The research program of the Canada Research Chair in decision making under uncertainty focuses on two major areas corresponding to two different sources of uncertainty that can plague a decision problem: uncertainty regarding operating conditions (parametric uncertainty) and uncertainty regarding the intentions of the decision maker (preference uncertainty).

### Area 1: Parametric uncertainty—Operating conditions

The first type of uncertainty considered is parametric uncertainty, i.e. when the decision maker is confronted with a situation in which he does not know the exact nature of the factors that will influence the performance of his decisions. The research focuses both on methodologies that are based on distributional assumptions about the characterization of uncertainty, namely as with the stochastic programming paradigm, and on characterization that is based on sets of potential realisation, namely with robust optimization. In particular, special attention is given to issues that arise in problems that are said to be “data-driven”, i.e. information about the uncertain factors take the shape of historical data, where hybrid forms of these two modeling paradigms naturally arise.

Here are some of our most recent contributions:

**Distributionally Robust Optimization**Although stochastic programming can effectively describe many decision-making problems, its use can be misleading in applications where there is inherent ambiguity in the choice of distribution for the random parameters. This is for example the case when information comes from historical data. In this work, we propose to use a robust form of the model which accounts for uncertainty in both the distributional form and moments of the random parameters. A new confidence region for the covariance matrix of a random vector was derived which provided probabilistic arguments for using our model in problems that rely heavily on historical data. The framework was applied to a number of fields.

**Distributionally robust stochastic knapsack problem**The quadratic knapsack problem is a simple combinatorial problem that has a wide range of real life applications (e.g. in logistics and finance where budget allocation problems arise) and as a subproblem for well-known combinatorial problems (e.g. combinatorial auctions, set covering problems, etc.). In practice, it is often the case that the reward and weight coefficients are uncertain at the time of making the decision. In contrast to the classical formulations, we assume that only part of the information on random data is known, i.e., the first and second moment of the random variables, their joint support and possibly an independence assumption.

**Robust partitioning for stochastic multi-vehicle routing**In this work, we consider the problem of assigning specific regions of a territory to each vehicle of a fleet prior to knowing the actual location of demand (requiring to be visited). Based on historical data, our approach suggests formulating a set known to contain the true distribution of demand and finds a partition which guarantees the best maximum route length traveled by any vehicle in the fleet.

**The Value of Stochastic Modeling**Although stochastic programming is an effective framework for handling decision problems that involve uncertain variables, it is always a costly task to formulate the stochastic model that accurately embodies our knowledge of these variables. In practice, this might require one to collect a large amount of observations, to consult with experts in the specialized field of practice, or to make simplifying assumptions about the underlying system. When none of these options seem feasible, a common heuristic method has been to simply seek the solution of a version of the problem where each uncertain variable takes on its expected value. In this work, we show that in many situations, this heuristic approach can actually be quite effective. We also provide tools to measure what might be the added value of developing a full stochastic model in order to improve on the performance of this simple strategy.

**Fleet mix optimization under uncertainty**We studied the problem of composing a fleet of aircrafts prior to knowing actual demand for the scheduled flights. Under limited information about the demand distribution, the mean value problem can be solved efficiently using readily available commercial software. In our experimental setup, the added value of designing a stochastic model for the uncertainty was under 7% and might not justify investing the resources in such modeling.

**Robust Optimization of a Class of Biconvex Functions**Although robust optimization has gained a lot of attention in recent years, its ease of resolution is usually lost when the function that needs to be “robustified” is not concave (or linear) with respect to the perturbing parameters. In this work, we proposed a new scheme for constructing conservative approximations for the robust optimization of a class of biconvex functions, specifically those that decompose as the sum of the maximum of functions that are biaffine with respect to the decisions and the perturbation. Such functions arise very naturally in a wide range of applications, including news-vendor, inventory, and classification problems and multiattribute utility theory.

### Area 2: Preference uncertainty—Intention of the decision maker

The second form of uncertainty to be studied is preference uncertainty. This is encountered when a decision has to be made and we have not completely determined the decision maker’s preferences regarding the compromise between different objectives or outcomes. This can occur for a variety of reasons. Decision support software products, for example, only have limited time to interact and assess the decision maker’s preference system. In fact, according to a number of studies, the information obtained about a subject is always limited and can suffer from many biases. Alternatively, the decision may need to satisfy a group of decision makers, such as a board of directors, causing ambiguity regarding the preference system to be applied. The aim of this area of the research program is to create a series of plausible preference systems and develop methods for identifying the decisions with the potential to best satisfy the decision makers based on information acquired about their preference systems.

Here are some of our most recent contributions:

**Decision Making under Uncertainty when Preference Information is Incomplete**We consider the problem of optimal decision making under uncertainty but assume that the decision maker’s utility function is not completely known. Instead, we consider all the utilities that meet some criteria, such as preferring certain lotteries over certain other lotteries and being risk averse, s-shaped, or prudent. This extends the notion of stochastic dominance. We then give tractable formulations for such decision-making problems. Numerical experiments on a portfolio selection problem illustrate the strengths and limitations of the method.

**Accounting for Risk Measure Ambiguity when Optimizing Financial Positions**Since the financial crisis of 2007-2009, there has been a renewed interest towards quantifying more appropriately the risks involved in financial positions. In this work, we show that one can account precisely for (neither more nor less than) what we know of the risk preferences of an investor/policy maker when comparing and optimizing financial positions. We assume that the decision maker can commit to some well-established properties of preferences (the use of a law-invariant convex risk measure for example) and that he can provide a series of assessments comparing pairs of potential risky returns. Given this information, we propose valuing the riskiness of a financial position according to the most pessimistic estimation of the potential level of risk perceived by the decision maker. We then give tractable formulations for such decision-making problems and experiment on a portfolio selection problem.