©UNESCO-EOLSS
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
THE ROLE OF SOFTWARE IN OPTIMIZATION AND
OPERATIONS RESEARCH
Harvey J. Greenberg
University of Colorado at Denver, USA
Keywords: optimization, mathematical programming, computational economics,
mathematical programming systems
Contents
1. Introduction
2. Historical Perspectives
3. Obtaining a Solution
4. Modeling
5. Computer-Assisted Analysis
6. Intelligent Mathematical Programming Systems
7. Beyond the Horizon
Acknowledgement
Glossary
Bibliography
Biographical Sketch
Summary
This chapter presents the many roles of software in optimization. Obtaining a solution
has been the focus of mathematical programming, and now one must consider different
architectures and algorithm environments. But the role goes beyond the numerical
computation, integrating concerns for supporting optimization modeling and analysis.
Beginning with a historical perspective this chapter describes modern developments,
including that of an intelligent mathematical programming system.
1. Introduction
The promise of operations research is to solve decision-making problems, and a large
part uses optimization. The mathematical program is given by the following:
optimize f ( x ) : x ∈ X , g ( x ) ≤ 0, h( x ) = 0,
(1)
where X ⊆ \ n , f : X 6 \, g : 6 \ m , h : X 6 \ M , and optimize is either minimize or
maximize.
In words, we seek to find a minimum or maximum of a real-valued function, possibly
subject to some mixture of inequality and equality constraints. There are variations of
this form, such as multiple objectives, goals, uncertainty, and logical expressions. Each
variation is approached by using the above standard form, and generally, this can be
done in more than one way. More discussion of this and particular mathematical
programs are given in the Mathematical Programming Glossary.
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
The basic role of software is to enable an expression of a mathematical program, obtain
a solution and provide tools to support model management and analysis of the results. In
this chapter, we begin with some historical perspectives about software and
optimization, followed by overviews of software designed to obtain a solution. Software
for modeling has its own history, but the focus here is on current systems and needs for
future designs. Then, computer-assisted analysis is explained, drawing from two bodies
of work and pointing towards opportunities for enhanced roles of software. This leads
naturally into a description of intelligent mathematical programming systems, marking
new roles for future systems. The concluding section discusses current developments
and the role of software in the broader field of operations research.
2. Historical Perspectives
Orchard-Hays gave an excellent historical account of the early years of mathematical
programming and its inseparable ties to the computing field. (Also see his companion
papers, as well as others, in the same proceedings.) Mathematical programming was
treated as a process, rather than an object. As such, people needed a way to express their
model, often with a general language like FORTRAN or with what were then called
matrix generators, which evolved into modeling languages. Some evolved further into
modeling systems.
This emerged in the early years, when system was used to denote the fact that it
contained complete optimization, taking data as an input file and producing a solution
file, rather than being a subroutine library. Currently, both the mathematical
programming system (MPS) and the library approach (e.g., IBM OSL) are used. With
the growth of object-oriented programming, there is a third paradigm: a shell that serves
as a main program with generic classes; the programmer adds particular classes within
this framework to represent a particular problem and algorithm. Current systems that
use this paradigm are ABACUS, MINTO and PICO.
There were significant developments in the 1980s, not the least of which was the
explosive growth of affordable computers. Waren, et al., give the status of nonlinear
programming (NLP), with the emergence of PCs cited as a significant change that
affects how algorithms are implemented and what has changed in importance, e.g.,
space (disk and memory) is no longer an issue (also see Fourer's guide in
Bibliography). This is in some contrast to most of the more recent surveys, which still
emphasize the algorithms apart from their implementation.
In integer programming, a recent plenary talk by Ralph Gomory suggested that modern
computers can make some of the group-theoretic ideas, which had been abandoned in
the 1970s, more practical. Following this note, Karla Hoffman gave a perceptive insight
using dinosaurs flying jets as a metaphor to emphasize the role of computing power
running old algorithms. Her primary assessment, however, was that it is the formulation
that has the most impact on solvability, and this is the theme to consider as the modern
role for software. In fact, this was a major goal of the project to develop an Intelligent
Mathematical Programming System (IMPS).
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
3. Obtaining a Solution
A solution is a global optimum, or a declaration that no optimum exists, preferably with
a reason, such as infeasible or unbounded. Software cannot guarantee that a correct
conclusion is reached. One reason is computational error, causing inaccurate results,
even for linear programs. Some problems use only integer arithmetic, where there is no
computational error, but they are typically hard problems that cannot guarantee an
accurate result due to how long it would take to consider the large number of possible
solutions. It is this latter source of inaccuracy, that is the cutting edge of optimization
software.
Several recent surveys provide details on the current state of the art in obtaining a global
optimum, drawing from classical methods based on local search (e.g., on calculus).
Because we cannot obtain exact solutions to hard problems in a practical amount of
time, two basic approaches have been taken: metaheuristics, such as a Tabu Search, and
approximation algorithms with guaranteed bounds. Software to implement these can
make a difference in their performance (see Global Optimization).
In discrete optimization some exact methods exist for special situations, like dynamic
programming, , but the branch and bound/cut remains the primary exact method. It is
also used in continuous (global) optimization, so branch and bound is a general exact
method. These typically use linear programming to solve relaxation problems for
bounds, cut generation, and search guidance. The relaxations drop integer restrictions,
and linear functions are used to approximate nonlinear ones (typically using the Taylor
expansion). A software issue is how fast the linear programs are solved when used this
way. The linear program (LP) relaxation of a combinatorial optimization problem has
very different characteristics than a true LP, such as one that represents product
distribution. This implies, for example, that the LP algorithm control parameters need
adjustments from their default values since they are typically tuned with test problems
that are true LPs (see Linear Programming, Dynamic Programming and Bellman's
Principle , Combinatorial Optimization and Integer Programming).
