Ban Kawas

A log-robust optimization approach to portfolio management

We present a robust optimization approach to portfolio management under uncertainty that builds upon insights gained from the well-known Lognormal model for stock prices, while addressing the model’s limitations, in particular, the issue of fat tails being underestimated in the Gaussian framework and the active debate on the correct distribution to use. Our approach, which we call Log-robust in the spirit of the Lognormal model, does not require any probabilistic assumption, and incorporates the randomness on the continuously compounded rates of return by using range forecasts and a budget of uncertainty, thus capturing the decision-maker’s degree of risk aversion through a single, intuitive parameter. Our objective is to maximize the worst-case portfolio value (over a set of allowable deviations of the uncertain parameters from their nominal values) at the end of the time horizon in a one-period setting; short sales are not allowed. We formulate the robust problem as a linear programming problem and derive theoretical insights into the worst-case uncertainty and the optimal allocation. We then compare in numerical experiments the Log-robust approach with the traditional robust approach, where range forecasts are applied directly to the stock returns. Our results indicate that the Log-robust approach significantly outperforms the benchmark with respect to 95 or 99% Value-at-Risk. This is because the traditional robust approach leads to portfolios that are far less diversified.

Value Interchangeability in Scenario Generation

Several types of symmetry have been identified and exploited in Constraint Programming, leading to large reductions in search time. We present a novel application of one such form of symmetry: detecting dynamic value interchangeability in the random variables of a 2-stage stochastic problem. We use a real-world problem from the literature: finding an optimal investment plan to strengthen a transportation network, given that a future earthquake probabilistically destroys links in the network. Detecting interchangeabilities enables us to bundle together many equivalent scenarios, drastically reducing the size of the problem and allowing the exact solution of cases previously considered intractable and solved only approximately.

Prescriptive Analytics for Allocating Sales Teams to Opportunities

For companies with large sales forces whose sellers approach business clients in teams, the problem of allocating sales teams to sales opportunities is a critical management task for maximizing the revenue and profit of the company. We approach this problem via predictive and prescriptive analytics, where the former involves data mining to learn the relationship between sales team composition and the revenue earned for different types of clients and opportunities, and the latter involves optimization to find the allocation of sales resources to opportunities that maximizes expected revenue subject to business constraints. In looking at the overall sales force problem, we focus on the interplay between the data mining and optimization components, making sure to formulate the two aspects in a jointly tractable and effective manner. We perform a sensitivity analysis of the optimization component to provide further insight into the interaction between prediction and prescription. Finally, we provide an empirical study using real-world data from a large technology companys sales force. Our results demonstrate that by using these analytics, we can increase revenue by 15%.

A robust optimization approach to enhancing reliability in production planning under non-compliance risks

Certain regulated industries are monitored by inspections that ensure adherence (compliance) to regulations. These inspections can often be with very short notice and can focus on particular aspects of the business. Failing such inspections can bring great losses to a company; thus, evaluating the risks of failure against various inspection strategies can help it ensure a robust operation. In this paper, we investigate a game-theoretic setup of a production planning problem under uncertainty in which a company is exposed to the risk of failing authoritative inspections due to non-compliance with enforced regulations. In the proposed decision model, the inspection agency is considered an adversary to the company whose production sites are subject to inspections. The outcome of an inspection is uncertain and is modeled as a Bernoulli-distributed random variable whose parameter is the mean of non-compliance probabilities of products produced at the inspected site and, therefore, is a function of production decisions. If a site fails an inspection, then all its products are deemed adulterated and cannot be used, jeopardizing the reliability of the company in satisfying customers’ demand. In the proposed framework, we address two sources of uncertainty facing the company. First, through the adversarial setting, we address the uncertainty arising from the inspection process as the company does not know a priori which sites the agency will choose to inspect. Second, we address data uncertainty via robust optimization. We model products’ non-compliance probabilities as uncertain parameters belonging to polyhedral uncertainty sets and maximize the worst-case expected profit over these sets. We derive tractable and compact formulations in the form of a mixed integer program that can be solved efficiently via readily available standard software. Furthermore, we give theoretical insights into the structure of optimal solutions and worst-case uncertainties. The proposed approach offers the flexibility of matching solutions to the level of conservatism of the decision maker via two intuitive parameters: the anticipated number of sites to be inspected, and the number of products at each site that are anticipated to be at their worst-case non-compliance level. Varying these parameters when solving for the optimal products allocation provides different risk-return tradeoffs and thus selecting them is an essential part of decision makers’ strategy. We believe that the robust approach holds much potential in enhancing reliability in production planning and other similar frameworks in which the probability of random events depends on decision variables and in which the uncertainty of parameters is prevalent and difficult to handle.

