Fabian Bastin

Convergence theory for nonconvex stochastic programming with an application to mixed logit

Monte Carlo methods have extensively been used and studied in the area of stochastic programming. Their convergence properties typically consider global minimizers or first-order critical points of the sample average approximation (SAA) problems and minimizers of the true problem, and show that the former converge to the latter for increasing sample size. However, the assumption of global minimization essentially restricts the scope of these results to convex problems. We review and extend these results in two directions: we allow for local SAA minimizers of possibly nonconvex problems and prove, under suitable conditions, almost sure convergence of local second-order solutions of the SAA problem to second-order critical points of the true problem. We also apply this new theory to the estimation of mixed logit models for discrete choice analysis. New useful convergence properties are derived in this context, both for the constrained and unconstrained cases, and associated estimates of the simulation bias and variance are proposed.

An adaptive Monte Carlo algorithm for computing mixed logit estimators

Abstract.Researchers and analysts are increasingly using mixed logit models for estimating responses to forecast demand and to determine the factors that affect individual choices. However the numerical cost associated to their evaluation can be prohibitive, the inherent probability choices being represented by multidimensional integrals. This cost remains high even if Monte Carlo or quasi-Monte Carlo techniques are used to estimate those integrals. This paper describes a new algorithm that uses Monte Carlo approximations in the context of modern trust-region techniques, but also exploits accuracy and bias estimators to considerably increase its computational efficiency. Numerical experiments underline the importance of the choice of an appropriate optimisation technique and indicate that the proposed algorithm allows substantial gains in time while delivering more information to the practitioner.

A retrospective trust-region method for unconstrained optimization

We introduce a new trust-region method for unconstrained optimization where the radius update is computed using the model information at the current iterate rather than at the preceding one. The update is then performed according to how well the current model retrospectively predicts the value of the objective function at last iterate. Global convergence to first- and second-order critical points is proved under classical assumptions and preliminary numerical experiments on CUTEr problems indicate that the new method is very competitive.

Evaluation of Optimization Methods for Estimating Mixed Logit Models

The performances of different simulation-based estimation techniques for mixed logit modeling are evaluated. A quasi-Monte Carlo method (modified Latin hypercube sampling) is compared with a Monte Carlo algorithm with dynamic accuracy. The classic Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimization algorithm line-search approach and trust region methods, which have proved to be extremely powerful in nonlinear programming, are also compared. Numerical tests are performed on two real data sets: stated preference data for parking type collected in the United Kingdom, and revealed preference data for mode choice collected as part of a German travel diary survey. Several criteria are used to evaluate the approximation quality of the log likelihood function and the accuracy of the results and the associated estimation runtime. Results suggest that the trust region approach outperforms the BFGS approach and that Monte Carlo methods remain competitive with quasi-Monte Carlo methods in high-dimensional problems, especially when an adaptive optimization algorithm is used.

Managing Hydroelectric Reservoirs Over an Extended Horizon Using Benders Decomposition With a Memory Loss Assumption

Traditional stochastic programming methods are widely used for solving hydroelectric reservoirs management problems under uncertainty. With these methods, random parameters are described using a scenario tree possessing an unstructured topology. Therefore, traditional methods can potentially handle high-order time-correlation effects, but their computational requirements grow exponentially with the branching level used to represent parameters (e.g., load, inflows, prices). Consequently, random parameters must be discretized very coarsely and, as a result, numerical solutions of mid-term optimization models can be quite sensitive to small perturbations to the tree parameters. In this paper, we propose a new approach for managing high-capacity reservoirs over an extended horizon (1-3 years). We partition the planning horizon in two stages and assume that a memory loss occurs at the end of the first stage. We propose a new Benders decomposition algorithm designed specifically to exploit this simplification. The low memory requirement of our method enables to consider a much higher branching level than would be possible with previous methods. The proposed approach is tested on a 104-week production planning problem with stochastic inflows.

Quantifying the social welfare loss in moral hazard models

The main aim of this paper is to measure the social welfare loss for a continuous moral hazard model when a set of minimal assumptions are fulfilled. By using a new approach, we are able to reproduce the results of Balmaceda, Balseiro, Correa, and Stier-Moses (2010) pertaining to the social welfare loss for discrete and continuous models respectively. Previous studies rely on the validity of the first-order approach at the expense of strong assumptions, in particular the convexity of the distribution function condition while we do not make such a restrictive assumption in our developments. In addition, we obtain new bounds for the social welfare loss that are both tight and easy to compute.

Formulation and solution strategies for nonparametric nonlinear stochastic programmes with an application in finance

We consider a class of stochastic programming models where the uncertainty is classically represented using parametric distribution families, but with unknown parameters that will be estimated together with the optimal value of the problem. However, misspecification of the underlying random variables often leads to unrealistic results when little is known about their true distributions. We propose to overcome this difficulty by introducing a nonparametric approach where we replace the estimation of the distribution parameters by that of cumulative distribution functions (CDFs). A practical algorithm is described which achieves this goal by using a monotonic spline representation of the inverse marginal CDFs and a projection-based trust-region globalization. Applications of the new algorithm to discrete choice theory are finally discussed, both with simulated data and in the context of a practical financial application related to interventions of the Bank of Japan in the foreign exchange market.

Transferring Time-Series Discrete Choice to Link-Based Route Choice in Space: Estimating Vehicle Type Preference using Recursive Logit Model

This paper considers a sequential discrete choice problem in a time domain, formulated and solved as a route choice problem in a space domain. Starting from a dynamic specification of time-series discrete choices, we show how it is transferrable to link-based route choices that can be formulated by a finite path choice multinomial logit model. This study establishes that modeling sequential choices over time and in space are equivalent as long as the utility of the choice sequence is additive over the decision steps, the link-specific attributes are deterministic, and the decision process is Markovian. We employ the recursive logit model proposed in the context of route choice in a network, and apply it to estimate time-series vehicle type choice based on Maryland Vehicle Stated Preference Survey data. The model has been efficiently estimated by a dynamic programming approach; the values of estimated coefficients provide important patterns on vehicle type preferences. Compared with a naive model based on sequential multinomial logit choices which are independent over time and a dynamic discrete choice model which considers agent’s future expectations, the smaller root mean square error of recursive logit model indicates that it has a better performance in estimating sequential choices over time. We also compare the predictive powers and find that the proposed model outperforms the basic approach and the dynamic approach.

Two-stage chance-constrained staffing with agent recourse for multi-skill call centers

We consider a stochastic staffing problem with uncertain arrival rates. The objective is to minimize the total cost of agents under some chance constraints, defined over the randomness of the service level in a given time period. In the first stage, an initial staffing must be determined in advance based on imperfect forecast of the arrival rates. At a later time, when the forecast becomes more accurate, this staffing can be corrected with recourse actions, by adding or removing agents at the price of some penalty costs. We present a method that combines simulation, mixed integer programming, and cut generation to solve this problem.

Chance-constrained scheduling with recourse for multi-skill call centers with arrival-rate and absenteeism uncertainty

We consider a chance-constrained two-stage stochastic scheduling problem for multi-skill call centers with uncertainty on arrival rate and absenteeism. We first determine an initial schedule based on an imperfect forecast on arrival rate and absenteeism. Then, this schedule is corrected applying recourse actions when the forecast becomes more accurate in order to satisfy the service levels and average waiting times constraints with some predefined probabilities. We propose a method that combines simulation with integer programming and cut generation to solve the problem.