AMO - Advanced Modeling and Optimization Volume 1, Number 1, 1999 The TOMLAB NLPLIB Toolb ox for Nonlinear Programming Kenneth Holmstrom and Mattias Bjorkman Center for Mathematical Mo deling Department of Mathematics and Physics Malardalen University PO Box SE Vasteras Sweden Abstract The pap er presents the to olb ox NLPLIB TB NonLinear Programming LIBrary a set of Matlab solvers test problems graphical and computational utilities for unconstrained and constrained optimization quadratic programming unconstrained and constrained nonlinear least squares b oxbounded global optimization global mixedinteger nonlinear programming and exp onential sum mo del tting NLPLIB TB like the to olb ox OPERA TB for linear and discrete optimization is a part of TOMLAB an environment in Matlab for research and teaching in optimization TOMLAB currently solves small and medium size dense problems Presently NLPLIB TB implements more than solver algorithms and it is p ossible to call solvers in the Matlab Optimization Toolb ox MEXle interfaces are prepared for seven Fortran and C solvers and others are easily added using the same type of interface routines Currently MEXle interfaces have b een developed for MINOS NPSOL NPOPT NLSSOL LPOPT QPOPT and LSSOL There are four ways to solve a problem by a direct call to the solver routine or a call to a multisolver driver routine or interactively using the Graphical User Interface GUI or a menu system The GUI may also b e used as a prepro cessor to generate Matlab co de for standalone runs If analytical derivatives are not available automatic dierentiation is easy using an interface to ADMATADMIT TB Furthermore ve types of numerical dierentiation metho ds are included in NLPLIB TB NLPLIB TB implements a large set of standard test problems Furthermore using MEXle interfaces problems in the CUTE test problem data base and problems dened in the AMPL mo deling language can b e solved TOMLAB and NLPLIB TB have b een used to solve several applied optimization problems New types of algorithms are implemented for the nonlinear least squares problem to approxi mate sums of exp onential functions to empirical data and for global optimization We present some preliminary test results which show very go o d p erformance for the NLPLIB TB solvers Keywords Nonlinear Programming MATLAB CUTE AMPL Graphical User Interface Soft ware Engineering Mathematical Software Optimization Algorithms Exp onential Sum Fitting Nonlinear Least Squares AMS Sub ject Classication C C C 1 Financed by the Malardalen University Research Board pro ject Applied Optimization and Modeling TOM 2 Email hkhmdhse URL httpwwwimamdhsetom 3 Email mbkmdhse The TOMLAB NLPLIB Toolb ox for Nonlinear Programming Introduction The to olb ox NLPLIB TB NonLinear Programming LIBrary Toolb ox is part of TOMLAB an environment in Matlab for research and teaching in optimization NLPLIB TB is a set of Matlab mles which solves nonlinear optimization problems and nonlinear parameter estimation problems in op erations research and mathematical programming The fo cus is on dense problems The to olb ox is running in Matlab x and works on b oth PC NT Windows Windows and UNIX systems SUN HP Currently NLPLIB TB consists of ab out lines of mle co de in les implementing algorithms utilities and predened problems all well do cumented in the Users Guide The Users Guide includes descriptions and examples of how to dene and solve optimization problems as well as detailed descriptions of the routines The optimization problem to b e solved is either selected using a interactive menu program or directly dened in a call to a multisolver driver routine The problem is solved using either a NLPLIB TB solver a solver in the Matlab Optimization Toolb ox or using a MEXle interface to call a Fortran or C optimization co de NLPLIB TB has interactive menu programs for unconstrained and constrained optimization un constrained and constrained nonlinear least squares quadratic programming b oxbounded global optimization and global mixedinteger nonlinear programming NLPLIB TB includes a graphical user interface GUI where all types of predened problems can b e solved Using the GUI the user has total control of all optimization parameters and variables TOMLAB MEXle interfaces for b oth PC and UNIX has b een developed for the commercial op timization co de MINOS In TOMLAB MINOS is used to solve nonlinear programs in NLPLIB TB and linear programs in OPERA TB TOMLAB MEXle interfaces working on b oth PC and UNIX have also b een developed for the commercial co des from the Systems Opti mization Lab oratory SOL Department of Op erations Research Stanford University California NPSOL NPOPT up dated version of NPSOL NLSSOL QPOPT LSSOL and LPOPT The aim is to expand