Distribution shaping and scenario bundling for stochastic programs with endogenous uncertainty

Stochastic programs are usually formulated with probability distributions that are exogenously given. Modeling and solving problems with endogenous uncertainty, where decisions can influence the probabilities, has remained a largely unresolved challenge. In this paper we develop a new approach to handle decision-dependent probabilities based on the idea of distribution shaping. It uses a sequence of distributions, successively conditioned on the influencing decision variables, and characterizes these by linear inequalities. We demonstrate the approach on a pre-disaster planning problem of finding optimal investments to strengthen links in a transportation network, given that the links are subject to stochastic failure. Our new approach solves a recently considered instance of the Istanbul highway network to optimality within seconds, for which only approximate solutions had been known so far.

Log-robust portfolio management with parameter ambiguity

We present a robust optimization approach to portfolio management under uncertainty when randomness is modeled using uncertainty sets for the continuously compounded rates of return, which empirical research argues are the true drivers of uncertainty, but the parameters needed to define the uncertainty sets, such as the drift and standard deviation, are not known precisely. Instead, a finite set of scenarios is available for the input data, obtained either using different time horizons or assumptions in the estimation process. Our objective is to maximize the worst-case portfolio value (over a set of allowable deviations of the uncertain parameters from their nominal values, using the worst-case nominal values among the possible scenarios) at the end of the time horizon in a one-period setting. Short sales are not allowed. We consider both the independent and correlated assets models. For the independent assets case, we derive a convex reformulation, albeit involving functions with singular Hessians. Because this slows computation times, we also provide lower and upper linear approximation problems and devise an algorithm that gives the decision maker a solution within a desired tolerance from optimality. For the correlated assets case, we suggest a tractable heuristic that uses insights derived in the independent assets case.

Symmetry breaking for exact solutions in adjustable robust optimisation

One of the key unresolved challenges in Adjustable Robust Optimisation is how to deal with large discrete uncertainty sets. In this paper we present a technique for handling such sets based on symmetry breaking ideas from Constraint Programming. In earlier work we applied the technique to a pre-disaster planning problem modelled as a two-stage Stochastic Program, and we were able to solve exactly instances that were previously considered intractable and only had approximate solutions. In this paper we show that the technique can also be applied to an adjustable robust formulation that scales up to larger instances than the stochastic formulation. We also describe a new fast symmetry breaking heuristic that gives improved results.

Risk-averse production planning

We consider a production planning problem under uncertainty in which companies have to make product allocation decisions such that the risk of failing regulatory inspections of sites - and consequently losing revenue - is minimized. In the proposed decision model the regulatory authority is an adversary. The outcome of an inspection is a Bernoulli-distributed random variable whose parameter is a function of production decisions. Our goal is to optimize the conditional value-atrisk (CVaR) of the uncertain revenue. The dependence of the probability of inspection outcome scenarios on production decisions makes the CVaR optimization problem non-convex.We give a mixed-integer nonlinear formulation and devise a branch-and-bound (BnB) algorithm to solve it exactly. We then compare against a Stochastic Constraint Programming (SCP) approach which applies randomized local search. While the BnB guarantees optimality, it can only solve smaller instances in a reasonable time and the SCP approach outperforms it for larger instances.