ENGINEERING AND ECONOMIC APPLICATIONS
OF COMPLEMENTARITY PROBLEMS
∗
M. C. FERRIS
†
AND J. S. PANG
‡
SIAM R
EV.
c
1997 Society for Industrial and Applied Mathematics
Vol. 39, No. 4, pp. 669–713, December 1997 005
Abstract. This paper gives an extensive documentation of applications of finite-dimensional
nonlinear complementarity problems in engineering and equilibrium modeling. For most applica-
tions, we describe the problem briefly, state the defining equations of the model, and give functional
expressions for the complementarity formulations. The goal of this documentation is threefold: (i) to
summarize the essential applications of the nonlinear complementarity problem known to date, (ii) to
provide a basis for the continued research on the nonlinear complementarity problem, and (iii) to sup-
ply a broad collection of realistic complementarity problems for use in algorithmic experimentation
and other studies.
Key words. complementarity problems, variational inequalities, frictional contact, structural
engineering, economic equilibria, transportation planning, Nash equilibria
AMS subject classifications. 90C33, 90C50, 70E15, 73K05, 90A14, 90D10
PII. S0036144595285963
1. Introduction. As a result of more than three decades of research, the subject
of complementarity problems, with its diverse applications in engineering, economics,
and sciences, has become a well-established and fruitful discipline within mathemat-
ical programming. Several monographs [34, 79, 135] and surveys [72, 148] have doc-
umented the basic theory, algorithms, and applications of complementarity problems
and their role in optimization theory.
An important reason why complementarity problems are so pervasive in engineer-
ing and economics is because the concept of complementarity is synonymous with the
notion of system equilibrium. The balance of supply and demand is central to all
economic systems; mathematically, this fundamental equation in economics is often
described by a complementary relation between two sets of decision variables. For
instance, the classical Walrasian law of competitive equilibria of exchange economies
[203] can be formulated as a nonlinear complementarity problem in the price and
excess demand variables. The complementarity condition expresses the fact that the
excess demand of a commodity must be zero if its price is positive; similarly, the price
of the commodity must be zero if there is positive excess supply. Complementarity
is also central to all constrained optimization problems. The well-known complemen-
tary slackness property in linear programming exemplifies the fundamental role of
complementarity in optimization; this property persists in nonlinear programs and
variational inequalities. Optimization is a recurring theme in numerous engineering
applications; however, many engineering systems involve the notion of equilibrium
without an objective being optimized. For instance, the renowned Wardrop principle
∗
Received by the editors May 12, 1995; accepted for publication (in revised form) January 14,
1997.
http://www.siam.org/journals/sirev/39-4/28596.html
†
Computer Sciences Department, University of Wisconsin, Madison, WI 53706 (ferris@cs.
wisc.edu). The work of this author was based on research supported by National Science Foun-
dation grant CCR-9157632, Department of Energy grant DE-FG03-94ER61915, and Air Force Office
of Scientific Research grant F49620-94-1-0036.
‡
Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD 21218-
2689 (jsp@vicp.mts.jhu.edu). The work of this author was based on research supported by National
Science Foundation grant CCR-9213739 and by Office of Naval Research grant N00014-93-1-0228.
669
670 M. C. FERRIS AND J. S. PANG
of user equilibrium in traffic theory [205] has a natural formulation as a nonlinear
complementarity system; in this case, the complementarity condition is a behavioral
statement about the users of a traffic network who are postulated to take short-
est paths in the network. Another example is the physical contact of mechanical
structures; here complementarity is between the contact force and the gap (i.e., the
distance) between the bodies in contact: the contact force is positive only if there is
contact (that is, if the gap is zero). A major objective of this paper is to elucidate
the pervasiveness of complementarity in these and other important engineering and
economic applications.
Among the many facets of research in complementarity problems, one that has
received wide attention in recent years is the development of robust and efficient
algorithms for solving the ever increasing applications of these problems. There is
presently a wide variety of computational methods for solving complementarity prob-
lems. Unlike the fixed-point homotopy methods [63, 197] some of which are known
to be practically deficient for solving realistic equilibrium applications of the com-
plementarity problem (see [72] and the references therein) most of the contemporary
computational schemes are designed with the goal of removing these deficiencies and
meeting the need of solving large-scale applications efficiently. These computational
methods include the following:
• extensions of Newton’s method for nonlinear equations [73, 84, 129, 147, 163]
that replace the direction finding routines with complementarity problems,
• a path search method [41, 42, 162] that uses a generalization of a line search
technique,
• quadratic programming-based algorithms [15, 59, 60, 149] that derive exten-
sions of the Gauss–Newton methodology,
• differentiable optimization based descent methods [55, 88, 123, 125, 199] that
reformulate the complementarity relationships as a nonlinear equation or pro-
gram,
• projection and proximal methods [8, 9, 48, 132, 182] that extend projected
gradient methods,
• smoothing techniques [22, 21, 23, 24, 89, 58, 160] that replace the nonsmooth
equations with differentiable approximations,
• and interior point methods [5, 14, 22, 21, 66, 81, 99, 100, 101, 102, 131, 180,
204, 207] based on removing inequalities by an interior penalty.
