Mathematics

Let G=(V,E) be a connected graph, where V and E represent, respectively, the node-set and the edge-set. Besides, let Q \subseteq V be a set of terminal nodes, and r \in Q be the root node of the graph. Given a weight c_{ij} \in \mathbb{N} associated to each edge (i,j) \in E, the Steiner Tree Problem in graphs (STP) consists in finding a minimum-weight subgraph of G that spans all nodes in Q. In this paper, we consider the Min-max Regret Steiner Tree Problem with Interval Costs (MMR-STP), a robust variant of STP. In this variant, the weight of the edges are not known in advance, but are assumed to vary in the interval [l_{ij}, u_{ij}]. We develop an ILP formulation, an exact algorithm, and three heuristics for this problem. Computational experiments, performed on generalizations of the classical STP instances, evaluate the efficiency and the limits of the proposed methods.

In this paper, we study an infeasible interior-point method for linear optimization with full-Newton step. The introduced method uses an algebraic equivalent transformation on the centering equation of the system which defines the central path. We prove that the method finds an ε -optimal solution of the underlying problem in polynomial time.

A basic idea in optimal transport is that optimizers can be characterized through a geometric property of their support sets called cyclical monotonicity. In recent years, similar "monotonicity principles" have found applications in other fields where infinite dimensional linear optimization problems play an important role. In this note, we observe how this approach can be transferred to non-linear optimization problems. Specifically we establish a monotonicity principle that is applicable to the Schrödinger problem and use it to characterize the structure of optimizers for target functionals beyond relative entropy. In contrast to classical convex duality approaches, a main novelty is that the monotonicity principle allows to deal also with non-convex functionals.

This manuscript discusses a scalable controller synthesis method for networked systems with a large number of identical subsystems based on the H-infinity control framework. The dynamics of the individual subsystems are described by identical linear time-invariant delay differential equations and the effect of transport and communication delay is explicitly taken into account. The presented method is based on the result that, under a particular assumption on the graph describing the interconnections between the subsystems, the H-infinity norm of the overall system is upper bounded by the robust H-infinity norm of a single subsystem with an additional uncertainty. This work will therefore briefly discuss a recently developed method to compute this last quantity. The resulting controller is then obtained by directly minimizing this upper bound in the controller parameters.

We consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form P(Ω) T q (Ω)|Ω | ??q??/2 and the class of admissible domains consists of two-dimensional open sets Ω satisfying the topological constraints of having a prescribed number k of bounded connected components of the complementary set. A relaxed procedure is needed to have a well-posed problem and we show that when q<1/2 an optimal relaxed domain exists. When q>1/2 the problem is ill-posed and for q=1/2 the explicit value of the infimum is provided in the cases k=0 and k=1 .

Accurate modeling of the patient flow within an Emergency Department (ED) is required by all studies dealing with the increasing and well-known problem of overcrowding. Since Discrete Event Simulation (DES) models are often adopted with the aim of assessing solutions for reducing the impact of this worldwide phenomenon, an accurate estimation of the service time of the ED processes is necessary to guarantee the reliability of the results. However, simulation models concerning EDs are frequently affected by data quality problems, thus requiring a proper estimation of the missing parameters. In this paper, a simulation-based optimization approach is used to estimate the incomplete data in the patient flow within an ED by adopting a model calibration procedure. The objective function of the resulting minimization problem represents the deviation between simulation output and real data, while the constraints ensure that the response of the simulation is sufficiently accurate according to the precision required. Data from a real case study related to a big ED in Italy is used to test the effectiveness of the proposed approach. The experimental results show that the model calibration allows recovering the missing parameters, thus leading to an accurate DES model.

Current integer programming solvers fail to decide whether 12 unit cubes can be packed into a 1x1x11 box within an hour using the natural relaxation of Chen/Padberg. We present an alternative relaxation of the problem of packing boxes into a larger box, which makes it possible to solve much larger instances.

We propose ADOM - an accelerated method for smooth and strongly convex decentralized optimization over time-varying networks. ADOM uses a dual oracle, i.e., we assume access to the gradient of the Fenchel conjugate of the individual loss functions. Up to a constant factor, which depends on the network structure only, its communication complexity is the same as that of accelerated Nesterov gradient method (Nesterov, 2003). To the best of our knowledge, only the algorithm of Rogozin et al. (2019) has a convergence rate with similar properties. However, their algorithm converges under the very restrictive assumption that the number of network changes can not be greater than a tiny percentage of the number of iterations. This assumption is hard to satisfy in practice, as the network topology changes usually can not be controlled. In contrast, ADOM merely requires the network to stay connected throughout time.

In this work, we study the computational complexity of reducing the squared gradient magnitude for smooth minimax optimization problems. First, we present algorithms with accelerated O(1/ k 2 ) last-iterate rates, faster than the existing O(1/k) or slower rates for extragradient, Popov, and gradient descent with anchoring. The acceleration mechanism combines extragradient steps with anchoring and is distinct from Nesterov's acceleration. We then establish optimality of the O(1/ k 2 ) rate through a matching lower bound.

We propose a distributed method to solve a multi-agent optimization problem with strongly convex cost function and equality coupling constraints. The method is based on Nesterov's accelerated gradient approach and works over stochastically time-varying communication networks. We consider the standard assumptions of Nesterov's method and show that the sequence of the expected dual values converge toward the optimal value with the rate of O(1/ k 2 ) . Furthermore, we provide a simulation study of solving an optimal power flow problem with a well-known benchmark case.