3.1. Data structures, Controls and Interfaces
Wirth's charming equation, Algorithms + Data Structures = Programs, is not quite
enough to be a system capable of solving optimization problems with computing
paradigms available in 2002. To it, we must add:
Programs + Controls + Interfaces = Systems
There have always been algorithm controls, but interfaces have become much more
sophisticated, especially with graphics.
Most publications in mathematical programming present algorithms, but very few
describe the data structures. Fundamentally, an algorithm is a precise sequence of steps.
We often refer to an algorithm family, or algorithm strategy, to mean some steps are not
completely defined, leaving room for tactical variations. For example, Figure 1 gives
the simplex algorithm strategy that uses the 2-phase method of getting an initial feasible
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
solution (or ascertaining that none exists), leaving such steps as pricing open to different
algorithm tactics.
Figure1: Two-Phase Simplex Algorithm Strategy
A data structure is how we store the information. Even a specific algorithm, such as
Dantzig's Specific Simplex Method, does not specify the type of arithmetic or other
things that depend upon the data structure (and the computing environment). In the case
of a simplex algorithm, a typical data structure is to store the data in a sparse format.
For example, the LP matrix is stored by columns with each column having a pair of data
entries for each nonzero:
row index | value
In particular, suppose we have a transportation problem with s sources and d
destinations. The dimensions of the matrix are m = s + d rows (one per node) and n = sd
columns (one per arc). Each column of the node-arc incidence matrix has exactly two
non-zeroes, -1 for the source and +1 for the destination. The total number of nonzeroes,
therefore, is 2n, and its density is
2n
mn
, which reduces to
2
m
. The greater the number of
nodes (m), the greater the sparsity, because the density becomes small as m becomes
large. To illustrate, consider the following node-arc incidence matrix, which represents
a transportation problem that has 2 sources and 3 destinations.
⎡ −1
⎢ 0
⎢
A=⎢ 1
⎢
⎢ 0
⎢⎣ 0
−1
0
0
1
−1
0
0
0
0
0
−1
1
0
0
−1
0
1
1
0
0
©Encyclopedia of Life Support Systems (EOLSS)
0⎤
−1⎥⎥
0⎥
⎥
0⎥
1⎥⎦
(2)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
There are 12 nonzero entries in this 5 × 6 matrix, so its density is
12
30
, which is 0.4. For
more realistic sizes, where m 1000 , the density is less than 1%. Many large LPs have
an embedded network structure, which is one reason they tend to be very sparse.
Nonlinear functions are represented by their parse tree, or something very similar.
Figure 2 shows an example.
Figure 2: Parse Tree for f ( x1 ; x2 ) = x12 − x1 x2 + x23
Solvers call upon this tree for function evaluation. Modern methods use automatic
differentiation, which stores nonlinear functions in a similar data structure, designed for
rapid calculations of the function and its derivatives (if they exist).
3.2. Parallel Algorithms
One of the current topics of research is how to design parallel algorithms. An early use
of parallel architecture was for Dantzig-Wolfe decomposition in LP. Many types of
architectures and algorithms are in the text by Bertsekas and Tsitsiklis, and more recent
ideas are in proceedings, notably those edited by Heath, et al. and by Pardalos, et al.
To illustrate how a parallel architecture can be used effectively, consider a
parallelization of the conjugate gradient method (see Nonlinear Programming). We are
given a quadratic form,
1 T
x Qx − cx
2
, where Q is an n × n symmetric, positive-definite
matrix. We know its (unique) minimum occurs at the solution to the equation, Qx = c,
and one way to compute this is by the following iterations:
xt +1 = xt + α t d t ,
(3)
where x0 is given, d0 = − g0 = c − Qx0, and
αt = −
gt d t
(d t )T Qd t
©Encyclopedia of Life Support Systems (EOLSS)
(4)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
d t = −( g t )T + β t d t−1
βt =
(5)
( g t −1 )T Qd t −1
(6)
(d t −1 )T Qd t −1
g t = ( xt )T Q − c = ∇
(
1 T
x Qx − cx
2
) x= x
t
(7)
Let us assess the computations at an iteration. At the beginning of iteration t, we have
already computed xt, dt-1 , and gt-1 (gt-1 )T. We must compute gt = Qxt − c and gt(gt)T ;
from these, we use equations (6) and (5) to compute βt and d t, respectively. Finally, we
use equation (3) to compute xt+1, thereby completing the iteration.
Suppose p ≥ n, so we can assign a processor to compute results pertaining to one
coordinate of the vectors. Let processor i be given the ith row of Q, denoted Qi•. The
vector Qxt - c is done completely in parallel, with processor i computing Qi• xt − ci.
These are communicated to a designated processor, which thus obtains gt. This
processor could compute gt(gt)T, or it could broadcast gt to the other processors and have
them compute ( git )2 and send its value back for accumulation. Which is better depends
on communication time versus the time to compute an inner product on the main
processor. Alternative schemes use hierarchies to accumulate inner products, and much
of the design depends not only upon the particular architecture, but also upon whether
the matrix and vectors are sparse.
This illustrates the general notion that one way to design a parallel algorithm is to use
parallel linear algebra. Many optimization algorithms can be parallelized this way, but
some do not use linear algebra centrally. In particular, combinatorial optimization
algorithms can take other advantage of parallel computation.
Consider the dynamic programming recursion for the 0-1 knapsack problem:
f j ( β ) = max{ f j −1 ( β ), c j + f j −1 ( β − a j )} for β ∈ {a j , a j + 1, ..., b}.
(8)
for j = 1, …, n, where f0 ≡ 0 (see Discrete Optimization, Dynamic Programming and
Bellman's Principle ). Suppose we decompose each iteration by having each processor
compute fj for an equal number of state values. For notational convenience, let b = kp
for some positive integer, k. In words, let the total size of the knapsack be an integer
multiple of the number of processors. This means that each processor has k state values
to compute, and each takes the same amount of time. This is clearly a logical
decomposition, but its merit depends on communication time, which depends upon the
particular parallel architecture.
Scheduling tasks to processors is the key problem in designing a parallel algorithm.
