Generalized Goal Programming:
Polynomial Methods and Applications
Emilio Carrizosa
Facultad de Matem´aticas, Universidad de Sevilla
Tarfia s/n, 41012 Seville, Spain
e-mail: ecarriz@cica.es
J¨org Fliege
Fachbereich Mathematik, Universit¨at Dortmund
44221 Dortmund, Germany
e-mail: fliege@math.uni-dortmund.de
January 8, 2001
Abstract
In this paper we address a general Goal Programming problem with linear
objectives, convex constraints, and an arbitrary componentwise nondecreasing
norm to aggregate deviations with respect to targets. In particular, classical
Linear Goal Programming problems, as well as several models in Location
and Regression Analysis are modeled within this framework.
In spite of its generality, this problem can be analyzed from a geometrical
and a computational viewpoint, and a unified solution methodology can be
given. Indeed, a dual is derived, enabling us to describe the set of optimal
solutions geometrically. Moreover, Interior-Point methods are described which
yield an ε-optimal solution in polynomial time.
Keywords: Goal Programming, Closest points, Interior point methods,
Location, Regression.
1
Version as of December 3, 2000 2
1 Introduction
1.1 Goal Programming
The origins of Goal Programming date back to the work of Charnes, Cooper and
Ferguson [7], where an l
1
-estimation regression model was proposed to estimate
executive compensation. Since then, and thanks to its versatility and ease of use, it
has become the by far most popular technique for tackling (linear) multiple-objective
problems, as evidenced by the bulk of literature on theory and applications of the
field. See, e. g., [40, 41, 44, 45] and the categorized bibliography of applications
therein.
By a Non-Preemptive Goal Programming problem one usually means some par-
ticular instance of the following model: a polyhedron K ⊆ IR
n
is given as the set of
decisions; there exist m criteria matrices, C
1
, . . . , C
m
, with C
j
in IR
n×n
j
; each deci-
sion x ∈ K is valued according to criterion C
j
by the vector C
>
j
x, to be compared
with a given target set T
j
⊆ IR
n
j
. With this, the deviation d
j
(x) of decision x with
respect to the target set T
j
is defined as
d
j
(x) = inf
z
j
∈T
j
γ
j
(C
>
j
x − z
j
)
for some given norm γ
j
, while the overall deviation at x is measured by
γ(d
1
(x), . . . , d
m
(x)),
where γ is a norm in IR
m
assumed to be monotonic in the nonnegative orthant IR
m
+
(see [4, 25]) i. e.
γ(u) ≤ γ(v) for all u, v ∈ IR
m
+
with 0 ≤ u
i
≤ v
i
for all i = 1, . . . , m.
Then, the solution(s) minimizing the overall deviation are sought. In other words,
one solves the convex program
inf
x∈K
γ(d
1
(x), . . . , d
m
(x)). (1)
As pointed out e. g. in [8, 39, 40], Non-Preemptive Goal Programming and
related models can be rephrased as minimum-distance problems. This follows from
Version as of December 3, 2000 3
the previous formulation, since (1) is equivalent to
min γ(γ
1
(C
>
1
x − z
1
), . . . , γ
m
(C
>
m
x − z
m
))
s.t. x ∈ K,
z
j
∈ T
j
∀j = 1, . . . , m.
(2)
Denoting by
e
γ the norm in IR
n
1
× . . . × IR
n
m
defined as
e
γ(u
1
, . . . , u
m
) = γ(γ
1
(u
1
), . . . , γ
m
(u
m
)),
problem (2) can be written as the minimum
e
γ-norm problem
min
e
γ(u
1
, . . . , u
m
)
s.t. u
j
= C
>
j
x − z
j
∀j = 1, . . . , m
(x, z) ∈ K ×
Q
1≤j≤m
T
j
(3)
In many applications, each criterion C
j
is assumed to be a vector c
j
∈ IR
n
, so it
values x through the scalar c
>
j
x; each target set T
j
is then a subset of IR of one the
forms
T
j
= [t
j
, +∞), (4)
T
j
= (−∞, t
j
], (5)
T
j
= {t
j
}, (6)
or, in Goal Range Programming [20], of the form
T
j
= [t
j
, t
j
]. (7)
This corresponds to a goal constraint of type c
>
j
x ≥ t
j
, c
>
j
x ≤ t
j
, c
>
j
x = t
j
, or
c
>
j
x ∈ [t
j
, t
j
], respectively. In other words, one desires to have c
>
j
x above t
j
, below t
j
,
exactly at t
j
, or between t
j
and t
j
, respectively.
