Last modified:  25 September 1999

This document describes some software packages for constrained optimization (linear and nonlinear programming). The software is maintained by research personnel at Stanford University and the University of California, San Diego. Here we give guidance on the choice of package, licensing, and technical support.

For licensing information, go to SBSI-SOL either now or further below.

Optimization Software and Problem Types

These packages are implemented in Fortran 77 and distributed as source code. They are intended for any machine with a reasonable amount of memory and a Fortran compiler. They may be called from a driver program (typically in Fortran, C or MATLAB). MINOS and SNOPT may also be used as stand-alone packages, reading data in the MPS format used by commercial mathematical programming systems.

MINOS and SNOPT are widely used as linear and nonlinear solvers for the GAMS and AMPL modeling systems (see below). NPSOL and LSSOL are part of the Optimization Chapter of the NAG Fortran library.

General Features

The nonlinear solvers make use of function and gradient values. The solution obtained will be a local optimum (which may or may not be a global optimum). If some of the gradients are unknown, they will be estimated by finite differences. If the functions or gradients are expensive to evaluate, SNOPT or NPSOL will usually be more efficient than MINOS.

For large problems, efficiency is improved if only some of the variables enter nonlinearly, or if the number of active constraints is nearly as large as the number of variables. MINOS can accommodate problems with many degrees of freedom (perhaps thousands), but a few hundred or less is preferable for both codes.

If the constraints have no feasible solution, the packages may terminate as soon as infeasibility is confirmed. Optionally, all except MINOS can continue to minimize the sum of infeasibilities, using implicit "elastic bounds". SQOPT allows the user to specify which constraints and bounds may be regarded as elastic. This allows SNOPT to deal methodically with infeasible nonlinear constraints.

Linear Programming

Given sufficient memory, MINOS, SNOPT and SQOPT can process large LP models similar to those solved by commercial packages such as OSL and CPLEX. They should be equally reliable, though slower for a variety of reasons. (For example, there is no Presolve to reduce the model size, and they do not employ steepest-edge pricing.) Warm starts are possible for solving a sequence of related problems (e.g., for a branch-and-bound algorithm).

LSSOL and QPOPT are suitable for LP and QP problems of arbitrary size if the constraint matrix is essentially dense, but the sparse solvers will usually be more efficient even for dense problems.

Quadratic Programming

MINOS, SNOPT and QPOPT are suitable for general QP problems (but may find just a local optimum). SQOPT and LSSOL require the quadratic to be positive definite or semidefinite (typically x'R'Rx + linear terms). A simple example is (x - x0)'(x - x0). With its "elastic bounds" feature, SQOPT can also solve "linear L1 problems" of two kinds:

where ||...|| is the 1-norm. Such problems arise in data-fitting.

All solvers except LSSOL allow the quadratic objective to be specified as an operator.

Nonlinear Objective

MINOS and SNOPT are highly effective for problems with a nonlinear objective function and large numbers of sparse linear constraints (as well as bounds on the variables). The algorithms follow different paths. MINOS may be more efficient if the objective and its gradients are cheap to evaluate (though the difference is usually not great).

Nonlinear Constraints

MINOS performs best if the constraints are "nearly linear". It solves a sequence of subproblems in which the constraints are linearized and the objective is an augmented Lagrangian (involving all nonlinear functions). Convergence is rapid near a solution.

SNOPT and NPSOL may be more reliable on highly nonlinear problems. They use an SQP algorithm in which the subproblems have the same linearized constraints as in MINOS, but the objective is a quadratic approximation to the Lagrangian. (Hence, no function or gradient values are needed during the solution of each QP.) A merit function promotes convergence from arbitrary starting points.

MATLAB Interfaces

F-MEX files are distributed with NPSOL, LSSOL and QPOPT. These are Fortran 77 links to MATLAB. (Currently they are for Unix systems. Inevitably, PC versions will be "similar but different".)

C-MEX files are available for MINOS (again for Unix). The following files may be viewed and saved locally: readme, report.dvi, report.ps. For Unix systems, the source code is in files.tar.gz. NOTE: Use SHIFT-CLICK to download this as a binary file. Otherwise, Netscape regards it as a text file and displays it!

C-MEX files exist for NPSOL (Unix and Apple PowerPC). Please enquire with Philip Gill (below).

C Interfaces

In many cases, a C program can call the optimization packages as Fortran subroutines. An example routine (driver.c) is provided with MINOS.

C source code for NPSOL is available from Philip Gill (see below). In general, C versions may be obtained via the Bell Labs f2c translator.

Licensing Information

All packages are licensed by OTL, the Office of Technology Licensing at Stanford University.
Tier 1 and Tier 2 licenses for in-house use (academic and commercial respectively). are handled by SBSI:

Specialized Tier 3 agreements are available for applications involving resale of SOL software as part of a larger package. These are negotiated by OTL:

Technical Support

For further information about the algorithms, please contact one of the following personnel. (Initial enquiries specific to MINOS should be directed to Michael Saunders. Those specific to SNOPT and the other packages should be sent to Philip Gill.)

Technical support is available for a fee from the following sources:

GAMS, AMPL and Other Interfaces

For many optimization applications, we recommend the use of high-level systems such as those listed below. They provide a convenient interface to MINOS, SNOPT, NPSOL and many other linear, integer and nonlinear solvers, and they extend the range of problem types that can be solved by traditional local optimizers.

• GAMS development and sales   (GAMS Development Corporation)

Algebraic modeling language: LP, NLP, MIP, MILP, MINLP, LCP, MCP, CNS.


• AMPL development   (Bell Labs)

Interactive algebraic modeling language: LP, NLP, MIP, MILP, MINLP.


• AMPL sales and AMPL Plus   (Compass Modeling Solutions Inc.)

GUI interface to AMPL.


• DECIS   (Dr Gerd Infanger, Dept of EESOR, Stanford University)

Large-scale stochastic programming.


• MINOPT   (Dept of Chemical Engineering, Princeton University)

Process design, optimal control, etc.: LP, NLP, NLP/DAE, MIP, MILP, MINLP. OCP, MIOCP.


• BARON   (Dept of Chemical Engineering, University of Illinois at Urbana-Champaign)

Global optimization (linear and nonlinear), concave programming, fractional programming, factorable programming, MILP, MIQP, etc.


• NEOS server   (Argonne National Laboratory)

Web-based access to AMPL and various solvers.