Load balancing is a proxy goal for maximizing the speedup, whose reciprocal is the
fraction of the best possible serial computation time. Let T*(s) be the best time it takes
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
to solve a problem in some class of size s on a serial computer. (The "size" is frequently
taken to mean the number of variables, but it could account for other data values.) Let
Tp(s) be the time it takes a parallel algorithm to solve the same problem with p
processors. Then,
T * ( s)
Speedup = S p ( s ) =
.
Tp ( s)
(9)
The best one can hope for is Sp(s) = p — equivalently, Tp(s) = T*(s)/p. This occurs if the
best serial algorithm can have its computations decomposed perfectly into p parts, each
taking the same amount of time. This could never be fully achieved, even with perfect
load balancing, due to message passing that is needed − that is, the information
produced by a processor must be communicated to at least one other processor.
The speedup of the dynamic programming parallelization is nearly linear. We have
S p (n, b) =
T * (n, b)
T * (n, k ) + M (n, k )
,
(10)
where M(n, k) is the communication time for n variables and k states. The size is given
as the number of variables (n) and the number of states (b), but we can consider the cost
per iteration that is, divide by n, since both algorithms perform n iterations. Then,
S p (n, kp) =
τ (kp)
.
τ (k ) + μ (k )
(11)
Further, each state requires a simple comparison of two values plus a storage of the
result, so τ(kp) = pτ (k). Substituting this into the above expression simplifies the
speedup, which is now independent of the number of variables:
S p ( n, b ) =
p
μ (k )
1+
τ (k )
(12)
,
The speedup depends on the communication time relative to the computation time,
which is the ratio:
μ (k )
τ (k )
. We can see that τ ( K ) ≈ α k for some constant α, but μ(k)
could be O(k) or O(log k), depending upon the architecture. A worst case has
S p (n, kp) ≈ γ p for some constant, γ, which is still very good for large state values.
Doubling the number of processors doubles the speedup. The more favorable case is
when μ (k) = O(log k), for then limk→∞ Sp(n, kp) → p. Currently, one of the most active
places for such research is Sandia National Laboratories, with the ASCI red machine
that consists of more than 9000 tightly-coupled Intel processors.
They have an object-oriented Parallel Integer and Combinatorial Optimization code,
called PICO. Despite its title, PICO is a framework designed for general parallel branch
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
and bound, including nonlinear programs with continuous-valued variables. While there
have been some efforts to use parallel algorithms for hard problems, this is still a
frontier with many design decisions and experiments ahead. One class of algorithms
that have a natural decomposition to balance the load among the processors is the class
of stochastic, population-based, evolutionary algorithms.
Some classes of hard nonlinear problems, such as in design optimization, involve
expensive functional evaluations. Often, f (x) is the solution to a differential equation or
a simulation, where x is a vector of design parameters. Although x is not large, obtaining
f (x) can take hours. Parallel algorithms are being developed for such problems, some
with a speedup on the order of p .
4. Modeling
Many modeling languages and systems have appeared over the past four decades; some
have disappeared. Each current language requires the modeler to adhere to a
predetermined way of expressing the model, and we shall discuss some of them in the
next section. An ideal system is one where the adaptation is the burden of the computer,
not the human. As we have different cognitive skills, some algebraic, some processoriented, some graphic, and various mixtures, the role of software is that the modeling
system ought to change its mode to suit the modeler, not the other way around.
Fundamentally, a model can be divided into its parts, as shown in Figure 3, where a
model is composed of objects and relations among those objects. In an optimization
model, there are two fundamental types of objects: data and decision. Data objects
include not only numerical values, like parameters, but also symbolic objects, like sets
of regions or a network topology, which can vary from one scenario to another. The
decision objects include not only variables, but also functionals, which could have other
objects in its domain.
There are three types of model relations: constraints, conditions and learning.
Constraints can be of the canonical form, g(x) ≤ 0, where x is a vector of decision
objects. This form hides the dependence of g on data objects, especially on "primary
data" objects used to compute what the solver sees. For example, the cost of capital
might be one of the primary data values, from which the modeling system is directed to
compute cost coefficients that appear in g. Logical constraints are prevalent, such as
"Select at most 1 from this set…". Most languages do not permit the expression to be in
English, and the modeler must bear the burden of how to express this in a way that is
eventually understood by the solver. An advantage of constrained logic programming
(CLP) is that it allows us to express logical relations more naturally than mathematical
programming languages. The conditions pertain to when certain objects or constraint
relations are in effect. For example, one might build a general network to accommodate
any scenario, then use data to determine whether a supply constraint or a transportation
variable should be generated. Admissibility conditions pertain to data values that are
checked to prevent erroneous scenarios.
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
Figure 3: Anatomy of a Model
Finally, learning is another level of relations that can be part of an artificially intelligent
environment for model formulation and management. It includes learning relations
among objects from experience with some scenarios, even without human feedback.
One example is when it detects an infeasibility and diagnoses its cause. This could lead
to an automatic check for those conditions in future scenarios.
The diagram in Figure 3 is simplified. It does not show, for example, whether an object
or relation is primitive or implied; it does not show attributes or other elements that
require discussion if we were to consider implementation. Its purpose is simply to
highlight an anatomy in tune with the role of software in optimization, starting with
model formulation. Also, the modeling process contains more than models; it also
contains guidance for solvers, model management aids, and rules to support analysis of
instances.
4.1. Expressive Power
The expressive power of a language is its ability to represent different kinds of objects
and relations. Theoretically, the languages in use today have the same expressive power,
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
but practically, they do differ. Some languages emphasize that they are algebraic, but
what does this mean? What languages are not algebraic? Essentially, an algebraic
language is one that allows you to express the optimization model the way you would
on a blackboard or in a textbook. It is some variant of the following:
Let x(i, j, k) denote the level of transportation of the kth material from i to j … There is
limited supply: ∑ j ,k x(i, j, k ) ≤ S (i) …
Languages like AMPL and GAMS are like this. Older languages, like MAGEN/OMNI
and RPMS/GAMMA, are process oriented. They would express activities, like the
following:
Let x(i, k) denote the level of operation of a distillation unit in region i at mode k … It
uses materials M(i, k) and has product yields Y (i, k); capacity is limited by C(i, k).