this list in the near future NLPLIB TB implements a large set of predened test problems It is easy to try to solve any of these problems using any of the solvers present The user can easily expand the set of test problems with his own problems NLPLIB TB was designed with the aim to simplify the solution of practical optimization problems After dening a new problem in the NLPLIB TB format it is then p ossible to try to solve the problem using any available solver or metho d For twodimensional nonlinear unconstrained problems the menu programs supp ort graphical dis play of the selected optimization problem as a mesh or contour plot The search directions together with marks of the trial step lengths are displayed on the contour plot For higherdimensional problems the contour plot is displayed in a twodimensional subspace Plots showing the estimated convergence rate and the sequence of function values are included The GUI has the same graphical options as the menu programs For nonlinear least squares problems a routine to plot the data against the starting mo del and the tted mo del is included Also included are new algorithms for the nonlinear parameter estimation problem of tting sums of exp onential functions to empirical data In Section the dierent optimization algorithms and solvers in NLPLIB TB are discussed Some other imp ortant utility routines are discussed in Section eg dierent types of dierentiation Some information ab out the MEXle interfaces the lowlevel routines and the test problems are given in Section The Section discusses the most frequently menu and plot options used In Section we present three areas where TOMLAB and NLPLIB TB have b een a valuable to ol constrained nonlinear least squares the sp ecial case of exp onential sum tting problems and b ox b ounded global optimization Some test results are presented for these application areas showing go o d p erformance for the NLPLIB TB solvers We end by some conclusions in Section The TOMLAB NLPLIB Toolb ox for Nonlinear Programming Optimization Algorithms and Solvers In this section we discuss the optimization problems that NLPLIB TB are able to solve In Table the optimization solvers in NLPLIB TB are listed The solver for unconstrained optimization ucSolve and the nonlinear least squares solvers lsSolve and clsSolve are all written as prototype routines ie the routines implements several optimization algorithms in one co de This simplies maintenance and further algorithm development Table Optimization solvers in NLPLIB TB Function Description ucSolve A prototype routine for unconstrained optimization with simple b ounds on the variables Implements Newton four quasiNewton and three conjugate gradient metho ds glbSolve A routine for b oxbounded global optimization gblSolve Standalone version of glbSolve Runs indep endently of NLPLIB TB glcSolve A routine for global mixedinteger nonlinear programming gclSolve Standalone version of glcSolve Runs indep endently of NLPLIB TB lsSolve A prototype algorithm for nonlinear least squares with simple b ounds Imple ments GaussNewton and hybrid quasiNewton and GaussNewton metho ds clsSolve A prototype algorithm for constrained nonlinear least squares Currently han dles simple b ounds and linear equality and inequality constraints using an active set strategy Implements GaussNewton and hybrid quasiNewton and GaussNewton metho ds conSolve Constrained nonlinear minimization solver using two dierent sequential quadratic programming metho ds nlpSolve Constrained nonlinear minimization solver using lter SQP sTrustR Solver for constrained convex optimization of partially separable functions using a structural trust region algorithm qpBiggs Solves a quadratic program qpSolve Solves a quadratic program qpe Solves a quadratic program restricted to equality constraints using a null space metho d qplm Solves a quadratic program restricted to equality constraints using Lagranges metho d The routine ucSolve implements a prototype algorithm for unconstrained optimization with simple b ounds on the variables uc ie solves the problem min f x x x x x st L U n where x x x R and f x R ucSolve includes several of the most p opular search step L U metho ds for unconstrained optimization Bound constraints are treated as describ ed in Gill et al The search step metho ds for unconstrained optimization included in ucSolve are the Newton metho d the quasiNewton BFGS and inverse BFGS metho d the quasiNewton DFP and inverse DFP metho d the FletcherReeves and PolakRibiere conjugategradient metho d and the Fletcher conjugatedescent metho d For the Newton and the quasiNewton metho ds the co de is using a subspace minimization technique to handle rank problems see Lindstrom The quasiNewton co des also use safe guarding