Along with the research in the design and analysis of algorithms comes the recog-
nition that the linkage of these algorithms with such mathematical programming
modeling languages as GAMS [19] and AMPL [51] is extremely important for the al-
gorithms to become easily accessible to practitioners and academic researchers. Mo-
tivated by the desire to solve complex economic equilibrium problems, Rutherford
[172] developed a modeling system for applied general equilibrium systems and sub-
sequently [174, 175] extended the original GAMS modeling language to enable the
treatment of complementarity constraints. Further work on modeling language inter-
faces of complementarity algorithms can be found in the Ph.D. thesis of Dirkse [39]
and the paper [43].
Numerical experimentation has always been an important part of algorithmic
development. The paper [73] is perhaps the first to report extensive computational
results with an algorithm for solving realistic nonlinear complementarity problems
arising from various economic applications. This set of test problems has since been
augmented in [59, 105, 149, 208]; a model library containing a summary of these
APPLICATIONS OF COMPLEMENTARITY 671
problems is formulated in the GAMS [40] and AMPL languages [39, Chapter 3].
Computational results comparing several algorithms on these test problems can be
found in [13].
Fueled by a desire to expand such a model library and to provide a basis for
the continued research on complementarity problems, we decided to undertake the
daunting task of uncovering all interesting applications of these problems known to
us. The result of our effort is a large collection of realistic complementarity problems
of various type, size, complexity, and computational difficulty. Collectively, these
problems are expected to pose new challenges for a general researcher in the field and
particularly for an algorithm designer.
The rest of this paper is divided into three major sections. The next section sets
up the notation used in the paper and describes the various types of complemen-
tarity problems that appear later. Section 3 describes the engineering problems and
gives their complementarity formulations. Section 4 does the same for the economic
equilibrium problems. Finally we give some concluding remarks in the fifth and last
section.
2. Types of complementarity problems. Complementarity problems come
in different types. In addition to the familiar ones—linear, nonlinear, generalized—
such adjectives as “mixed, horizontal, vertical, extended” have been coined to describe
a complementarity problem of a particular type. In this section we give an overview
of the various problems that we consider collectively as complementarity problems.
Many of the applications that we detail in the sequel have naturally occurring comple-
mentarity forms, and in our description of the problems we will maintain this natural
form.
2.1. Nonlinear complementarity problems. This classical problem, defined
by a nonlinear function F : R
n
→ R
n
,istofindanx∈R
n
such that
NCP(F ): 0≤x ⊥ F(x)≥0,
where we use the perp notation “⊥” to signify that in addition to the stated inequal-
ities, the equation x
T
F (x) = 0 also holds. Note that since x
T
F (x)=
P
n
i=1
x
i
F
i
(x),
this can be equivalently stated as
x ≥ 0,F(x)≥0,x
i
F
i
(x)=0,i=1,2,...,n.
In effect then complementarity states that either x
i
or F
i
(x) must be zero for each
i =1,2,...,n. It is easy to see that NCP(F ) is equivalent to the following problem
of solving the nonsmooth equation
min(x, F (x))=0,
where the min operation is taken componentwise. Complementarity problems of this
form arise as the Karush–Kuhn–Tucker (KKT) conditions of a constrained nonlinear
program
minimize θ(x)
subject to g(x) ≤ 0,x≥0,
where θ : R
n
→ R is a continuously differentiable real-valued function and g : R
n
→
R
m
is a continuously differentiable vector-valued function.
672 M. C. FERRIS AND J. S. PANG
2.2. Variational inequalities. In practice, many problems have lower and/or
upper bounds on the variables, instead of the standard nonnegativity shown above.
Most generally, a problem may have both lower and upper bounds on some variables,
only lower or upper bounds on other variables, and no such bounds at all on the re-
maining variables. To accommodate such a general problem, the following variational
inequality format has been frequently used in the literature:
VI (F, [l, u]) : find x ∈ [`, u] such that (y − x)
T
F (x) ≥ 0 ∀y ∈ [`, u],
where ` and u are n-dimensional vectors with `
i
∈ [−∞, ∞) and u
i
∈ (`
i
, ∞], and
[`, u] ≡{x∈R
n
:`≤x≤u}.
If `
i
=0,u
i
=∞for each i =1,2,...,n, it follows easily that VI (F, [0, ∞]) is
precisely NCP(F ). In several places the VI (F, [`, u]) is termed a box-constrained or
a rectangular variational inequality, the box referring to the set [`, u].
