As the title indicates, the proposed research activity aims to study and define new methodologies and algorithms for solving particular classes of difficult optimization problems. In particular, the considered problems belong to the classes of black box and large scale optimization problems.
Difficult Black Box Optimization Problem.
These complex problems are characterized by the fact that there exists only a ¿black box¿ (a simulation tool or an approximation technique) which is able to provide sufficiently good approximations of the relationships between the control variables and the values of the objective functions /constraints.In particular, the aim of the research activity will be the definition of efficient methods for solving particular black box optimization problems which arise in fields of optimal design in engineering and management of services in healthcare. These problems present one or more of the following challenging features: some of the variables are restricted to take integer values, the values of the objective function and constraints can be computed with different precision (fidelity) and different computational times, the values of the objective function and constraints are affected by random noise following an unknown probability distribution.
Difficult Large Scale Optimization Problems.
The distinguishing characteristic of these problems is the large number of variables and constraints .The recent developments in the field of data mining and machine learning need to deal with large scale problems in which the objective functions have particular structures that make them very difficult to minimize. In particular, these functions are very expensive from the computational point of view and have surfaces with large regions where their gradients(or parts of them) are vanishingly small.The proposed activity will try to identify possible methodological approaches to tackle the previous computational difficulties.
The research group carries out an extensive research activity on nonlinear optimization methods and has a consolidated experience on the use of these methods to tackle new difficult real world problems. These characteristics of the research group are perfectly reflected in the proposed project where recent instances of real world problems need further research activity on particular new methodologies.
Black Box Optimization Problems/Optimal Design Problems-Management of Healthcare Services.
The study and the definition of methods for Black Box Optimization Problems is one of the main research arguments of the group. Furthermore, due to its consolidated collaborations, the group has great experience in applying optimization algorithms for the optimal designs of electrical motors/electrical magnetic apparatus, the optimal ship design problems and the health care service management. This methodological and applied research activities can be a sound basis for the following objectives:
-the definitions of new algorithms for multifidelity optimization problems,. By extending the results proposed in [3] or [7] it should be possible to define algorithms with stronger theoretical properties and better computational performances with respect the few algorithms proposed in literature in this framework. The idea is that the proposed algorithms should efficiently selects the fidelity used for the objective function/constraints evaluations starting from the lowest-fidelity and moving to a higher fidelities as it converges towards the minimum points.
-the use of the new multifidelity optimization algorithms for solving the optimal ship design problems which need high-fidelity computational tools, especially
for innovative configurations and extreme/off-design conditions.
-the definitions of algorithms for mixed integer nonsmooth optimization problems. The frameworks proposed in [6],[7] and [9] could be the starting points for define efficient and globally convergent algorithm for these difficult class of problems. Few attempts to define such algorithms have been proposed in literature. In any case, none of these is able to efficiently deal with both the difficulty that the objective functions and the constraints are nonsmooth with respect to the continuous variables and the difficulty that the number of the discrete variables is not small.
-the definitions of new algorithms for stochastic optimization problems. The linesearch and trust region approaches proposed in [7] and [8] can be adapted in order to force the convergence of the procedure to a stationary point even if the evaluations of the objective functions/constraints are affected by noise.
-the use of the new mixed integer nonsmooth optimization algorithms and stochastic algorithms to provide methodological tools and effective decision support systems in the management of hospital services. (in particular for Emergency Departments).
Large Scale Optimization Problems/Training of Machine Learning Systems.
In this field, the research group can take advantage of the research activity carried out in previous years and concerning the large-scale unconstrained optimization methods and training algorithms for machine learning systems. The aims of the proposed activity are:
-the definitions of new algorithms for large scale unconstrained optimization problems. The idea is to exploit and adapt the techniques proposed in [10][11] in order to tackle the difficult class of problems where some/all components of the objective functions gradients are very small in a large regions and where the whole gradients are not computationally available.
- the use of the new large scale unconstrained optimization algorithms for training complex machine learning system needed by recent data mining problems.
[1] Liuzzi, Lucidi, Rinaldi (2016). A derivative-free approach to constrained multiobjective nonsmooth optimization. SIAM J. on Optim., 26,
[6] Liuzzi, Lucidi, Rinaldi (2015). Derivative-Free Methods For Mixed-Integer Constrained Optimization Problems. JOTA, 164,
[7] Fasano, Liuzzi, Lucidi, Rinaldi (2014). A Linesearch-Based Derivative-Free Approach For Nonsmooth Constrained Optimization. SIAM J. on Optim., 24
[8] Liuzzi. Lucidi, Rinaldi, Vicente (2019). Trust-region methods for the derivative-free optimization of nonsmooth black-box functions, SIAM J. on Optim., 26
[9] Liuzzi, Lucidi, Rinaldi (2020). An algorithmic framework based on primitive directions and nonmonotone line searches for black-box optimization problems with integer variables. Math. Prog. Computation.
[10] Fasano, Lucidi (2009). A nonmonotone truncated Newton-Krylov method exploiting negative curvature directions, for large scale unconstrained optimization. Opt. Letter, 3.
[11] Al-Baali, Caliciotti, Fasano, Roma (2020). A Class of Approximate Inverse Preconditioners Based on Krylov-Subspace Methods, for Large-Scale Nonconvex Optimization, SIAM J. on Optim., 26