229 research outputs found
On global minimizers of quadratic functions with cubic regularization
In this paper, we analyze some theoretical properties of the problem of
minimizing a quadratic function with a cubic regularization term, arising in
many methods for unconstrained and constrained optimization that have been
proposed in the last years. First we show that, given any stationary point that
is not a global solution, it is possible to compute, in closed form, a new
point with a smaller objective function value. Then, we prove that a global
minimizer can be obtained by computing a finite number of stationary points.
Finally, we extend these results to the case where stationary conditions are
approximately satisfied, discussing some possible algorithmic applications.Comment: Optimization Letters (2018
A Fast Active Set Block Coordinate Descent Algorithm for -regularized least squares
The problem of finding sparse solutions to underdetermined systems of linear
equations arises in several applications (e.g. signal and image processing,
compressive sensing, statistical inference). A standard tool for dealing with
sparse recovery is the -regularized least-squares approach that has
been recently attracting the attention of many researchers. In this paper, we
describe an active set estimate (i.e. an estimate of the indices of the zero
variables in the optimal solution) for the considered problem that tries to
quickly identify as many active variables as possible at a given point, while
guaranteeing that some approximate optimality conditions are satisfied. A
relevant feature of the estimate is that it gives a significant reduction of
the objective function when setting to zero all those variables estimated
active. This enables to easily embed it into a given globally converging
algorithmic framework. In particular, we include our estimate into a block
coordinate descent algorithm for -regularized least squares, analyze
the convergence properties of this new active set method, and prove that its
basic version converges with linear rate. Finally, we report some numerical
results showing the effectiveness of the approach.Comment: 28 pages, 5 figure
Analysis of flow cytometric aneuploid DNA histograms: validation of an automatic procedure against ad hoc experimental data
In this paper we present an improved version of a method for the automatic analysis of flow cytometric DNA histograms from samples containing a mixture of two cell populations. The procedure is tested against two sets of ad hoc experimental data, obtained by mixing cultures of cell lines in different known proportions. The potentialities of the method are enlightened and discussed with regard to its capability of recovering the population percentages, the DNA index and the G0/G1, S, G2+M phase fractions of each population. On the basis of the obtained results, the procedure appears to be a promising tool in the flow cytometric data analysis and, in particular, in problems of diagnosis and prognosis of tumor diseases
An Active-Set Algorithmic Framework for Non-Convex Optimization Problems over the Simplex
In this paper, we describe a new active-set algorithmic framework for
minimizing a non-convex function over the unit simplex. At each iteration, the
method makes use of a rule for identifying active variables (i.e., variables
that are zero at a stationary point) and specific directions (that we name
active-set gradient related directions) satisfying a new "nonorthogonality"
type of condition. We prove global convergence to stationary points when using
an Armijo line search in the given framework. We further describe three
different examples of active-set gradient related directions that guarantee
linear convergence rate (under suitable assumptions). Finally, we report
numerical experiments showing the effectiveness of the approach.Comment: 29 pages, 3 figure
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
Solving non-monotone equilibrium problems via a DIRECT-type approach
A global optimization approach for solving non-monotone equilibrium problems
(EPs) is proposed. The class of (regularized) gap functions is used to
reformulate any EP as a constrained global optimization program and some bounds
on the Lipschitz constant of such functions are provided. The proposed global
optimization approach is a combination of an improved version of the
\texttt{DIRECT} algorithm, which exploits local bounds of the Lipschitz
constant of the objective function, with local minimizations. Unlike most
existing solution methods for EPs, no monotonicity-type condition is assumed in
this paper. Preliminary numerical results on several classes of EPs show the
effectiveness of the approach.Comment: Technical Report of Department of Computer Science, University of
Pisa, Ital
A derivative-free approach to constrained multiobjective nonsmooth optimization
open3noopenLiuzzi, G.; Lucidi, S.; Rinaldi, F.Liuzzi, G.; Lucidi, S.; Rinaldi, Francesc
An interior point method for nonlinear constrained derivative-free optimization
In this paper we consider constrained optimization problems where both the
objective and constraint functions are of the black-box type. Furthermore, we
assume that the nonlinear inequality constraints are non-relaxable, i.e. their
values and that of the objective function cannot be computed outside of the
feasible region. This situation happens frequently in practice especially in
the black-box setting where function values are typically computed by means of
complex simulation programs which may fail to execute if the considered point
is outside of the feasible region. For such problems, we propose a new
derivative-free optimization method which is based on the use of a merit
function that handles inequality constraints by means of a log-barrier approach
and equality constraints by means of a quadratic penalty approach. We prove
convergence of the proposed method to KKT stationary points of the problem
under quite mild assumptions. Furthermore, we also carry out a preliminary
numerical experience on standard test problems and comparison with a
state-of-the-art solver which shows efficiency of the proposed method.Comment: We dropped the convexity assumption to take into account that
convexity is no longer required, we changed the theoretical analysis,
exposition of the main algorithm has changed. We first present a simpler
method and then the main algorithm. Numerical results have been a lot
extended by adding some compariso
A clustering heuristic to improve a derivative-free algorithm for nonsmooth optimization
In this paper we propose an heuristic to improve the performances of the recently
proposed derivative-free method for nonsmooth optimization CS-DFN. The heuristic
is based on a clustering-type technique to compute an estimate of Clarke’s generalized
gradient of the objective function, obtained via calculation of the (approximate)
directional derivative along a certain set of directions. A search direction
is then calculated by applying a nonsmooth Newton-type approach. As such, this
direction (as it is shown by the numerical experiments) is a good descent direction
for the objective function. We report some numerical results and comparison with
the original CS-DFN method to show the utility of the proposed improvement on a
set of well-known test problems
Derivative-free methods for mixed-integer nonsmooth constrained optimization
In this paper, we consider mixed-integer nonsmooth constrained optimization
problems whose objective/constraint functions are available only as the output
of a black-box zeroth-order oracle (i.e., an oracle that does not provide
derivative information) and we propose a new derivative-free linesearch-based
algorithmic framework to suitably handle those problems. We first describe a
scheme for bound constrained problems that combines a dense sequence of
directions (to handle the nonsmoothness of the objective function) with
primitive directions (to handle discrete variables). Then, we embed an exact
penalty approach in the scheme to suitably manage nonlinear (possibly
nonsmooth) constraints. We analyze the global convergence properties of the
proposed algorithms toward stationary points and we report the results of an
extensive numerical experience on a set of mixed-integer test problems
- …