Mixed Integer Linear Programming in R

Transcription

1 Mixed Integer Linear Programming in R Stefan Theussl Department of Statistics and Mathematics Wirtschaftsuniversität Wien July 1, 2008

2 Outline Introduction Linear Programming Quadratic Programming Mixed Integer Programming COIN-OR Initiative R Packages Interfaces to Open Source Solvers Commercial Solvers Optimization Infrastructure for R Benchmarks Outlook and Future Work

3 Introduction

4 Applications in Finance Portfolio selection and asset allocation (Markowitz) Pricing and hedging of options Asset/liability management Risk management

5 Mathematical Programming Linear Programming (LP) Quadratic Programming (QP) Nonlinear Programming (NLP) Additionally if variables have to be of type integer (integer programming IP) Mixed Integer Linear Programming (MILP) Mixed Integer Quadratic Programming (MIQP) Nonlinear Mixed INteger Programming (NMINP)

6 Linear Programming Mathematical formulation objective min c t x subject to constraints Ax = b x 0 objective variables x i, i = 1,..., N Examples: Asset/liability cash flow matching, asset pricing and arbitrage

7 Quadratic Programming Mathematical formulation objective min x t Qx subject to constraints Ax = b x 0 objective variables x i, i = 1,..., N Examples: Portfolio selection, asset allocation

8 Mixed Integer Programming Some variables have to be of type integer E.g., we cannot buy 3.4 shares but 3 or 4 Open source MILP solvers available but currently no open source MIQP solver in R

9 COIN-OR Initiative

10 Organization of COIN-OR The COmputational INfrastructure for Operations Research (COIN-OR) project promotes the development and use of interoperable, open source software for operations research Furthermore, it is a repository consisting of several projects Each has its own project manager, wiki, website, mailing list (similar to R-Forge or Sourceforge) Stable releases or snapshots as well es direct source code access through svn are provided Pre-compiled binaries (Linux, Windows) the CoinAll distribution includes a large subset of COIN-OR

11 The COIN-OR Philosophy Reuse instead of reinvent Reduce development time and increase robustness Increase interoperability Peer review of software Free distribution of ideas Reproducability

12 COIN-OR Projects Open Solver Interface (OSI) Cut Generator Library (CGL) Branch, Cut and Price Library (BCP) Interior Point Optimization (IPOPT) COIN-OR LP (CLP) SYMPHONY and many more

13 R Packages

14 Interfaces to Open Source Solvers Interfaces to the GNU Linear Programming Kit (GLPK) via packages Rglpk and glpk Interfaces to lp solve via packages lpsolve and lpsolveapi and Rsymphony, an interface to the COIN-OR SYMPHONY solver

15 Interfaces to Open Source Solvers How to solve MILPs in R? Rsymphony solve LP(obj, mat, dir, rhs, bounds = NULL, types = NULL, max = FALSE) Rglpk solve LP(obj, mat, dir, rhs, types = NULL, max = FALSE, bounds = NULL, verbose = FALSE) lp(direction = "min", objective.in, const.mat, const.dir, const.rhs, transpose.constraints = TRUE, int.vec, presolve=0, compute.sens=0, binary.vec, all.int=false, all.bin=false, scale = 196, dense.const, num.bin.solns=1, use.rw=false)

16 Commercial Solvers Currently an R interface to the CPLEX callable library from ILOG is available Package Rcplex already on CRAN Rcplex is capable of solving linear as well as mixed integer quadratic programs Supports sparse matrices ( simple triplet matrix, Matrix package) CPLEX can be called via Rcplex(cvec, Amat, bvec, Qmat = NULL, lb = 0, ub = Inf, control = list(), objsense = c("min", "max"), sense = "L", vtype = NULL)

17 Optimization Infrastructure for R

18 Optimization Infrastructure for R We plan to offer a package for optimization which allows to solve MILPs (and other programs) using a unique interface: OI solve(problem, solver = "lp solve", control = NULL) The main function takes 3 arguments: problem represents an object containing the description of the MILP solver specifies the solver to be used (e.g., GLPK, lp solve, SYMPHONY,... ) control is a list containing additional control arguments to the corresponding solver

19 Benchmarks

20 MIPLIB2003 MIPLIB2003 is a collection of real-world mixed integer programs standard test set used to compare the performance of MILP solvers available online at maintained by Alexander Martin, Tobias Achterberg and Thorsten Koch instances in MPS file format can be read in R via Rglpk s MPS file reader

21 Results of Benchmark Experiment GLPK Rglpk lp solve lpsolve SYMPHONY Rsymphony aflow30a air air cap danoint disctom fiber fixnet gesa manna mas misc modglob nw p pk pp08acuts qiu rout Inf vpm Table: Objective values command line solvers and R interfaces

22 Results of Benchmark Experiment GLPK Rglpk lp solve lpsolve SYMPHONY Rsymphony aflow30a air air cap danoint disctom fiber fixnet gesa manna mas misc modglob nw p pk pp08acuts qiu rout vpm Table: Comparison of runtimes [min] command line solvers vs. R interfaces

23 Outlook and Future Work Optimization infrastructure package Interfaces to NMINP solvers Bonmin and LaGO (project RINO on R-Forge) Investigation of SYMPHONY s parallel solver Applications: Consensus ranking of journal ratings, portfolio optimization

24 Thank you for your attention Further reading: Gerard Cornuejols and Reha Tütüncü. Optimization Methods in Finance. Cambridge University Press, 2006.