In many applications, some of the underlying conditions are defined by a system
of nonlinear equations, while the complementarity conditions are only applied to some
of the variables and functions. This leads to a type of problem generally termed a
mixed nonlinear complementarity problem and can be written in the following form:
F
I
(x)=0,x
I
free,
0 ≤ x
J
⊥ F
J
(x) ≥ 0,
where I and J form a partition of {1, 2,...,n}. To incorporate a free variable x
i
into
a model, we would set `
i
= −∞ and u
i
= ∞. The mixed nonlinear complementarity
problem can be recovered as a special case of VI (F, [`, u]) by setting `
i
= −∞,
u
i
= ∞ for i ∈I, and `
i
=0,u
i
=∞for i ∈J. Due to this fact, the box-constrained
variational inequality is typically termed a mixed complementarity problem. Notice
that the bounds ` and u on the variables implicitly define the constraints associated
with the function F .
A special case of a mixed complementarity problem is where I = {1, 2,...,n},
resulting in a system of nonlinear equations:
NE : F (x)=0.
The VI (F, [l, u] ∩ X), where X = {x : Ax = b} and A is an m × n matrix, is
also easily transformed into a box-constrained variational inequality by introducing
multipliers for the linear equality constraints. It is easy to show that VI (F, [l, u]∩X)
is equivalent to VI (H, [l, u] ×R
m
), where
H(x, λ)=
"
F(x)+A
T
λ
−Ax + b
#
.(2.1)
More generally, if X = {x : g(x) ≤ 0,h(x)=0}, where g : R
n
→ R
m
and h :
R
n
→ R
s
are continuously differentiable functions, then under a suitable constraint
qualification, for any solution x of the VI (F, [l,u] ∩X), there must exist multipliers
λ ∈ R
m
and η ∈ R
s
such that the triple (x, λ, η) is a solution of the VI (H, [l, u] ×
[0, ∞) × R
s
), where
H(x, λ, µ)=
L(x, λ, µ)
−g(x)
h(x)
APPLICATIONS OF COMPLEMENTARITY 673
with
L(x, λ, µ) ≡ F (x)+
m
X
i=1
λ
i
∇g
i
(x)+
s
X
j=1
µ
j
∇h
j
(x)
is the vector-valued Lagrangian function for the VI (F, [l, u] ×X).
2.3. Vertical complementarity problems. There is a certain lack of sym-
metry in the nonlinear complementarity problem as can be seen in the formulation
min(x, F (x)) = 0. One of the functions in this formulation is quite arbitrary while
the other is the identity. Many commonly occurring problems actually have a more
general form
min(F
1
(x),F
2
(x))=0(2.2)
for two given functions F
1
,F
2
: R
n
→ R
n
. Of course, it is possible to have more
than two functions in the above equation; the resulting problem is called the vertical
nonlinear complementarity problem:
VCP(F ) : min(F
1
(x),F
2
(x),...,F
m
(x))=0.
Clearly, this means that F
j
i
(x) ≥ 0 for all i =1,2,...,n and j =1,2,...,m and for
each component i, F
j
i
(x) = 0 for at least one j. The affine version of this problem
(that is when all the functions F
i
are affine) was introduced by Cottle and Dantzig
[31]; it has been treated in several studies [44, 65, 80, 124].
The VCP can be equivalently cast as a box-constrained variational inequality
(more precisely, a mixed nonlinear complementarity problem) by introducing extra
variables z
j
∈ R
n
,j =2,3,...,m; the equivalent formulation is
P
m
k=2
z
k
= F
1
(x),
0 = min
z
j
,F
j
(x)−
P
m
k=j+1
z
k
,j=2,...,m−1,
0 = min(z
m
,F
m
(x)).
(2.3)
It can be seen that if x solves the VCP(F), then the above equations are satisfied
with
z
j
≡ min(F
1
(x),...,F
j−1
(x)) −
P
m
k=j+1
z
k
,j=2,...,m−1,
z
m
≡min(F
1
(x),...,F
m−1
(x));
conversely, if (2.3) holds, then x solves the VCP(F ). (Danny Ralph at the University
of Melbourne had previously communicated to the authors a related formulation of
the VCP as a normal equation.)
Notice that the number of variables in problem (2.3) is mn, compared with just
n in the original min formulation of the VCP(F ). Some algorithms for solving the
VCP (e.g., the Gauss–Newton method for nonsmooth equations in [150]) may exploit
the particular structure of the min formulation much more effectively than treating
the variational inequality directly. However, many of the variables introduced in (2.3)
arise in a purely linear fashion, so a general purpose solver for the VI may be able to
exploit this fact.
