A Class of Preconditioners for Large Indefinite Linear Systems, as by-product of Krylov subspace Methods: Part II

In this paper we consider the parameter dependent class of preconditioners M(a,d,D) defined in the companion paper The latter was constructed by using information from a Krylov subspace method, adopted to solve the large symmetric linear system Ax = b. We first estimate the condition number of the preconditioned matrix M(a,d,D). Then our preconditioners, which are independent of the choice of the Krylov subspace method adopted, proved to be effective also when solving sequences of slowly changing linear systems, in unconstrained optimization and linear algebra frameworks. A numerical experience is provided to give evidence of the performance of M(a,d,D).preconditioners; large indefinite linear systems; large scale nonconvex optimization; Krylov subspace methods

A Class of Preconditioners for Large Indefinite Linear Systems, as by-product of Krylov subspace Methods: Part I

We propose a class of preconditioners, which are also tailored for symmetric linear systems from linear algebra and nonconvex optimization. Our preconditioners are specifically suited for large linear systems and may be obtained as by-product of Krylov subspace solvers. Each preconditioner in our class is identified by setting the values of a pair of parameters and a scaling matrix, which are user-dependent, and may be chosen according with the structure of the problem in hand. We provide theoretical properties for our preconditioners. In particular, we show that our preconditioners both shift some eigenvalues of the system matrix to controlled values, and they tend to reduce the modulus of most of the other eigenvalues. In a companion paper we study some structural properties of our class of preconditioners, and report the results on a significant numerical experience.preconditioners; large indefinite linear systems; large scale nonconvex optimization; Krylov subspace methods

A derivative-free approach for a simulation-based optimization problem in healthcare

Hospitals have been challenged in recent years to deliver high quality care with limited resources. Given the pressure to contain costs,developing procedures for optimal resource allocation becomes more and more critical in this context. Indeed, under/overutilization of emergency room and ward resources can either compromise a hospital's ability to provide the best possible care, or result in precious funding going toward underutilized resources. Simulation--based optimization tools then help facilitating the planning and management of hospital services, by maximizing/minimizing some specific indices (e.g. net profit) subject to given clinical and economical constraints. In this work, we develop a simulation--based optimization approach for the resource planning of a specific hospital ward. At each step, we first consider a suitably chosen resource setting and evaluate both efficiency and satisfaction of the restrictions by means of a discrete--event simulation model. Then, taking into account the information obtained by the simulation process, we use a derivative--free optimization algorithm to modify the given setting. We report results for a real--world problem coming from the obstetrics ward of an Italian hospital showing both the effectiveness and the efficiency of the proposed approach

A Simulation--Based Optimization approach for analyzing the ambulance diversion phenomenon in an Emergency-Department network

Ambulance Diversion (AD) is one of the possible strategies for relieving the worldwide phenomenon of Emergency Department (ED) overcrowding. It can be carried out when an ED is overloaded and consists of redirecting incoming by ambulance patients to neighboring EDs. Properly implemented, AD should result in reducing delays of patient treatment, ensuring safety and rescue of life-threatening patients. From an operational point of view, AD corresponds to a resource pooling policy among EDs in a network. In this paper we propose a novel model for studying the effectiveness of AD strategies, based on the Simulation-Based Optimization (SBO) approach. In particular, we developed a discrete event simulation model for reproducing the ED network operation. Then, for each AD policy considered, we formulate and solve an optimal resources allocation problem consisting of a bi-objective SBO problem where the target is the minimization of the non-value added time spent by patients and the overall cost incurred by the ED network. A set of optimal points belonging to the Pareto frontier is obtained for each policy. To show the reliability of the proposed approach, a real case study consisting of six large EDs in the Lazio region of Italy is considered, analyzing the effects of adopting different AD policies.Comment: 22 page

Bridging the gap between Trust–Region Methods (TRMs) and Linesearch Based Methods (LBMs) for Nonlinear Programming: quadratic sub–problems

We consider the solution of a recurrent sub–problem within both constrained and unconstrained Nonlinear Programming: namely the minimization of a quadratic function subject to linear constraints. This problem appears in a number of LBM frameworks, and to some extent it reveals a close analogy with the solution of trust–region sub–problems. In particular, we refer to a structured quadratic problem where five linear inequality constraints are included. We show that our proposal retains an appreciable versatility, despite its particular structure, so that a number of different real instances may be reformulated following the pattern in our proposal. Moreover, we detail how to compute an exact global solution of our quadratic sub–problem, exploiting first order KKT conditions

The admission experience survey italian version (I-AES). a factor analytic study on a sample of 156 acute psychiatric in-patients

