Decision-making
in most applications involves dealing with uncertainty, from random stock
returns in finance to random demand in inventory management to random arrival
and service times in a service center. This leads to the following two
questions:
- how should this randomness be
modelled?
- how should system performance
be evaluated, and ultimately optimized?
When
assessing possible answers, it is fair to ask what makes a decision-making
methodology more appealing than another. Intuitively, an “ideal” approach
exhibits strengths in both dimensions outlined above, in the sense that (1) it
allows for a modelling of randomness that does not require many additional
assumptions, especially in settings where the validity of such assumptions is
hard to check in practice, and (2) it allows for efficient optimization
procedures for a wide range of problem instances, which in these days of “big
data” most likely includes instances of very large scale. In other words, an
“ideal” approach should build directly upon the information at hand –
specifically, historical observations of the randomness – without forcing this
information into rigid constructs that are difficult to validate even with the
benefit of hindsight, and while preserving computational tractability to remain
of practical relevance for the decision maker.
Interestingly,
the method of choice to model uncertainty has long scored poorly on not just
one but both criteria above. Indeed, probability theory assigns likelihoods to
events that, when defined as a function of several sources of uncertainty, may
occur in many different ways (think for instance of the sum of ten random
variables taking a specific value, and all the possible combinations of those
ten random variables that achieve the desired value as their sum). To assign a
probability to this complex event, it is then necessary to resort to an
advanced mathematical technique called convolution, made even more complex when
random variables are continuous – and this is just to compute a probability, not
optimize it. Additional hurdles include the fact that the manager observes
realizations of the random variables but never the probability distributions
themselves, and that such distributions are difficult to estimate accurately in
many practical applications.
So why have
probabilities become the standard tool to model uncertainty? A possible answer
is that they were developed to describe random events in the realm of pure
mathematics and have since been used beyond their intended purpose to guide business
managers in their optimization attempts. This might have happened because
academicians and practitioners alike lacked other tools to describe uncertainty
in a satisfactory manner, or because the systems under early consideration
remained simple enough to fit easily within the probabilistic framework. But as
the size of the problems considered by today’s managers increases and today’s
business environments are defined more and more by fast-changing market
conditions, it seems particular urgent to bring forward a viable alternative to
probabilistic decision-making that exhibits the attributes outlined above. This
is precisely what researchers in a field called “robust optimization” have
done. (Disclosure: robust optimization is also one of my key research areas.)
By now
robust optimization has been a hot topic in operations research for almost two
decades, and I won’t go over its distinguished contributions to fields as
varied as portfolio management, logistics and shortest path problems, among
many others, in this post – for that purpose, the interested reader is referred
to one of the several overview papers available on the Internet.
For today’s
post I will focus on a specific aspect of robust optimization that is currently
gaining much traction, specifically, the use of robust optimization as a
“one-stop” decision-making methodology that (a) builds directly upon the data
at hand, (b) remains tractable, intuitive and insightful in a wide range of
settings where probabilistic models become intractable and (c) leads to the
axioms of probability theory as a consequence of the modelling framework.
While
(a) and (b) have been standard arguments in favor of robust optimization for at
least a decade, the connection with the world of probabilities provided in (c)
is a novel development that should appeal to many decision makers. The
resulting framework is at the core of a paper by Chaithanya Bandi and Dimitris
Bertsimas – my former dissertation adviser – published last summer in Mathematical Programming.
They suggest
modelling sources of uncertainty not as random variables with probability
distributions but as parameters belonging to uncertainty sets that are
consistent with the limit laws of probabilities (which bound the deviation of a
sum of independently and identically distributed random variables from their
mean, using their standard deviation and the square root of the number of
random variables). Other asymptotic laws can be incorporated to the uncertainty
set in specific circumstances.
The objective of the problem then becomes to
optimize the worst-case value of some criterion, instead of its expected value
as would have been the case in stochastic optimization.
Bandi and
Bertsimas discuss:
- Using historical data and the
central limit theorem,
- Modeling correlation
information,
- How to use stable laws to
construct uncertainty sets for heavy-tailed distributions that have
infinite variance,
- Incorporating distributional
information using “typical sets”, first introduced in the context of
information theory, which exhibit the properties that (i) the probability
of the typical set is nearly 1 and (ii) all elements of the typical set
are nearly equiprobable, and are defined in the Bandi and Bertsimas paper
in terms of uncertainty sets with examples drawn from common
distributions.
The rest of
the paper shows the results obtained by applying this framework to three
applications.
Performance analysis of queueing networks. (Some of the work described in this
section is joint work between Bandi, Bertsimas and Nataly Youssef - download their paper here.) The authors
introduce the concept of a robust queue where arrival and service processes are
modelled by uncertainty sets instead of assigning probability distributions and
derive formulas on the worst-case waiting time for the n-th customer in the
queue. They connect their results with those obtained in traditional queueing
theory and derive results for systems with heavy-tailed behavior such as
Internet traffic that are not believed to have been previously available.
Further, they analyze single-class queueing networks using their framework and
present computational results where they compare their approach, dubbed the
Robust Queueing Network Analyzer (RQNA), with results obtained using simulation
and the Queueing Network Analyzer (QNA) developed in traditional queueing
theory. Their observation is that RQNA’s results are often significantly closer
to simulated values than QNA’s.
Optimal mechanism design for multi-item
auctions. (Full paper.) In this
problem, an auctioneer is interested in selling multiple items to multiple
buyers with private valuations for the items. The auctioneer’s goal in the
robust optimization approach is to maximize the worst-case revenue, with his
beliefs on buyers’ valuations modelled through uncertainty sets. His decisions
are the allocation of items and the payment rules, which should satisfy the
following properties:
- Individual Rationality: Bidders do not derive negative
utility by participating in the auction, assuming truthful bidding, i.e.,
bidding their true valuation of an item.
- Budget Feasibility: Each buyer is charged within his
budget constraints.
- Incentive Compatibility: the total utility of the i-th
buyer under truthful bidding is greater than the total utility that Buyer i
derives by bidding any other bid vector.
Bandi and
Bertsimas provide a robust optimization mechanism (ROM) that solves the problem
and consists of (1) an algorithm to compute the worst-case revenue before a bid
vector is realized, and (2) an algorithm to compute allocations and payments
afterward, which also uses the worst-case revenue as input.
In addition, they
investigate the special case where the buyers do not have any budget
constraints, for which they compare the resulting algorithms in their model
with a mechanism called Myerson auction. They argue that their method in that
setting exhibits stronger robustness properties when the distribution or the
standard deviation of the valuations is misspecified.
Pricing multi-dimensional options. In this finance problem, Bandi and
Bertsimas (along with their collaborator Si Chen) propose to model the
underlying price dynamics with uncertainty sets and then apply robust
optimization rather than dynamic programming to solve the pricing problem. They
illustrate their approach in the context of European call options and
reformulate the problem as a linear problem.
The approach has the flexibility
to incorporate transaction costs and liquidity effects. It also captures a
phenomenon known in finance as the implied volatility smile, which can be
explained in the context of robust optimization using risk aversion arguments.
Bandi and Bertsimas further give examples of the accuracy of their method
relative to observed market prices. You can download that paper here.
The authors’ central argument is that “modelling
stochastic phenomena with probability theory is a choice” and that “given the
computational difficulties in high dimensions, we feel we should consider
alternative, computationally tractable approaches in high dimensions.” They
provide compelling evidence that robust optimization is well-suited for that
purpose.