Whereas the choice of the aggregating norm γ is crucial, (although, in applica-
tions, mostly reduced to the cases l
1
or l
∞
) the choice of the norms γ
j
to measure
deviations in the case n
j
= 1 ∀j is irrelevant, and we can consider each γ
j
to be
equal to the absolute value function. Then, the deviations take on the more familiar
form
d
j
(x) =
max
n
t
j
− c
>
j
x, 0
o
if T
j
= [t
j
, +∞),
max
n
c
>
j
x − t
j
, 0
o
if T
j
= (−∞, t
j
],
|c
>
j
x − t
j
| if T
j
= {t
j
},
max
n
t
j
− c
>
j
x, 0
o
+ max
n
c
>
j
x − t
j
, 0
o
if T
j
= [t
j
, t
j
].
Version as of December 3, 2000 4
From these expressions, it should become clear that target sets of type (7), (thus
also of type (6)) are used only for modeling convenience, since they can be derived
from sets of types (4) and (5): splitting criterion j into criteria j
1
, j
2
, and defining
T
1
j
= [t
j
, +∞) and T
2
j
= (−∞, t
j
], the deviation d
j
(x) is simply the sum of the
deviations with respect to T
1
j
and T
2
j
.
1.2 Examples
Applications of Goal Programming abound in the literature; see e. g. the list of 351
applications papers cited in [40]. However, the range of applicability of (1) is by no
means reduced to what is usually classified as Goal Programming: a vast series of
important models in different fields of Optimization can also be seen as particular
instances of (1), mainly from the perspective of minimum-distance problems. Some
of them are briefly discussed below.
Overdetermined systems of (in)equalities
If a system of linear equalities and inequalities
a
>
1
x ≥ b
1
a
>
2
x ≥ b
2
.
.
. ≥
.
.
.
a
>
p
x ≥ b
p
a
>
p+1
x = b
p+1
.
.
. =
.
.
.
a
>
p+q
x = b
p+q
(8)
is infeasible, one can look for a so-called least infeasible solution, i. e. a point x
∗
solving
min
x
γ(max(0, b
1
− a
>
1
x), . . . , max(0, b
p
− a
>
p
x), |b
p+1
− a
>
p+1
x|, . . . , |b
p+q
− a
>
p+q
x|)
for some norm γ monotonic in IR
p+q
+
. This is simply a Goal Programming problem in
which the vectors a
i
(i = 1, . . . , p+q) play the role of the criteria and the components
b
i
(i = 1, . . . , p+q) of the right hand side vector represent the targets, see Example 4
in Section 3.
Version as of December 3, 2000 5
When only equalities appear in (8), one obtains the problem of solving an overde-
termined system of linear equations, classical in Approximation Theory [33, 43], or,
equivalently, the Linear Regression problem [42]. Usually, γ is assumed to be an l
p
norm, mainly p = 2, (yielding the well-known Least Squares problem [3]) p = 1, or
p = ∞ [1].
Multifacility location
In Continuous Location [29, 36], distances are usually measured by gauges. For
simplicity, we will consider throughout this paper only gauges γ of the form
γ(x) = inf{t ≥ 0 : x ∈ tB}
for some nonempty convex compact B ⊂ IR
m
(its unit ball) containing the origin
in its interior. In applications, this additional assumption is usually fulfilled, see,
e. g. [10, 29]. Observe that norms correspond to symmetric gauges. Moreover, since
the origin is assumed to be an interior point, the gauge takes always finite values.
See e. g. [17] for the case of gauges with values on IR
+
∪ {+∞}.
Let F be a nonempty finite set and let ∅ 6= E ⊆ F ×F . Then (F, E) is a directed
graph. Following e. g. [13, 27], F represents the set of facilities (some of which may
have fixed locations in IR
n
), whereas E represents the interactions between these
facilities.
For each edge e := (f, g) ∈ E, let γ
e
be a given gauge in IR
n
, which measures
the cost of the interaction between facility f and facility g. Let γ be a gauge in IR
E
monotonic in the non-negative orthant.
For a nonempty closed convex set K ⊆ (IR
n
)
F
, consider the optimization problem
inf
(x
f
)
f ∈F
∈K
γ((γ
(f,g)
(x
f
− x
g
))
(f,g)∈E
). (9)
The most popular instance of (9) is the continuous minisum multifacility location
problem, see [36, 46, 47] and the references therein. There, the node set F is
partitioned into two sets A and V , representing respectively the fixed and the free
locations, and a family (a
f
)
f∈A
∈ (IR
n
)
A
of fixed locations is given. The feasible
region K is then defined by
K =
n
x = (x
f
)
f∈F
∈ (IR
n
)
F
| x
f
= a
f
for all f ∈ A
o
, (10)