These have been unnecessarily contentious, and now there is another paradigm, based
on logical expressions, called logic programming. We commonly refer to constrained
logic programming to mean logical expressions of what to do, subject to constraints
like:
Choose 1 from the set...
4.2. Logic Programming and Optimization
There has been a gradual merger of operations research methodology with logic
programming. The role of the software is to provide a different paradigm for expressing
combinatorial optimization problems, like scheduling. Rather than require the modeler
to specify some system of inequalities; constraints, such as job i must precede job j, can
be expressed more naturally.
To illustrate, consider the traveling salesman problem. There are many integer
programming formulations, but constraint logic programming can represent it
differently. Let yi be from the set of cities, {1, …, n}. To express the constraint that
every city be visited exactly once, the logical constraint is all different {yi}, and that is
precisely all one must specify! This is beyond the expressive power of integer
programming, adding new capabilities to optimization modeling, as well as new
opportunities to express bad formulations.
5. Computer-Assisted Analysis
Obtaining a solution is not enough, and we have just described the role of software in
formulating and managing models. In addition, the role of optimization software must
support analysis. At least in linear programming, we often solve larger problems than
we can understand. It is not only post-optimal analysis that is important, but also
debugging tools to deal with anomalies and infeasibility diagnosis. This is a underdeveloped area, which began to emerge in the late 1970s. The only general-purpose
software currently available are ANALYZE, GMSCHK and MProbe. Some modeling
systems, like ASCEND, MIMI and XPRESS-LP, have some analysis aids.
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
In this section, this function is divided into some overlapping parts and elaboration is
given along the lines that are in a recent survey by Chinneck and Greenberg.
5.1. Query and Reporting
A fundamental part of optimization software is to be able to browse a model instance
and its solution obtained by the optimizer. (The solution is not necessarily optimal, as in
debugging an infeasible instance.) Modeling languages provide only very basic report
writing capabilities, with little or no debugging aids. ANALYZE enables multiple-view
query and syntax-directed report writing for mixed-integer linear programs, but it does
not have an interface with all modeling languages. Rutherford provides extensive report
writing tools for GAMS.
To appreciate the need for query and reporting, consider the LP:
min cx : x ≥ 0, Ax = b,
(13)
where A is m × n and the vectors conform to those dimensions. When m and n are small
enough to see A on one screen (or a sheet of paper), we have no problem. When,
realistically, the LP has many thousands of rows and columns, we face problems of
what to see and how to see it.
The functions of query and reporting extend to verification and validation. The former
pertains to knowing that what is in the computer-resident model is what is believed to
be there. The latter pertains to how well the model represents the world for which it is
intended. Both are fundamental to model management.
5.2. Debugging
Debugging is finding out what went wrong when the result is infeasible or anomalous.
The only software systems designed to assist with this are ANALYZE, GMSCHK,
MPRobe, and a suite of Chinneck's IIS codes. The approaches to infeasibility diagnosis
are as follows:
Navigation:
This is simply interactive query with a system like ANALYZE,
generally by someone who is expert in knowing the model.
Parametric
Programming: This begins with a feasible system for a parameter, θ, such as
Ax = θb, x ≥ 0. This is feasible for θ = 0 (possibly for θ > 0) and
parametric programming is used to drive θ to 1. It will reach a
feasibility threshold in (0,1), showing blockage, such as insufficient
supply or capacity.
Phase I
This applies only to linear programs. It rests on the fact that the dual
Aggregation:
variables associated with a phase I solution comprise a solution to the
alternative system. If {Ax = b, L ≤ x ≤ U} is infeasible and π is a
Phase I dual solution, the single equation, πAx = πb is inconsistent
with the bounds, L ≤ x ≤ U. This reveals which activities are involved
those with nonzero coefficients, πAj ≠ 0 and a blocking bound.
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
Logical Testing: This applies to when there are binary variables that represent logical
truth-values. Topological sorting is used to reveal inconsistencies,
such as a cycle of precedence relations. These tests extend to Horn
clauses.
Exploiting
Structures:
Special structures include block and link (which includes netforms)
and inventory equations.
Partitioning:
This is a powerful approach, which can be fully automated. The idea
is to find a partition of the original system of constraints and explore
at least one of those subsystems for probable cause. One type of
subsystem is the irreducible infeasible subsystem (IIS). This is
infeasible but the removal of any one constraint renders the
subsystem feasible. A necessary condition to become feasible is that
at least one constraint in each IIS be deleted, or adjusted in a way that
makes the subsystem feasible. A sufficient condition to become
feasible is that at least one constraint from every IIS be deleted. (The
IISs can overlap, so the removal of one constraint that is in every IIS
would render a feasible subsystem.)
There are tactical variations within these approaches, and the IIS approach extends to
general mathematical programs, where variables can be integer-valued and functions
can be nonlinear. Of course, the subproblems to solve in order to obtain IISs can then be
NP-hard. The role of software is to support all of these approaches because how they
behave depends upon the problem, and how experienced the analyst is in mathematical
programming. A rule-based intelligence is a level of ANALYZE that can automate the
diagnostic process and use heuristics to guide it.
To illustrate, Figure 4 shows a network with a maximum flow of 32 units, but the total
demand is 60 units. One could tell this to the analyst, but there are much smaller
isolations that are causal. In a realistic network, the cut set could have many thousands
of arcs, far too many for a person to visualize (see Graph and Network Optimization).
