A randomized concave programming method for choice network revenue management

Models incorporating more realistic models of customer behavior, as customers choosing from an offer set, have recently become popular in assortment optimization and revenue management. The dynamic program for these models is intractable and approximated by a deterministic linear program called the CDLP which has an exponential number of columns. However, when the segment consideration sets overlap, the CDLP is difficult to solve. Column generation has been proposed but finding an entering column has been shown to be NP-hard. In this paper we propose a new approach called SDCP to solving CDLP based on segments and their consideration sets. SDCP is a relaxation of CDLP and hence forms a looser upper bound on the dynamic program but coincides with CDLP for the case of non-overlapping segments. If the number of elements in a consideration set for a segment is not very large (SDCP) can be applied to any discrete-choice model of consumer behavior. We tighten the SDCP bound by (i) simulations, called the randomized concave programming (RCP) method, and (ii) by adding cuts to a recent compact formulation of the problem for a latent multinomial-choice model of demand (SBLP+). This latter approach turns out to be very effective, essentially obtaining CDLP value, and excellent revenue performance in simulations, even for overlapping segments. By formulating the problem as a separation problem, we give insight into why CDLP is easy for the MNL with non-overlapping considerations sets and why generalizations of MNL pose difficulties. We perform numerical simulations to determine the revenue performance of all the methods on reference data sets in the literature.assortment optimization, randomized algorithms, network revenue management.

Data-Driven Estimation in Equilibrium Using Inverse Optimization

Equilibrium modeling is common in a variety of fields such as game theory and transportation science. The inputs for these models, however, are often difficult to estimate, while their outputs, i.e., the equilibria they are meant to describe, are often directly observable. By combining ideas from inverse optimization with the theory of variational inequalities, we develop an efficient, data-driven technique for estimating the parameters of these models from observed equilibria. We use this technique to estimate the utility functions of players in a game from their observed actions and to estimate the congestion function on a road network from traffic count data. A distinguishing feature of our approach is that it supports both parametric and \emph{nonparametric} estimation by leveraging ideas from statistical learning (kernel methods and regularization operators). In computational experiments involving Nash and Wardrop equilibria in a nonparametric setting, we find that a) we effectively estimate the unknown demand or congestion function, respectively, and b) our proposed regularization technique substantially improves the out-of-sample performance of our estimators.Comment: 36 pages, 5 figures Additional theorems for generalization guarantees and statistical analysis adde

An enhanced concave program relaxation for choice network revenue management

The network choice revenue management problem models customers as choosing from an offer set, and the firm decides the best subset to offer at any given moment to maximize expected revenue. The resulting dynamic program for the firm is intractable and approximated by a deterministic linear program called the CDLP which has an exponential number of columns. However, under the choice-set paradigm when the segment consideration sets overlap, the CDLP is difficult to solve. Column generation has been proposed but finding an entering column has been shown to be NP-hard. In this paper, starting with a concave program formulation called SDCP that is based on segment-level consideration sets, we add a class of constraints called product constraints (σPC), that project onto subsets of intersections. In addition we propose a natural direct tightening of the SDCP called ESDCPκ, and compare the performance of both methods on the benchmark data sets in the literature. In our computational testing on the benchmark data sets in the literature, 2PC achieves the CDLP value at a fraction of the CPU time taken by column generation. For a large network our 2PC procedure runs under 70 seconds to come within 0.02% of the CDLP value, while column generation takes around 1 hour; for an even larger network with 68 legs, column generation does not converge even in 10 hours for most of the scenarios while 2PC runs under 9 minutes. Thus we believe our approach is very promising for quickly approximating CDLP when segment consideration sets overlap and the consideration sets themselves are relatively small

Tolling, Capacity Selection and Equilibrium Problems with Equilibrium Constraints

An Equilibrium problem with an equilibrium constraint is a mathematical construct that can be applied to private competition in highway networks. In this paper we consider the problem of finding a Nash Equilibrium regarding competition in toll pricing on a network utilising 2 alternative algorithms. In the first algorithm, we utilise a Gauss Siedel fixed point approach based on the cutting constraint algorithm for toll pricing. In the second algorithm, we extend an existing sequential linear complementarity approach for finding Nash equilibrium subject to Wardrop Equilibrium constraints. Finally we consider how the equilibrium may change between the Nash competitive equilibrium and a collusive equilibrium where the two players co-operate to form the equivalent of a monopoly operation