Coercive treatments are often regarded as an inevitable and yet highly debated feature of psychiatric care. Perceived coercion is often reported by patients involuntarily committed as well as their voluntary counterparts. The Admission Experience Survey (AES) is a reliable tool for measuring perceived coercion in mental hospital admission. We developed the Italian AES (I-AES) through translation back-translation and administered it to 156 acutely hospitalized patients (48% women, 69% voluntarily committed) in two university hospitals in Rome (Policlinico Umberto I, Sant'Andrea Hospital). A principal component analysis (PCA) with equamax rotation was conducted. The I-AES showed good internal consistency (Cronbach's alpha = 0.90); Guttmann split-half relia- bility coefficient was 0.90. AES total score significantly differed between voluntary and involuntary committed patients (5.08 ± 4.1 vs. 8.1 ± 4.9, p < .05). PCA disclosed a three-factor solution explaining 59.3 of the variance. Some discrepancies were found between the factor structure of the I-AES and the original version. I- AES total score was positively associated with numbers of previous involuntarily hospitalization (r = 0.20, p < .05) and psychiatric symptoms' severity (r = 0.22, p < .02). I-AES and its proposed new factor structure proved to be reliable to assess perceived coercion in mental hospital admission. Consequently, it may represent a helpful instrument for the study and reduction of patients' levels of perceived coercion

On the use of the SYMMBK algorithm for computing negative curvature directions within Newton-Krylov methods

In this paper, we consider the issue of computing negative curvature directions, for nonconvex functions, within Newton-Krylov methods for large scale unconstrained optimization. This issue has been widely investigated in the literature, and different approaches have been proposed. We focus on the well known SYMMBK method proposed for solving large scale symmetric possibly inde finite linear systems [3, 5, 7, 20], and show how to exploit it to yield an effective negative curvature direction. The distinguishing feature of our proposal is that the computation of such negative curvature direction is iteratively carried out, without storing no more than a couple of additional vectors. The results of a preliminary numerical experience are reported showing the reliability of the novel approach we propose

Issues on the use of a modified Bunch and Kaufman decomposition for large scale Newton’s equation

In this work, we deal with Truncated Newton methods for solving large scale (possibly nonconvex) unconstrained optimization problems. In particular, we consider the use of a modified Bunch and Kaufman factorization for solving the Newton equation, at each (outer) iteration of the method. The Bunch and Kaufman factorization of a tridiagonal matrix is an effective and stable matrix decomposition, which is well exploited in the widely adopted SYMMBK [2, 5, 6, 19, 20] routine. It can be used to provide conjugate directions, both in the case of 1 × 1 and 2 × 2 pivoting steps. The main drawback is that the resulting solution of Newton’s equation might not be gradient–related, in the case the objective function is nonconvex. Here we first focus on some theoretical properties, in order to ensure that at each iteration of the Truncated Newton method, the search direction obtained by using an adapted Bunch and Kaufman factorization is gradient–related. This allows to perform a standard Armijo-type linesearch procedure, using a bounded descent direction. Furthermore, the results of an extended numerical experience using large scale CUTEst problems is reported, showing the reliability and the efficiency of the proposed approach, both on convex and nonconvex problems

A two-objective optimization of ship itineraries for a cruise company

This paper deals with the problem of cruise itinerary planning which plays a central role in worldwide cruise ship tourism. In particular, the Day-by-day Cruise Itinerary Optimization (DCIO) problem is considered. Assuming that a cruise has been planned in terms of homeports and journey duration, the DCIO problem consists in determining the daily schedule of each itinerary so that some Key Performance Indicators are optimized. The schedule of an itinerary, i.e. the sequence of visited ports, the arrival and departure time at each port, greatly affect cruise operative costs and attractiveness. We propose a Mixed Integer Linear Programming (MILP) formulation of the problem with the objective of minimizing the itinerary cost due to fuel and port costs, while maximizing an itinerary attractiveness index. This latter is strongly related to the ports visited as well as to the overall schedule of the itinerary. Therefore the problem turns out to be a bi-objective optimization problem. We provide its solution in terms of Pareto optimal solution points. Each single objective MILP problem which arises is solved by using an exact algorithm,i mplemented in a commercial solver. We consider the day-by-day itineraries of a major luxury cruise company in many geographical areas all over the world. Here we report, as illustrative examples, the results obtained on some of these real instances

Preconditioning Strategies for Nonlinear Conjugate Gradient Methods, Based on Quasi-Newton Updates

This paper reports two proposals of possible preconditioners for the Nonlinear Conjugate Gradient (NCG) method, in large scale unconstrained optimization. On one hand, the common idea of our preconditioners is inspired to L-BFGS quasi-Newton updates, on the other hand we aim at explicitly approximating in some sense the inverse of the Hessian matrix. Since we deal with large scale optimization problems, we propose matrix-free approaches where the preconditioners are built using symmetric low-rank updating formulae. Our distinctive new contributions rely on using information on the objective function collected as by-product of the NCG, at previous iterations. Broadly speaking, our first approach exploits the secant equation, in order to impose interpolation conditions on the objective function. In the second proposal we adopt and ad hoc modified-secant approach, in order to possibly guarantee some additional theoretical properties