Search

In this paper, we consider stochastic optimal control of systems driven by stochastic differential equations with irregular drift coefficient. We establish a necessary and sufficient stochastic maximum principle. To achieve this, we first derive an explicit representation of the first variation process (in the Sobolev sense) of the controlled diffusion. Since the drift coefficient is not smooth, the representation is given in terms of the local time of the state process. Then we construct a sequence of optimal control problems with smooth coefficients by an approximation argument. Finally, we use Ekeland’s variational principle to obtain an approximating adjoint process from which we derive the maximum principle by passing to the limit. The work is notably motivated by the optimal consumption problem of investors paying wealth tax.

In this article, we consider the motion planning of a rigid object on the unit sphere with a unit speed. The motion of the object is constrained by the maximum absolute value, $U_{max}$ of geodesic curvature of its path; this constrains the object to change the heading at the fastest rate only when traveling on a tight smaller circular arc of radius $r \frac{1}{2}$, while paths of the above type may cease to exist depending on the boundary conditions and the value of $r$, optimal paths may be concatenations of more than three circular arcs.

Semi-regulated, privately-operated public transport (PT) systems are common in low and middle-income countries of the Global South. Several cities are now planning or implementing formalized schemes intended to improve quality of service, safety, security of PT services and to reduce negative externalities, with limited success. Bogotá, which embarked on a progressive citywide PT reform process starting in 2009, replicates the negative experience of other large cities introducing citywide transit reforms. The new formalized scheme was implemented gradually between 2012 and 2021. As a result two types of services coexisted in the city for nine years: semiformal and formalized services. This resembled a natural experiment that provided a unique situation to evaluate the restructuring reform in Bogotá from the user’s perspective. This study analyzes passenger satisfaction surveys for both systems by means of ordered Probit models and Importance-Performance Analysis. Despite the good intentions behind the formalization process, results show that users provide better ratings to the traditional semi-regulated services than the new formalized system, mainly because bus frequency and fares are worse and expected improvements in road safety, personal security and comfort have not been delivered. The reform in Bogotá allowed the city to have resilience during the pandemic, though; and provided a framework for inclusion of zero and low-emission buses. The analysis suggests some recommendations for the improvement of transit reform processes: putting the focus on service quality, as well as assigning sufficient funding and developing institutional capacities to undertake the required changes.

In this work, in the context of Linear and convex Quadratic Programming, we consider Primal Dual Regularized Interior Point Methods (PDR-IPMs) in the framework of the Proximal Point Method. The resulting Proximal Stabilized IPM (PS-IPM) is strongly supported by theoretical results concerning convergence and the rate of convergence, and can handle degenerate problems. Moreover, in the second part of this work, we analyse the interactions between the regularization parameters and the computational footprint of the linear algebra routines used to solve the Newton linear systems. In particular, when these systems are solved using an iterative Krylov method, we are able to show—using a new rearrangement of the Schur complement which exploits regularization—that general purposes preconditioners remain attractive for a series of subsequent IPM iterations. Indeed, if on the one hand a series of theoretical results underpin the fact that the approach here presented allows a better re-use of such computed preconditioners, on the other, we show experimentally that such (re)computations are needed only in a fraction of the total IPM iterations. The resulting regularized second order methods, for which low-frequency-update of the preconditioners are allowed, pave the path for an alternative class of second order methods characterized by reduced computational effort.

We consider a continuous-time Robbins–Monro-type stochastic approximation procedure for a system described by a (multidimensional) stochastic differential equation driven by a general Lévy process, and we find sufficient conditions for its convergence in terms of Lyapunov functions. While the jump part of the noise may spoil convergence to the root of the drift in some cases, we show that by a suitable choice of noise coefficients we obtain convergence under hypotheses on the drift weaker than those used in the diffusion case or convergence to a selected root in the case of multiple roots of the drift.

We develop an exact cutting plane solution algorithm for a special class of bilevel programming models utilized for optimal price-bidding of energy producers in day-ahead electricity markets. The proposed methodology utilizes a suitable reformulation in which a key prerequisite for global optimality, termed bilevel feasibility, is relaxed. Solving the problem to global optimality involves finding the price-offers of the strategic producer (upper-level decision variables) which maximize his self-profit upon clearing of the market and identification of the optimal energy quantity distribution (lower-level decision variables). To exclude from consideration the encountered bilevel infeasible solutions, the algorithm employs a special type of valid cuts drawn from the theory of integer parametric programming. The generation of these cuts involves finding the truly optimal lower-level solution using the strategic price-offers at the bilevel infeasible solution subject to exclusion and devising range intervals for these offers such that the optimality of this solution is retained when each of them lies in its corresponding interval. Each cut imposes a suitable part of this solution, under the condition that each price-offer belongs to its associated interval, which renders the bilevel infeasible solution invalid. We establish the theoretical framework for the development of the proposed algorithm, we illustrate its application on a small case study, and we present extensive computational results demonstrating its behavior and performance on random problem instances. These results indicate that the algorithm is capable of solving to global optimality considerably larger problems than those that a previous elementary version of the same algorithm could solve. This constitutes significant research contribution, considering the lack of generic optimization software for bilevel programming, as well as the fact that the applicability of specialized algorithms on problems of realistic size is rather limited.

