INFORMS Journal on Computing
Vol. 16, No. 3, Summer 2004, pp. 255–265
issn 0899-1499 eissn 1526-5528 04 1603 0255
inf orms
®
doi 10.1287/ijoc.1030.0046
©2004 INFORMS
The Constraint Consensus Method for
Finding Approximately Feasible Points in
Nonlinear Programs
John W. Chinneck
Department of Systems and Computer Engineering, Carleton University, Ottawa, Ontario, K1S 5B6, Canada,
chinneck@sce.carleton.ca
T
his paper develops a method for moving quickly and cheaply from an arbitrary initial point at an extreme
distance from the feasible region to a point that is relatively near the feasible region of a nonlinearly con-
strained model. The method is a variant of a projection algorithm that is shown to be robust, even in the
presence of nonconvex constraints and infeasibility. Empirical results are presented.
Key words : nonlinear programming; feasibility; approximate algorithms
History : Accepted by W. David Kelton; received October 2002; revised May 2003; accepted June 2003.
1. Introduction
There are numerous nonlinear-programming applica-
tions in which it is necessary to move very quickly
fromaninitialpoint,oftenveryfarfromfeasibility,to
a final point that is approximately feasible for a set
of nonlinear constraints. A method for doing this is
a valuable “nonlinear crash start” in the solution of
any nonlinear program, for example. It is also a nec-
essary first step in global optimization using exhaus-
tive search algorithms (Kearfott and Dian 2000). See
Pardalos and Resende (2002) for further information
on techniques of nonlinear programming and global
optimization.
A generic nonlinear program consisting of m con-
straints in n variables is shown in Equation 1. Our
main interest is the case in which one or more of the
constraint functions are nonlinear and the bounds on
the variables are very wide.
Minimize or maximize fx
s.t. gx≤ ≥ = b
l ≤ x ≤ u
(1)
The major motivation here is to find reasonable
bounds on variables when these are not supplied by
the user. The MProbe software (Chinneck 2001a, b,
2002) analyzes functions for characteristics such as
their shape and distribution of values. It does so
by random sampling in the multidimensional box
definedbythevariablebounds.Usersoftenomitvari-
able bounds when they are unsure as to how to set
them. The resulting unbounded sampling box is so
large that it is extremely unlikely that any sample
points will be placed in the region of interest around
the feasible region.
For example, consider a three-variable model in
which the feasible region is approximately a cube
with sides of length 100, for a volume of 1 × 10
6
.
If the bounds are not set by the user, MProbe will
assume bounds of ±1 × 10
10
on each variable, for a
total sampling box volume of 8 × 10
30
. The probabil-
ity of placing a random point in the feasible region is
then 1 × 10
6
/8 × 10
30
= 125 × 10
−25
, which is vanish-
ingly small. This difficulty worsens as the number of
unbounded variables in the model increases. For this
reasonMProbeneedstobeabletoidentifyquicklythe
approximatelocationofthefeasibleregionbymoving
from arbitrary initial points in the original sampling
box to points that are relatively near to the feasible
region. These approximately feasible points are used
asaguidetosetvariableboundsthatdefineasuitable
box for productive sampling.
In the MProbe context it is important that near-
feasibility be achieved relatively quickly since this is
only a preliminary step towards the core sampling
work. On the other hand, it is acceptable for the
method to fail occasionally since many random ini-
tial points are supplied. However, the method should
converge to an approximate feasible point quickly for
a reasonable fraction of the initial points.
A note on nomenclature: We assume throughout
that constraints have the form LHS ≤ ≥ = RHS,
whereRHSisaconstant.TheLHSisoftenreferredto
as the “constraint body” and includes the functional
part of the constraint.
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