Figure 4: An Infeasible Network Flow
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
Optimization software begins to process this problem by solving its Phase I:
16
min ∑ vi : v ≥ 0,
i =1
Supply at node 1:
x14
+ v1 ≤ 20
Supply at node 2:
x24 + x25
+ v2 ≤ 20
Supply at node 3:
x35
+ v3 ≤ 20
Supply at node 4:
x14 + x24 − x46 − x47 + v4 = 0
Supply at node 5:
x25 + x35 − x57 − x58 + v5 = 0
Supply at node 6:
x46
+ v6 ≥ 10
Supply at node 7:
x47 + x57
+ v7 ≥ 20
Supply at node 8:
x58
+ v8 ≥ 30
Flow bounds:
0 ≤ x14 + v 9 ≤ 30
0 ≤ x24 + v10 ≤ 20
0 ≤ x25 + v11 ≤ 10
0 ≤ x35 + v12 ≤ 10
0 ≤ x46 + v13 ≤ 10
0 ≤ x47 + v14 ≤ 2
0 ≤ x57 + v15 ≤ 20
0 ≤ x58 + v16 ≤ 30
(14)
The analyst must now debug this infeasible LP by diagnosing its cause. Conventional
solvers and most modeling languages provide only displays of values, such as what you
see in Figure 5. In a realistic problem this table would be far too large to be useful, so
the analyst needs some screening aids.
Figure 5: Display of Solution Values for Network in Figure 4
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
There are several approaches to help analysts diagnose an infeasibility. One is to use the
Phase I dual prices to form one infeasible aggregate constraint:
x25 + 2 x35 + x47 ≥ 70.
(15)
This inequality is implied by the original system by taking a weighted sum. Letting pi
denote the dual price of the i-th supply constraint, and noting that p ≤ 0, we first obtain
the following:
p1 ( x14 + p2 ( x24 + x25 ) + p3 x35 ≥ ( p1 + p2 + p3 )20.
(16)
Since p = (−1,−1, 0), this becomes the aggregate supply limit: − x14 − x24 − x25 ≥ −40 .
Combining the balance equations in the same manner, and using its price vector (1,2)
(shown in Figure 1: Display of Solution Values for Network in Figure 4) we obtain the
aggregate balance equation:
x14 + x24 − x46 − x47 + 2( x25 + x35 − x57 − x58 = 0 .
The demand requirements use the price vector (1, 2, 2) to obtain the following aggregate
demand requirement: x46 + 2( x47 + x57 ) + 2 x58 ≥ 1 × 10 + 2 × 20 + 2 × 30 = 110 .
Adding these three aggregates produces the overall aggregate constraint, and this
approach notes that the greatest aggregate value on the left-hand side, for the given
bounds, is less than the aggregate requirement:
max{x25 + 2 x35 + x47 : 0 ≤ x25 ≤ 10, 0 ≤ x35 ≤ 10, 0 ≤ x47 ≤ 2} = 32 < 70.
(17)
The ANALYZE system does this and can provide an English response, such as what is
shown in Figure 6. The system can continue to help the analyst by explaining the
aggregate coefficients and right-hand side.
Figure 6: ANALYZE Diagnosis Using Phase I Price Aggregation
An IIS approach could produce the subnetworks shown in Figure 7. In 7(a), the only
constraints included are the flow bounds on x25, x35 and x4t . The sources of those arcs
are not shown because it does not matter where those flows come from, only that they
are limited to the values shown. Nodes 5, 7 and 8 are shown because the constraints
associated with them are in this IIS. Only the demand arcs are oriented; the others are
undirected, which means that the non-negativity constraints on those flows are not in the
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
IIS. The demands at nodes 7 and 8 dominate the direction of flow. By its definition, the
IIS is an infeasible subsystem such that dropping any one constraint renders the proper
subsystem feasible. If, for example, we drop the demand requirement at node 7, we can
ship 10 units from 7 to 5 (i.e., x57 = −10), then send it to node 8, thus rendering the
proper subsystem feasible. In 7(b), the demand at node 7 is dropped, but non-negativity
is added to the flow variable, x57.
Figure 7: Two IISs for the Infeasible Network in Figure 4
ANALYZE can import IISs created by Chinneck's code, which allows separate
treatment of logical constraints, such as non-negativity. If we retain x ≥ 0 in our
subsystem, we obtain a smaller subnetwork, shown in Figure 8. Node 7 plays no role,
and this subnetwork presents the analyst with the two relevant flow bounds and the one
demand requirement. Those constraints, themselves, with flow balance at node 5,
comprise an infeasible subsystem.
Figure 8: Good Diagnosis = Small Isolation + Succinct Explanation
6. Intelligent Mathematical Programming Systems
What makes an MPS intelligent? In artificial intelligence, we would say the computer
performs reasoning, arriving at conclusions that we normally attribute to human
problem-solving. Here, this is broadened to include discourse models, such as
visualization aids. The main focal points, analysis, formulation, and discourse, comprise
building blocks for the IMPS. The concepts on which they are built, the functions they
serve, and the techniques employed comprise the cornerstone of its foundation.
6.1. Formulation
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
Very few optimization modeling languages or systems provide intelligent aids for
formulation. One exception is the XPRESS-LP system. There have been approaches
taken, and the general idea is illustrated in Figure 9.
Figure 9: Formulation Component of IMPS
There have been experiments with the form of the model assistant, ranging from model
libraries to jigsaw puzzle reasoning. In all cases the role of the software was to help
someone who is not an expert in mathematical programming formulate a model. This
has so far not succeeded, and if it is to be developed, the formulation phase must be
viewed more broadly. In addition to some expression of the model objects and relations,
there needs to be support generated for model management, database auditing, solver
selection and subsequent analysis support.
6.2. Model and Scenario Management
Here we expand the model assistant functions with two components: solver selector and
scenario manager. These are two fundamental functions, shown in Figure 10 in
conjunction with the formulation assistants. The figure is already rather busy, so other
functions and files are omitted. Model management aids include verification and
validation tasks. Verification is confirming that what we think is in the computer is
indeed there. Validation is determining the extent to which the model, or some scenario,
is sufficiently consistent with what it is supposed to model that results are not
misleading.