We show that the asymptotic regularity and the strong convergence of the modified Halpern iteration due to T.-H. Kim and H.-K. Xu and studied further by A. Cuntavenapit and B. Panyanak and the Tikhonov–Mann iteration introduced by H. Cheval and L. Leuştean as a generalization of an iteration due to Y. Yao et al. that has recently been studied by Boţ et al. can be reduced to each other in general geodesic settings. This, in particular, gives a new proof of the convergence result in Boţ et al. together with a generalization from Hilbert to CAT(0) spaces. Moreover, quantitative rates of asymptotic regularity and metastability due to K. Schade and U. Kohlenbach can be adapted and transformed into rates for the Tikhonov–Mann iteration corresponding to recent quantitative results on the latter of H. Cheval, L. Leuştean and B. Dinis, P. Pinto, respectively. A transformation in the converse direction is also possible. We also obtain rates of asymptotic regularity of order O(1/n) for both the modified Halpern (and so in particular for the Halpern iteration) and the Tikhonov–Mann iteration in a general geodesic setting for a special choice of scalars.

We consider the problem of computing the \emph{distance-based representative skyline} in the plane, a problem introduced by Tao, Ding, Lin and Pei [Proc. 25th IEEE International Conference on Data Engineering (ICDE), 2009] and independently considered by Dupin, Nielsen and Talbi [Optimization and Learning - Third International Conference, OLA 2020] in the context of multi-objective optimization. Given a set $P$ of $n$ points in the plane and a parameter $k$, the task is to select $k$ points of the skyline defined by $P$ (also known as Pareto front for $P$) to minimize the maximum distance from the points of the skyline to the selected points. We show that the problem can be solved in $O(n\log h)$ time, where $h$ is the number of points in the skyline of $P$. We also show that the decision problem can be solved in $O(n\log k)$ time and the optimization problem can be solved in $O(n \log k + n \operatorname{loglog} n)$ time. This improves previous algorithms and is optimal for a large range of values of $k$., 25 pages

We introduce two desirable properties concerning the allocative efficiency term in the decomposition of economic efficiency. Since Farrell, economic efficiency, defined in terms of the cost, revenue, profitability, or profit functions, is decomposed into a technical efficiency measure and allocative efficiency. Resorting to duality theory, allocative efficiency is calculated as a residual. In this framework, we show that this residual is numerically inconsistent for several economic efficiency decompositions: those based on the Russell Efficiency Measures, the Slack-Based Measure and the Weighted Additive Measures. It should be expected that a technical inefficient firm, if projected to the optimal economic benchmark, e.g., that maximizing profit, should be allocative efficient, yet we show that the above decompositions may signal that it is allocative inefficient. Our first property, called ‘essential,’ demands that economic efficiency decompositions satisfy the above criterion. We also extend this property by requiring that the allocative efficiency of a technically inefficient firm, evaluated at the projected benchmark on the production frontier, coincides with the allocative efficiency of the benchmark itself. Regarding this extension of the property, we show that, besides the measures mentioned above, the Directional Distance Function and the Hölder Distance Function fail in general to comply with it. However, we also prove that, thanks to the flexibility of these two approaches in choosing a common directional vector for all the assessed firms or a specific Hölder norm, these measures may satisfy the extended essential property. We conclude that unless both properties are fulfilled, the (residual) allocative component of many previously published decompositions cannot be correctly interpreted as price inefficiency.

Multi-objective optimization problems and their solution algorithms are of great importance as single-objective optimization problems are not usually a true representation of many real-world problems. In general, multi-objective optimization problems result in a large set of Pareto optimal solutions. Each solution in this set is optimal with some trade-offs. Therefore, it is difficult for the decision-maker to select a solution, especially in the absence of subjective or judgmental information. Moreover, an analysis of all the solutions is computationally expensive and, hence, not practical. Thus, researchers have proposed several techniques such as clustering and ranking of Pareto optimal solutions to reduce the number of solutions. The ranking methods are often used to obtain a single solution, which is not a good representation of the entire Pareto set. This paper deviates from the common approach and proposes a novel graph-theoretical clustering method. The quality of the clustering based on the Silhouette score is used to determine the number of clusters. The connectivity in the objective space is used to find representative solutions for clusters. One step forward, we identify ‘extreme solutions’. Hence, the reduced set contains both extreme solutions and representative solutions. We demonstrate the performance of the proposed method by using different 3D and 8D benchmark Pareto fronts as well as Pareto fronts from a case study in Royal Australian Navy. Results revealed that the reduced set obtained from the proposed method outperforms that from the K-means clustering, which is the most popular traditional clustering approach in Pareto pruning.