Tri-Dung Nguyen

Robust ranking and portfolio optimization

The portfolio optimization problem has attracted researchers from many disciplines to resolve the issue of poor out-of-sample performance due to estimation errors in the expected returns. A practical method for portfolio construction is to use assets’ ordering information, expressed in the form of preferences over the stocks, instead of the exact expected returns. Due to the fact that the ranking itself is often described with uncertainty, we introduce a generic robust ranking model and apply it to portfolio optimization. In this problem, there are n objects whose ranking is in a discrete uncertainty set. We want to find a weight vector that maximizes some generic objective function for the worst realization of the ranking. This robust ranking problem is a mixed integer minimax problem and is very difficult to solve in general. To solve this robust ranking problem, we apply the constraint generation method, where constraints are efficiently generated by solving a network flow problem. For empirical tests, we use post-earnings-announcement drifts to obtain ranking uncertainty sets for the stocks in the DJIA index. We demonstrate that our robust portfolios produce smaller risk compared to their non-robust counterparts.

Optimizing prices in descending clock auctions

A descending (multi-item) clock auction (DCA) is a mechanism for buying items from multiple potential sellers. In the DCA, bidder-specific prices are decremented over the course of the auction. In each round, each bidder might accept or decline his offer price. Accepting means the bidder is willing to sell at that price. Rejecting means the bidder will not sell at that price or a lower price. DCAs have been proposed as the method for procuring spectrum from existing holders in the FCCs imminent incentive auctions so spectrum can be repurposed to higher-value uses. However, the DCA design has lacked a way to determine the prices to offer the bidders in each round. This is a recognized, important, and timely problem. We present, to our knowledge, the first techniques for this. We develop a percentile-based approach which provides a means to naturally reduce the offer prices to the bidders through the bidding rounds. We also develop an optimization model for setting prices so as to minimize expected payment while stochastically satisfying the feasibility constraint. (The DCA has a final adjustment round that obtains feasibility after feasibility has been lost in the final round of the main DCA.) We prove attractive properties of this, such as symmetry and monotonicity. We develop computational methods for solving the model. (We also develop optimization models with recourse, but they are not computationally practical.) We present experiments both on the homogeneous items case and the case of FCC incentive auctions, where we use real interference constraint data to get a fully faithful model of feasibility. An unexpected paradox about DCAs is that sometimes when the number of rounds allowed increases, the final payment increases. We provide an explanation for this.

A fast approximation algorithm for solving the complete set packing problem

We study the complete set packing problem (CSPP) where the family of feasible subsets may include all possible combinations of objects. This setting arises in applications such as combinatorial auctions (for selecting optimal bids) and cooperative game theory (for finding optimal coalition structures). Although the set packing problem has been well-studied in the literature, where exact and approximation algorithms can solve very large instances with up to hundreds of objects and thousands of feasible subsets, these methods are not extendable to the CSPP since the number of feasible subsets is exponentially large. Formulating the CSPP as an MILP and solving it directly, using CPLEX for example, is impossible for problems with more than 20 objects. We propose a new mathematical formulation for the CSPP that directly leads to an efficient algorithm for finding feasible set packings (upper bounds). We also propose a new formulation for finding tighter lower bounds compared to LP relaxation and develop an efficient method for solving the corresponding large-scale MILP. We test the algorithm with the winner determination problem in spectrum auctions, the coalition structure generation problem in coalitional skill games, and a number of other simulated problems that appear in the literature.

Finding the nucleoli of large cooperative games

The nucleolus is one of the most important solution concepts in cooperative game theory as a result of its attractive properties - it always exists (if the imputation is non-empty), is unique, and is always in the core (if the core is non-empty). However, computing the nucleolus is very challenging because it involves the lexicographical minimization of an exponentially large number of excess values. We present a method for computing the nucleoli of large games, including some structured games with more than 50 players, using nested linear programs (LP). Although different variations of the nested LP formulation have been documented in the literature, they have not been used for large games because of the large size and number of LPs involved. In addition, subtle issues such as how to deal with multiple optimal solutions and with tight constraint sets need to be resolved in each LP in order to formulate and solve the subsequent ones. Unfortunately, this technical issue has been largely overlooked in the literature. We treat these issues rigorously and provide a new nested LP formulation that is smaller in terms of the number of large LPs and their sizes. We provide numerical tests for several games, including the general flow games, the coalitional skill games and the weighted voting games, with up to 100 players.

Modelling infrastructure interdependencies, resiliency and sustainability

The three key concepts of interdependency, resiliency and sustainability of a complex system have appeared in a number of studies and in various contexts. Nevertheless, little has been done to define and analyse them, especially the latter two, in a unified quantitative framework for engineering infrastructures. In this paper, we propose overarching mathematical modelling frameworks to quantify these three key concepts in the context of complex infrastructure systems with multiple interdependent subsystems (i.e., the system of systems). We show how interdependencies between subsystems can affect the resiliency and sustainability of the entire system. We provide a case study in the context of biofuel development and use different dynamical models to demonstrate these concepts.

Task assignment with controlled and autonomous agents

We analyse assignment problems in which not every agent is controlled by the central planner. The autonomous agents search for vacant tasks guided by their own preference orders over available tasks. The goal of the central planner is to maximise the total value of the assignment, taking into account the behaviour of the uncontrolled agents. Such optimisation problems arise in numerous real-world situations, ranging from organisational economics to “crowdsourcing” and disaster response. We show that the problem faced by the central planner can be transformed into a mixed integer bilevel optimisation problem. Then we demonstrate how this program can be reduced to a disjoint bilinear program, which is much more manageable computationally.

Optimal trading under non-negativity constraints using approximate dynamic programming

Abstract In this paper, we develop an extended dynamic programming (DP) approach to solve the problem of minimising execution cost in block trading of securities. To make the problem more practical, we add non-negativity constraints to the model and propose a novel approach to solve the resulting DP problem to near-optimal results. We also include time lags in the problem state to account for the autoregressive behaviour of most financial securities as a way of increasing problem sensitivity to variability of prices and information. The computation times achieved for the proposed algorithm are fast and allow for the possibility of live implementation. We demonstrate the benefits offered by the new approach through numerical analysis and simulation runs in comparison to the classic model without the non-negativity constraints.

Optimization techniques for multivariate least trimmed absolute deviation estimation

Given a dataset an outlier can be defined as an observation that does not follow the statistical properties of the majority of the data. Computation of the location estimate is of fundamental importance in data analysis, and it is well known in statistics that classical methods, such as taking the sample average, can be greatly affected by the presence of outliers in the data. Using the median instead of the mean can partially resolve this issue but not completely. For the univariate case, a robust version of the median is the Least Trimmed Absolute Deviation (LTAD) robust estimator introduced in Tableman (Stat Probab Lett 19(5):387–398, 1994), which has desirable asymptotic properties such as robustness, consistently, high breakdown and normality. There are different generalizations of the LTAD for multivariate data, depending on the choice of norm. Chatzinakos et al. (J Comb Optim, 2015) we present such a generalization using the Euclidean norm and propose a solution technique for the resulting combinatorial optimization problem, based on a necessary condition, that results in a highly convergent local search algorithm. In this subsequent work, we use the

The Interplay of Ranks of Submatrices

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