Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming

This paper deals with an analysis of the Conjugate Gradient (CG) method (Hestenes and Stiefel in J Res Nat Bur Stand 49:409-436, 1952), in the presence of degenerates on indefinite linear systems. Several approaches have been proposed in the literature to issue the latter drawback in optimization frameworks, including reformulating the original linear system or recurring to approximately solving it. All the proposed alternatives seem to rely on algebraic considerations, and basically pursue the idea of improving numerical efficiency. In this regard, here we sketch two separate analyses for the possible CG degeneracy. First, we start detailing a more standard algebraic viewpoint of the problem, suggested by planar methods. Then, another algebraic perspective is detailed, relying on a novel recently proposed theory, which includes an additional number, namely grossone. The use of grossone allows to work numerically with infinities and infinitesimals. The results obtained using the two proposed approaches perfectly match, showing that grossone may represent a fruitful and promising tool to be exploited within Nonlinear Programming

Emergency logistics for wildfire suppression based on forecasted disaster evolution

This paper aims to develop a two-layer emergency logistics system with a single depot and multiple demand sites for wildfire suppression and disaster relief. For the first layer, a fire propagation model is first built using both the flame-igniting attributes of wildfires and the factors affecting wildfire propagation and patterns. Second, based on the forecasted propagation behavior, the emergency levels of fire sites in terms of demand on suppression resources are evaluated and prioritized. For the second layer, considering the prioritized fire sites, the corresponding resource allocation problem and vehicle routing problem (VRP) are investigated and addressed. The former is approached using a model that can minimize the total forest loss (from multiple sites) and suppression costs incurred accordingly. This model is constructed and solved using principles of calculus. To address the latter, a multi-objective VRP model is developed to minimize both the travel time and cost of the resource delivery vehicles. A heuristic algorithm is designed to provide the associated solutions of the VRP model. As a result, this paper provides useful insights into effective wildfire suppression by rationalizing resources regarding different fire propagation rates. The supporting models can also be generalized and tailored to tackle logistics resource optimization issues in dynamic operational environments, particularly those sharing the same feature of single supply and multiple demands in logistics planning and operations (e.g., allocation of ambulances and police forces). © 2017 The Author(s

Iterative Grossone-Based Computation of Negative Curvature Directions in Large-Scale Optimization

We consider an iterative computation of negative curvature directions, in large-scale unconstrained optimization frameworks, needed for ensuring the convergence toward stationary pointswhich satisfy second-order necessary optimality conditions. We show that to the latter purpose, we can fruitfully couple the conjugate gradient (CG) method with a recently introduced approach involving the use of the numeral called Grossone. In particular, recalling that in principle the CG method is well posed onlywhen solving positive definite linear systems, our proposal exploits the use of grossone to enhance theperformance of the CG, allowing the computation of negative curvature directions in the indefinite case, too. Our overall method could be used to significantly generalize the theory in state-of-the-art literature. Moreover, it straightforwardly allows the solution of Newton’s equation in optimization frameworks, even in nonconvex problems. We remark that our iterative procedure to compute a negative curvature direction does not require the storage of any matrix, simply needing to store a couple of vectors. This definitely represents an advance with respect to current results in the literature