Trust-region methods are a broad class of methods for continuous optimization that found application in a variety of problems and contexts. In particular, they have been studied and applied for problems without using derivatives.The analysis of trust-region derivative-free methods has focused on global convergence, and they have been proved to generate a sequence of iterates converging to stationarity independently of the starting point. Most of such an analysis is carried out in the smooth case, and, moreover, little is known about the complexity or global rate of these methods.In this paper, we start by analyzing the worst case complexity of general trust-region derivative-free methods for smooth functions. For the non-smooth case, we propose a smoothing approach, for which we prove global convergence and bound the worst case complexity effort. For the special case of non-smooth functions that result of the composition of smooth and non-smooth/convex components, we show how to improve the existing results of the literature and make them applicable to the general methodology.

In the context of the derivative-free optimization of a smooth objective function, it has been shown that the worst case complexity of direct-search methods is of the same order as the one of steepest descent for derivative-based optimization, more precisely that the number of iterations needed to reduce the norm of the gradient of the objective function below a certain threshold is proportional to the inverse of the threshold squared.Motivated by the lack of such a result in the non-smooth case, we propose, analyze, and test a class of smoothing direct-search methods for the unconstrained optimization of nonsmooth functions. Given a parameterized family of smoothing functions for the non-smooth objective function dependent on a smoothing parameter, this class of methods consists of applying a direct-search algorithm for a fixed value of the smoothing parameter until the step size is relatively small, after which the smoothing parameter is reduced and the process is repeated.One can show that the worst case complexity (or cost) of this procedure is roughly one order of magnitude worse than the one for direct search or steepest descent on smooth functions.The class of smoothing direct-search methods is also showed to enjoy asymptotic global convergence properties. Some preliminary numerical experiments indicates that this approach leads to better values of the objective function, pushing in some cases the optimization further, apparently without an additional cost in the number of function evaluations.

Direct Multisearch is a well-established class of algorithms, suited for multiobjective derivative-free optimization. In this work, we analyze the worst-case complexity of this class of methods in its most general formulation for unconstrained optimization. Considering nonconvex smooth functions, we show that to drive a given criticality measure below a specific positive threshold, Direct Multisearch takes at most a number of iterations proportional to the square of the inverse of the threshold, raised to the number of components of the objective function. This number is also proportional to the size of the set of linked sequences between the first unsuccessful iteration and the iteration immediately before the one where the criticality condition is satisfied. We then focus on a particular instance of Direct Multisearch, which considers a more strict criterion for accepting new nondominated points. In this case, we can establish a better worst-case complexity bound, simply proportional to the square of the inverse of the threshold, for driving the same criticality measure below the considered threshold.

Since the 1990’s, studies and pilot tests have been conducted to reduce traffic, accidents, and pollution due to urban freight transport (UFT). These ended up in several policies, regulations, and restrictions for UFT, such as low emission zones, delivery time windows, and vehicle size and weight restrictions. However, issues in UFT under regulatory measures still persevere. This study introduces an optimization framework for deriving an optimal combination of various types of vehicles with different capacities for vehicle replacement in UFT. This framework allows an understanding of how an urban freight company with a limited budget efficiently satisfies its freight demand within an urban area in the presence of regulatory measures by urban administrators. The introduced formulation, which is a mixed-integer linear programming, will assist the operator in choosing the best investment strategy for introducing new vehicles of certain types and sizes, for operation in different zones, into its fleet while gaining economic benefits and having a positive impact on the liveability of the urban area. Furthermore, a sensitivity analysis is performed to consider the effects of specific uncertain parameters on the total cost. The numerical results show that the share of electric vehicles in the fleet increases, and they are more competitive than diesel vehicles.

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