One element of verification is shown in Figure 10, which is the data. Checks are
produced by the model assistant, and this file is read by the Verifier. For example, the
model assistant determines that a necessary condition for feasibility is that some
function of the data must satisfy a relation, f (D) ≥ 0 (viz., total supply - total demand ≥
0). Whenever the data objects in D change, the verifier checks that the relation holds. Its
role is to prevent infeasible scenarios, giving causal information whenever it happens.
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
Figure 10: Formulation Component of IMPS
6.3. Discourse
The objective of a discourse model is to communicate with a user, to which we add: in a
manner that is most natural for him or her. Natural language includes both text and
graphics. Algebraic and functional forms are what we are used to seeing in textbooks,
but constrained logic programming (CLP) is changing that. We now enlarge our
expressive powers to include many mathematical and logical forms.
The idea is to have one fundamental structure from which many different views can be
constructed at the will of the analyst. Some of the early ideas from neural networks and
concept graphs have not quite fulfilled their potential in discourse for optimization. We
primarily rely on syntax, but there is a growing development of visualization,
particularly for analysis support.
6.4. Analysis
So far, intelligent analysis support has been rule-driven. To illustrate, we shall show
some substructures found by ANALYZE, followed by a hypothetical discourse. Figure
11 shows a portion of a linear program that traces the flow of activities that generate
baseload electricity in a particular region. There are 26 activities that could generate this
electricity. Of these, 13 have zero reduced cost, including 9 basic activities. Each of the
13 activities is a candidate for being in a margin-setting path, so other paths were traced.
Each of the 12 paths must be blocked by some activity at bound; otherwise, a less
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
expensive path would have produced more, displacing the margin-setting flows in
Figure 11.
Figure 11: A Blocked Path in a Linear Program
Figure 12 shows a possible discourse with an analyst after the blocking path is found.
The English translations are made possible by the syntax maps recognized by
ANALYZE.
Figure 12: Discourse Explaining Blocked Path in Figure
Figure 13 gives an overview of an entire IMPS, with three places for intelligent support:
during model formulation (or revision), selecting a solver, and analyzing the results.
The level of detail is minimal, as one notes upon looking at Figure 10. The knowledge
bases could be distributed or united, and the model manager maintains all components.
7. Beyond the Horizon
Many of the underlying principles that connect software and optimization apply to other
techniques in operations research. Simulation is notable because the old concept of
languages evolved into systems (or platforms), and intelligent modeling aids began to
emerge in the 1980s. Animation was introduced to help debug simulation models, and
Jones showed how animation can be used for related insights into sensitivity analysis.
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
Figure 13: IMPS Components
More visualization has been a key to modern software systems, and modelers have
come to expect graphic user interfaces (GUI). Algorithm animation has not fully
penetrated optimization software, but this is also coming.
As visualization increases in importance, we can expect to see more on the web.
Websim is the current term that refers to web-based simulation, and it is one of the
current research areas. There are Java Applets presently on the web to allow people to
model and solve optimization problems. These are small, designed for pedagogical
purposes, but we can expect this to develop further into remote capabilities for
production.
Acknowledgement
The author is grateful to Professor, Dr. U. Derigs, for the opportunity to contribute this
chapter to the EOLSS and for his patience while I learned the formatting requirements.
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
Glossary
Algorithm:
Algorithm
Controls:
Approximation
Algorithm:
CLP:
Data structure:
GUI:
IMPS:
Infeasibility:
Load Balancing:
Message Passing:
Metaheuristics:
Model Assistant:
MPS:
NP-Hard:
Speedup:
A sequence of precisely specified computations to reach a
solution.
Parameters that affect tactics within an algorithm class, such as
whether to conduct some test.
For an NP-hard problem, this is a procedure that takes a
polynomial amount of time to obtain a solution that is guaranteed
to be within some percent of optimal.
Constrained Logic Programming.
How we store information in a computer.
Graphic user interface.
Intelligent Mathematical Programming System.
An instance of a mathematical program for which there is no
feasible solution.
The goal of having each of many processors perform the same
amount of computation.
The transfer of information from one processor to another in a
parallel computing environment.
A general framework for searching a space to solve hard
problems (meta is a prefix, as in a meta-language being a
language to describe languages).
A component of an IMPS designed to assist model formulation.
Mathematical Programming System.
A problem class for, which there is no known algorithm that can
solve any member in polynomial time.
The fraction of how close to perfect distribution of computation
an algorithm implementation comes.
Bibliography
Ashford R.W. and Daniel R.C. (1986). LP-MODEL: XPRESS-LP's model builder. IMA Journal of
Mathematics in Management 1(163), 176. [Downloads are available at <http:// www.dash.co.uk.>]
Bentley P.J. ed. (1992, 1999). Evolutionary Design by Computers, 446 pp. San Francisco, CA: Morgan
Kaufman. [This is a basic reference for evolutionary algorithms/programming.]
Bertsekas D.P. and Tsitsiklis J.N. (1989). Parallel and Distributed Computation, 715 pp. Englewood
Cliffs, NJ: Prentice-Hall. [This presents the fundamentals of parallel algorithms, with special attention to
optimization.]
Chandru V. and Hooker J.N. (1953, 1999). Optimization Methods for Logical Inference, 365 pp. New
York: John Wiley & Sons. [This organizes the developments in applying integer programming to the sort
of logical inference done in artificial intelligence. Emphasis is given to the authors' work during the past
decade.]
Chinneck J.W. (1997). Computer codes for the analysis of infeasible linear programs. Journal of
Operational Research Society 47(1), 61-72. [This describes the author's development of computing IISs
in various optimization programs.]
Chinneck J.W. (1997). Feasibility and viability. Advances in Sensitivity Analysis and Parametric
Programming, (ed. T. Gal and Greenberg H.J.), pp. 14-1-14-41, Boston, MA: Kluwer Academic
Publishers. [This surveys the state-of-the-art in practical methods for helping an analyst understand why a
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
problem is infeasible, including non-linear programs. This addresses the related question of whether some
variables must be zero in every feasible solution.]
