A model for inventory systems with multiple products is studied. Demands of items are time-dependent and follow power patterns. Shortages are allowed and fully backlogged. For this inventory system, our findings provide the efficient inventory policy that helps decision-makers to obtain the initial inventory levels and the reorder points that maximize the profit per unit time. Moreover, when it is assumed that the warehouse used for the storage of products has a limited capacity, the optimal inventory policy is also developed. The model presented here extends some inventory systems studied by other authors. Numerical examples are introduced to illustrate the applicability of the theoretical results presented.

Wildfires burn annually across the United States (US), which threaten those in close proximity to them. Due to drastic alterations of soil properties and to the land surfaces by these fires, risks of flash floods, debris flows, and severe erosion increases for these areas, which can have catastrophic consequences for biota, people and property. Computational tools, such as the WildfireRain algorithm, have been designed and implemented to assess the potential occurrence of debris flows over burn scars. However, in order to efficiently operate these tools, they require independent, non-overlapping buffers around burned areas to be defined, which is not a trivial task. In this paper we consider the problem of efficiently subsetting the conterminous US (CONUS) domain into optimal subdomains around burn scars, aiming to enable domain-wide WildfireRain product outputs to be used for operations by the National Weather Service (NWS). To achieve this, we define the Object Encapsulation Problem, where burn scars are represented by single-cell objects in a gridded domain, and circular buffers must be constructed around them. We propose a Linear Programming (LP) model that solves this problem efficiently. Optimal results produced using this model are presented for both a simplified synthetic data set, as well as for a subset of burn scars produced by severe wildfires in 2012 over the CONUS.

Lasserre’s moment-SOS hierarchy consists in approximating instances of the generalized moment problem (GMP) with moment relaxations and sums-of-squares (SOS) strenghtenings that boil down to convex semidefinite programming problems. Due to the generality of the initial GMP, applications of this technology are countless, and one can cite among them the polynomial optimization problem, the optimal control problem, the volume computation problem, stability sets approximation problems, and solving nonlinear partial differential equations. The solution to the original GMP is then approximated with finite truncatures of its moment sequence. For each application, proving convergence of these truncatures towards the optimal moment sequence gives valuable insight on the problem, including convergence of the relaxed values to the original GMP’s optimal value. This note proposes a general proof of such convergence, regardless the problem one is faced with, under simple standard assumptions. As a byproduct of this proof, one also obtains strong duality properties both in the infinite dimensional GMP and its finite dimensional relaxations.

Classical facility location models can generate solutions that do not maintain consistency in the set of utilized facilities as the number of utilized facilities is varied. We introduce the concept of nested facility locations, in which the solution utilizing p facilities is a subset of the solution utilizing q facilities, for all i ≤ p < q ≤ j, given some lower limit i and upper limit j on r, the number of facilities that will be utilized in the future. This approach is demonstrated with application to the p-median model, with computational testing showing these new models achieve reductions in both average regret and worst-case regret when r ≠ p facilities are actually utilized.

In this paper, the convergence of the fundamental alternating minimization is established for non-smooth non-strongly convex optimization problems in Banach spaces, and novel rates of convergence are provided. As objective function a composition of a smooth, and a block-separable, non-smooth part is considered, covering a large range of applications. For the former, three different relaxations of strong convexity are considered: (i) quasi-strong convexity; (ii) quadratic functional growth; and (iii) plain convexity. With new and improved rates benefiting from both separate steps of the scheme, linear convergence is proved for (i) and (ii), whereas sublinear convergence is showed for (iii).