An Empirical Quantile Estimation Approach to Nonlinear Optimization Problems with Chance Constraints

We investigate an empirical quantile estimation approach to solve chance-constrained nonlinear optimization problems. Our approach is based on the reformulation of the chance constraint as an equivalent quantile constraint to provide stronger signals on the gradient. In this approach, the value of the quantile function is estimated empirically from samples drawn from the random parameters, and the gradient of the quantile function is estimated via a finite-difference approximation on top of the quantile-function-value estimation. We establish a convergence theory of this approach within the framework of an augmented Lagrangian method for solving general nonlinear constrained optimization problems. The foundation of the convergence analysis is a concentration property of the empirical quantile process, and the analysis is divided based on whether or not the quantile function is differentiable. In contrast to the sampling-and-smoothing approach used in the literature, the method developed in this paper does not involve any smoothing function, and hence the quantile-function gradient approximation is easier to implement, and there are fewer accuracy-control parameters to tune. Numerical investigation shows that our approach can also identify high-quality solutions, especially with a relatively large step size for the finite-difference estimation, which works intuitively as an implicit smoothing. Thus,the possibility exists that an explicit smoothing is not always necessary to handle the chance constraints. Just improving the estimation of the quantile-function value and gradient itself likely could already lead to high performance for solving the chance-constrained nonlinear programs

Service Center Location with Decision Dependent Utilities

We study a service center location problem with ambiguous utility gains upon receiving service. The model is motivated by the problem of deciding medical clinic/service centers, possibly in rural communities, where residents need to visit the clinics to receive health services. A resident gains his utility based on travel distance, waiting time, and service features of the facility that depend on the clinic location. The elicited location-dependent utilities are assumed to be ambiguously described by an expected value and variance constraint. We show that despite a non-convex nonlinearity, given by a constraint specified by a maximum of two second-order conic functions, the model admits a mixed 0-1 second-order cone (MISOCP) formulation. We study the non-convex substructure of the problem, and present methods for developing its strengthened formulations by using valid tangent inequalities. Computational study shows the effectiveness of solving the strengthened formulations. Examples are used to illustrate the importance of including decision dependent ambiguity.Comment: 29 page

Coordinated Vehicle Platooning with Fixed Routes: Adaptive Time Discretization, Strengthened Formulations and Approximation Algorithms

We consider the coordinated vehicle platooning problem over a road network with time constraints and the routes of vehicles are given. The problem is to coordinate the departure time of each vehicle to enable platoon formation hence maximizing the total fuel saving. We first focus on the case that the routes form a tree. In this case, the flexibility time window of each vehicle admits a unified representation, under which the synchronization of departure time is equivalent to the management of time-window overlapping. Based on this observation, a time-window-overlapping induced mixed-integer linear program formulation has been established which is equivalent to the continuous-time formulation established in the previous work. By imposing an adaptive time-discretization procedure, we can further reformulate the time coordination problem using a vehicle-to-time-bucket assignment induced mixed-integer linear program. Compared to the continuous-time formulation, the assignment formulation is free of big-M coefficients, and it is demonstrated by numerical experiments that the assignment formulation is indeed more effective in computational performance. As an independent interest, three approximation algorithms have been derived with provable competitive ratios under a certain regularity assumptions, which is for the first time on this discrete optimization problem. The solution approach has been extended to general cases in which the graph formed by vehicle routes can have loops, by adding a systematic loop breaking scheme. An insightful discovery is that whether there exists an exact equivalent assignment formulation for a general problem instance has a close connection to the pattern of lattice groups.Comment: 74 pages, 12 figure