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Convex Over and Under Envelopes for Nonlinear Estimation


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Couenne (Convex Over and Under ENvelopes for Nonlinear Estimation) is a branch&bound algorithm to solve Mixed-Integer Nonlinear Programming (MINLP) problems of the form:

      min f0(x,y)
             fi(x,y) ≤ 0     i=1,2..., m
             x in Rn, y in Zp

where all fi(x,y) are, in general, nonlinear functions.

Couenne aims at finding global optima of nonconvex MINLPs. It implements linearization, bound reduction, and branching methods within a branch-and-bound framework. Its main components are:

  • an expression library;
  • separation of linearization cuts;
  • branching rules;
  • bound tightening methods.

It is distributed on COIN-OR under the Eclipse Public License (EPL). The EPL is a license approved by the Open Source Initiative (OSI), thus Couenne is OSI Certified Open Source Software.

Download, installation and usage

Couenne is found on the COIN-OR project page. It can be downloaded with Subversion -- see also some instructions on using svn. Run the command:

svn co Couenne

to get the source code of the stable version. Before building and installing Couenne, some third party packages are needed. These cannot be downloaded from COIN-OR, and have to be obtained independently. These packages are: ASL, Blas, Lapack, and HSL or MUMPS. The user is referred to the INSTALL file in each of the subdirectories of Couenne/ThirdParty for instructions on how to obtain them.

In general, the stable version of Couenne is subject to slight changes such as bug fixes. In order to be up-to-date with such changes, you may run the command svn update within the Couenne directory. All releases are also available as archive at A release is not subject to change.

To install Couenne, we refer to general installation instructions for COIN-OR projects. We also suggest the excellent Ipopt compilation hints page for compiling on non-Linux systems, such as Mac and Windows. The impatient may want to issue the following commands:

cd Couenne
cd ThirdParty      # Read INSTALL.* file in each subdirectory and get third party software
cd ..
mkdir build
cd build
../configure -C
make install

The above commands place Couenne in the Couenne/build/bin/ directory, libraries in Couenne/build/lib/, and include files in Couenne/build/include/. An alternative directory can be specified with the --prefix option of configure. For instance, when replacing "../configure -C" above with

../configure -C --prefix=/usr/local

the Couenne executable will be installed in /usr/local/bin/, the libraries in /usr/local/lib/, and the include files in /usr/local/include/. Couenne is run as follows:


where is an AMPL stub (.nl) file. Such files can be generated from AMPL with the command "write gfilename;" (notice the "g" before the file name), for example.

You may also specify a set of options to tweak the performance of Couenne. These are found in the couenne.opt option file. A sample option file is given in the Couenne/src/ directory.


A user manual is available, with explanations on most options available in Couenne. Doxygen documentation is also available, and it can be generated by running

make doxydoc

from the same build/ directory where you ran configure, make, and make install. Documentation in both html and LaTeX format can be found in the Doc/ subdirectory. Fire up your browser and take a look at Doc/html/index.html for documentation of Couenne.

Resources and links

Couenne is maintained by Pietro Belotti.

Web page:

Dependencies: CoinUtils, Cbc, Cgl, Clp, Ipopt, and Osi (from COIN-OR); ASL (the Ampl Solver Library), Lapack, Blas, HSL, MUMPS, SCIP, and SoPlex.

External resources: COIN-OR, Eclipse Public License.

Report a bug, contribute to Couenne

As an open-source code, contributions to Couenne are welcome. To submit a contribution to Couenne, please follow the COIN-OR guidelines.

In order to report a bug, use the issue system.

In order to ensure that your issue is addressed in a timely fashion, please try to be as exhaustive as you can in the bug report, for instance by reporting what version of Couenne you have downloaded and what operating system you are using, and again by attaching the model/data files with which the crash occurred.



This project was initiated in 2006 within a collaboration between IBM and Carnegie Mellon University, aimed at developing algorithms for MINLP.

Credit should be given to our colleagues in this collaboration: Andreas, François, Pierre, Stefan, and Timo, who developed part of Couenne, and Larry T. Biegler, Gérard Cornuéjols, Ignacio E. Grossmann, and Jon Lee. Each has contributed an essential part of the development of Couenne.

