more robust ear non-convex optimization problems with inequality con- and efficient than its ”pure” counterparts for critical prob- straints. Trial steps are composed of two components; a lems. Moreover, an additional hybrid algorithm is defined, Cauchy globalization step and an SQP correction step. A in which the Interior Point method is replaced by the New- single linear artificial constraint is incorporated that en- tonian resolution of a KKT system identified by the Aug- sures non-accent in the SQP correction step, thus ”guiding” mented Lagrangian algorithm. the algorithm through areas of indefiniteness. A salient feature of our approach is feasibility of all subproblems. Ernesto G. Birgin IME-USP Daniel Robinson Department of Computer Science Oxford University [email protected]