Chinneck J.W. Analyzing mathematical programs using MProbe. Annals of Operations Research, To
appear. Downloads are available at <http://www.sce.carleton.ca/faculty/chinneck/mprobe.html>. [This
applies to any mathematical program, but its main use is to provide shape information for nonlinear
functions. It probes AMPL source code and uses a sampling technique.]
Chinneck J.W. and Greenberg H.J. (1999). Intelligent mathematical programming software: Past, present,
and future. Canadian Operations Research Society Bulletin 33(2), 14-28. Also appeared in INFORMS
Computing Society Newsletter 20(1), 1-9, 1999. [This surveys optimization software that is intelligent in
its ability to help users formulate and analyze mathematical programs and generally manage the process
of doing so.]
Crescenzi P. and Kann V., ed.. (2000). A compendium of NP optimization problems.
<http://www.nada.kth.se/~viggo/problemlist/compendium.html>. [The editors maintain this site, giving
latest bounds and time complexity of approximation algorithms for NP-hard problems.]
Dutta A., Siegel H.J., and Whinston A.B.. (1983). On the application of parallel architectures to a class of
operations research problems. R.A.I.R.O. Recherche opérationanelle/Operations Research 17(4),317-341.
[This is one of the first parallel algorithms for linear programming. It applies to Dantzig-Wolfe
decomposition.]
Eckstein J., Hart W.E., and Phillips C.A. (2000). PICO: An object-oriented framework for parallel
branch and bound. Research report, Albuquerque, NM: Sandia National Laboratories. Also available as a
RUTCOR technical report. [This is a primer for the entitled object-oriented system where the user
supplies the classes for a particular problem and algorithm.]
Fourer R. (1983). Modeling languages versus matrix generators for linear programming. ACM
Transactions On Mathematical Software 9, 143-183. [This was an early description that clarified the
entitled distinction.]
Fourer R. (1996). Software for optimization: A buyer's guide, Parts 1 & 2. INFORMS Computing Society
Newsletter 17(1, 2). [This gives a broad view of what is available in all areas of optimization software.]
Glover F. and Laguna M. (1997). Tabu Search, 382 pp. Boston, MA: Kluwer Academic Publishers. [This
is the basic book about Tabu Search, written by its pioneers.]
Greenberg H.J. (1993) A Computer-Assisted Analysis System for Mathematical Programming Models and
Solutions: A User's Guide for ANALYZE. Boston, MA: Kluwer Academic Publishers. Downloads are
available at <http://www.cudenver.edu/~hgreenbe/imps/software.html>. [This describes and illustrates the
ANALYZE software system, which is part of the author's intelligent mathematical programming system.]
Greenberg H.J. (1994). Syntax-directed report writing in linear programming. European Journal of
Operational Research 72(2), 300-311 [This shows how the rules of composition the syntax of a model
can be used to drive the report writing process, including interactive query.]
Greenberg H.J. (1996). A bibliography for the development of an intelligent mathematical programming
system. Annals of Operations Research 65, 55-90. Online versions are available at:
<http://www.cudenver.edu/~hgreenbe/imps/impsbib/impsbib.html>
and
<http://orcs.bus.okstate.edu/itorms>. [This gives hundreds of citation from early works through 1996.]
Greenberg
H.J.(1996-2000).
Mathematical
Programming
Glossary.
<http://www.
cudenver.edu/~hgreenbe/glossary/>. [This is an ongoing glossary with about 700 entries and
supplements.]
Greenberg H.J. and Maybee J.S. ed. (1981). Computer-Assisted Analysis and Model Simplification, 522
pp. New York: Academic Press. [This introduced computer-assisted analysis as a formal foundation for
what evolved into an intelligent aid.]
Greenberg H.J. and Murphy F.H. (1991). Approaches to diagnosing infeasible linear programs. ORSA
Journal on Computing 3(3), 253-261. Note: the journal has been renamed INFORMS Journal on
Computing. [This describes the main approaches to the entitled problem, pointing out that Diagnosis =
Isolation + Explanation.]
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
Greenberg H.J. and Murphy F.H. (1992). A comparison of mathematical programming modeling systems.
Annals of Operations Research 38,177-238. [This uses specific examples to compare and contrast three
types of systems: process, algebraic and schematic.]
Greenberg H.J. and Murphy F.H. (1995). Views of mathematical programming models and their
instances. Decision Support Systems 13(1), 3-34. [This shows many ways to view optimization models,
with focus on LP. A central data structure is proposed to enable the computer to produce any view that
best meets the cognitive skill of the user.]
Heath M.T., Ranade A., and Schreiber R.S. ed. (1999). Algorithms for Parallel Processing. New York:
Springer. [This is a modern account, with a variety of subjects by different authors.]
Hochbaum D.S., ed. (1997). Approximation Algorithms for NP-Hard Problems. Boston, MA: PWS
Publishing Co. [This is the fundamental reference to learn about approximation algorithms and their
application to optimization problems.]
Hoffman K.L. Combinatorial optimization: Current successes and directions for the future. Journal of
Computational and Applied Mathematics, (In Press). [This gives perceptive insights about the importance
of modeling combinatorial optimization problems. It was also a keynote talk at the 1999 SIOPT meeting.]
Hooker J. (2000). Logic-Based Methods for Optimization: Combining Optimization and Constraint
Satisfaction. New York: John Wiley & Sons. [This combines the strengths of the entitled topics into a
coherent theory of problem expression and algorithm design.]
Jones C.V. (1995). Visualization and Optimization. Boston, MA: Kluwer Academic Publishers,. An
online version is at <http://www.chesapeake2.com/cvj/itorms/>. [This is a complete compendium of the
many visualizations that have been developed, including many by the author.]
McAloon K. and Tretko C. ( 1995). Logic and Optimization. New York: John Wiley & Son. [This is an
important introduction, taking problem descriptions from the view of artificial intelligence, and solution
methodology from integer programming.]