Project Links

Options for BonCouenne

Linearization options

convexification_cuts <num>

Specify the frequency (in terms of nodes) at which linearization cuts are generated. Default: 1. If 0, linearization cuts are never separated.

convexification_points <num>

Specify the number of points at which to convexify. Default: 1.

violated_cuts_only <yes|no>

If set to yes (default), only violated convexification cuts will be added.

art_lower <num>

Set artificial lower bound (for minimization problems), useful when a lower bound is known or for testing purposes. Default value is -1050.

opt_window <num>

Multiplier for restricting variable bounds around known optimum (to be read from file with method CouenneProblem::readOptimum()). If the optimal value x,,i,, of the i-th variable is known, before starting Couenne its bounds will be intersected with interval [xi-K(1+|xi|),xi+K(1+|xi|)], where K is the value of the option. Default value is infinity.

use_quadratic <yes|no>

Use quadratic expressions and related exprQuad class. Still in testing, so default is "no".

Bound tightening options

feasibility_bt <yes|no>

Use feasibility-based bound tightening (strongly recommended). Default value is "yes".


Optimality-based (expensive) bound tightening. Only recommended for problems with few variables and/or at the initial nodes of the B&B tree. Default is "no". If set to "yes", we recommend to couple it with a value of log_num_obbt_per_level of 0 (see below).

log_num_obbt_per_level <num>

Specify the frequency (in terms of nodes) for optimality-based bound tightening. Default is 0.

  • If -1, apply at every node (expensive!).
  • If 0, apply at root node only.
  • If k>0, apply with probability 2(k - level), level being the current depth of the B&B tree.
aggressive_fbbt <yes|no"

Aggressive feasibility-based bound tightening (to use with NLP points). Default value is "yes". This is also computationally expensive.

log_num_abt_per_level <num>

Specify the frequency (in terms of nodes) for aggressive bound tightening (similar to log_num_obbt_per_level).

  • If -1, apply at every node (expensive!);
  • If 0, apply at root node only;
  • If k>0, apply with probability 2(k - level), level being the current depth of the B&B tree.

Branching options

branch_fbbt <yes|no>

Apply bound tightening before branching. default: yes

branch_conv_cuts <yes|no>

Apply convexification cuts before branching (not active yet). Default: no.

branch_pt_select <string>

Chooses branching point selection strategy. Possible values are

  • "lp-clamped": LP point clamped in [k,1-k] of the bound intervals (k defined by lp_clamp);
  • "lp-central": LP point if within [k,1-k] of the bound intervals, middle point otherwise (k defined by branch_lp_clamp);
  • "balanced": minimizes max distance from curve to convexification;
  • "min-area": minimizes total area of the two convexifications;
  • "mid-point": convex combination of current point and mid point;
  • "no-branch": do not branch, return null infeasibility; for testing purposes only.

Default is mid-point.

branch_lp_clamp <num>

Defines a threshold for selecting an LP point as the branching point; is between 0 and 0.5 and defaults to 0.2. Suppose variable x,,i,, with bounds [li,ui] is chosen for branching. If the lp-central or lp-clamp strategies are selected, the branching point is projected into the interval [Li,Ui] with Li = li+ a(ui - li) and Ui = li+ (1-a)(ui - li).

branch_midpoint_alpha <num>

Defines convex combination of mid point and current LP point: branching point will be alpha xi + (1-alpha) (li+ui)/2. Default value is 0.25.

Options branch_pt_select and branch_lp_clamp above are also available for the following set of operators, and are applied to each of these operators independently: "prod", "div", "exp", "log", "trig", "pow", "negpow", "sqr", "cube".

For instance, the following settings:

branch_pt_select balanced

branch_pt_select_prod lp-clamp
branch_lp-clamp_prod 0.15

branch_pt_select_log lp-central
branch_lp-clamp_log 0.1

specify balanced strategy for all operators except products and logarithms, lp-clamp with parameter 0.15 for products and lp-central with parameter 0.1 for logarithms.

Upper bounding options

local_optimization_heuristic <yes|no>

Search for local solutions of NLPs. Default: yes.

log_num_local_optimization_per_level <num>

Specify the logarithm of the number of local optimizations to perform on average for each level of given depth of the tree.

If equal to -1, solve as many nlp's at the nodes for each level of the tree.

Nodes are randomly selected. If for a given level there are less nodes than this number nlp, are solved for every nodes. For example, if parameter is 8, nlp's are solved for all node until level 8, then for half the node at level 9, 1/4 at level 10.

art_cutoff <num>

Set artificial cutoff useful when a feasible solution is known or for testing purposes. Default value is 1050.


This is a feasibility tolerance for candidate feasible solutions. Default value is 10-7.


Convex Over and Under Envelopes for Nonlinear Estimation







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