McCarl B. (1996.). GMSCHK User Documentation. <http://agrinet.tamu.edu/mccarl/gamssoft.htm>.
Lubbock, TX: Texas A&M University. [This is one-of-a-kind software that applies directly to GAMS
source code.]
Nemhauser G.L., Savelsbergh M.W.P., and Sigismondi G.C. (1994). MINTO, A Mixed INTeger
Optimizer. Operations Research Letters 15, 47-58. Downloads available at <http:
//akula.isye.gatech.edu/~mwps/projects/minto.html>. [This is an object-oriented system where the user
supplies the classes for a particular problem and algorithm.]
Orchard-Hays W. (1978). History of mathematical programming systems. Design and Implementation of
Optimization Software, (ed. H.J. Greenberg) 551 pp., The Netherlands: Alphen aan den Rijn : Sijthoff &
Noordhoff. [This gives the early history of mathematical programming and how it was intertwined with
the history of computer science during the years 1950-75.]
Pardalos P.M., Resende M.G.C., and Ramakrishnan K.G. ed. (1995). Parallel Processing of Discrete
Optimization Problems, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 22.
Providence, RI: American Mathematical Society. [This has an important collection of papers and an
insightful introduction.]
Robbins T. (1970). Antenna Design. Doctoral Thesis, Southern Methodist University, Dallas, TX. [This
was an early example of applying optimization to a very difficult objective, requiring significant
computer time for each evaluation.]
Rumelhart D.E. and McClelland J.L. ed. (1986). Parallel Distributed Processing, Explorations in the
Microstructure of Cognition, Vol. 1: Foundations. Cambridge, MA: MIT Press, [This was a very inspiring
book that helped to create major research efforts in neural network computers.]
Rutherford
T.
(1999).
Some
GAMS
Programming
Utilities.
<http://nash.
colorado.edu/tomruth/inclib/tools.htm>. Boulder, CO: University of Colorado. [This is a useful suite of
tools, including interfaces with building output tables.]
©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J.
Greenberg
Sowa J.F. (1984). Conceptual Information Processing, 374 pp. Amsterdam; New York: American
Elsevier. [This presented new concepts connecting language and intelligence. It led to new
implementations of natural language processing.]
Spedicato E. ed. (1994). Algorithms for Continuous Optimization: The State of the Art. NATO ASI
Series, 565 pp. Dordrecht, The Netherlands: Kluwer Academic Publishers. [This has an excellent
collection of papers, focused on numerical methods to solve nonlinear programs.]
Stasko J.T., Dominque J.B., Brown M.H., and Price B.A., ed. (1998). Software Visualization;
Programming as a Multimedia Experience, 562 pp. Cambridge, MA: MIT Press. [This is a general book
on visualization that includes algorithm animation.]
Thienel S. (1995). ABACUS A Branch-And-CUt System. Doctoral Thesis, Universität zu Köln, Germany.
More information is available at <http://www.informatik.uni-koeln.de/ls_juenger/ projects/abacus/>.
[This is an object-oriented system where the user supplies the classes for a particular problem and
algorithm.]
Vanderplaats G. (1995). DOT Users Manual. Colorado Springs, CO: Vanderplaats Research and
Development, Inc. Downloads available at <http://www.vrand.com>. [This describes a Design
Optimization System, which addresses problems that are hard because the objective function requires a
significant amount of time for each evaluation. Generally, f (x) is the numerical solution to a PDE, where
x is a vector of design parameters.]
Waren A.D., Hung M.S., and Lasdon L.S. (1987). The status of nonlinear programming software: An
update. Operations Research 35(4), 489-503. [One of the significant advances addressed in this survey is
the emergence of PCs, making computer storage space no longer an issue, as it had been.]
Wirth N. (1976). Algorithms + Data Structures = Programs, 366 pp. Englewood Cliffs, N.J.: PrenticeHall. [This is a classic book about Algorithms + Data Structures = Programs.]
Biographical Sketch
Harvey Greenberg is Professor of Mathematics at the University of Colorado at Denver. After receiving
his Ph.D. in Operations Research from The Johns Hopkins University in 1968, he joined the Faculty of
Computer Science and Operations Research at Southern Methodist University. Dr. Greenberg has
pioneered the interfaces between operations research and computer science since that time. He was a
founder of what is now the INFORMS Computing Society, having served as its Chairman twice, and he
was the founding Editor of the INFORMS Journal of Computing. While at the U.S. Department of
Energy, 1976-83, Dr. Greenberg developed Computer Assisted Analysis (CAA), dealing with advanced
analysis techniques, including debugging scenarios that might be infeasible or anomalous. Shortly after
joining CU-Denver in 1983, he formed a Consortium to develop an Intelligent Mathematical
Programming System, which extended CAA in breadth and depth. During that time, the project generated
several software systems and about 50 papers. One of the systems, ANALYZE, was recognized with the
first “ORSA/CSTS Award for Excellence in the Interfaces Between Operations Research and Computer
Science.” The system is still used today, and it was a major part of Dr. Greenberg’s plenary talk at the
Canadian Operational Research Society, when he accepted their Harold Lardner Prize for having
“achieved international distinction in Operational Research.” Since 1996, Dr. Greenberg has been an
active developer of web materials to support teaching. In addition to his own site, which contains the
Mathematical Programming Glossary, he developed a site as a service to the Mathematics Department,
called “Using the Web to Teach Mathematics.” Most recently, Dr. Greenberg established a collaborative
relationship with Sandia National Laboratories, helping to develop parallel algorithms for NP-hard
problems, notably in computational biology, such as protein folding. Dr. Greenberg has published 104
papers and four books; he has (co)edited seven additional books or proceedings. He serves on six editorial
boards, and has been Guest Editor for the Annals of Mathematics and Artificial Research on such topics
as “Reasoning about Mathematical Models” and “Representations of Uncertainty”.
©Encyclopedia of Life Support Systems (EOLSS)