The research activity will be devoted to the definition of new methodologies for solving two classes of difficult nonlinear optimization problems deriving from important real-world problems.
The first class consists of problems where the analytic expressions of the objective and the constraints functions are unknown and their values are obtained by direct measurements or by using complex approximations or simulation programs. Therefore for such problems no first order information is available. From the methodological point of view, the aim of the research is the definitions of derivative-free methods for difficult optimization problems such as constrained global multiobjective optimization problems, nonlinear mixed optimization problems, nonsmooth optimization problems, problems where the objective function and the constraints can be approximated with a variable precision. In the practical part of the activity some of the new algorithms will be used for tackling difficult real problems deriving from optimal designs of electrical motors, optimal designs of electrical magnetic apparatus, optimal ship design problems, managements of healthcare services, workforce management, definitions of optimal trading strategies.
The second class considered consists of problems which have the difficulty of having an extremely large number of variables and constraints of the problems and, in some cases, also an extremely large number of terms in the objective function. The methodological part of the research will consider the definition of new truncated Newton methods or Frank-Wolfe type algorithms for large scale unconstrained nonlinear optimization problems or large scale simple constrained nonlinear optimization problems and new Newton-Type algorithms for large scale constrained nonlinear optimization problems. Then some of these new algorithms will be used for solving particular optimization problem arising in the machine learning and the data mining fields.
From the methodological point of view, the research activity will be carried out along the following new approaches:
-Derivative-free methods for constrained global multiobjective optimization problems.
The few derivative-free methods for constrained multiobjective optimization problems converge toward stationary points and, hence, they can get trapped in a local not global pareto solutions. New methods could be obtained by combining the global strategies proposed in [Di Pillo, Liuzzi, Lucidi, Piccialli, Rinaldi, COAP (2016)] e [Liuzzi, Lucidi, Piccialli, COAP (2016)] with the local algorithm proposed in [Liuzzi, Lucidi, Rinaldi, SOPT (2016)]. These new methods could exploit the smart partitions of an hyperinterval containing the feasible set of the problem produced by the global strategies to produce points sufficiently close to global pareto points. Starting from these points, a local minimization algorithm could efficiently determine the corresponding global pareto point. This approach should guarantee good theoretical and practical properties.
- Derivative-free methods for nonlinear mixed optimization problems.
The aim is to improve the methods proposed in [Liuzzi, Lucidi, Rinaldi, COAP (2012)] and [Liuzzi, Lucidi, Rinaldi, JOTA (2015)] both in term of efficiency and in term of quality of the produced points. The approaches proposed could be improved by using more efficient search directions and nonmonotone line search techniques. These tools should guarantee that the algorithm converges quickly towards better solution.
- Derivative-free methods for nonsmooth optimization problems.
The idea is to define a new approach which use search directions computed by suitable local model of the objective function. This distinguishing should guarantee a better efficiency with respect the others derivative free algorithm proposed in literature for the same class of problems.
- Derivative-free methods for variable precision optimization problems.
The research activity try to define a first globally convergent derivative free algorithm for problems where both the objective functions and the nonlinear constraints can be approximated with a variable precision. Furthermore to guarantee a good numerical efficiency the proposed algorithm will based on a linesearch approach.
- Nonlinear optimization methods for large scale unconstrained optimization problems.
The aim is to exploit the results proposed in [Fasano, Roma, COAP (2016)] for defining new preconditioned truncated Newton method which are able to solve efficiently large scale highly ill-conditioned unconstrained problems.
- Nonlinear optimization methods for large scale simply constrained optimization problems.
New Frank-Wolfe type algorithms will be proposed for tackling problems with a huge number of variable. These new algorithm should be a good numerical behavior by using an approach based on an efficient active set strategy similar to the one proposed in [De Santis, Lucidi, Rinaldi, SIOP (2016)].
- Nonlinear optimization methods for large scale nonlinear constrained optimization problems.
For these difficult class of optimization problems the research activity will study new algorithms based on the approach of transforming the original nonlinear constrained problem into an equivalent box constrained minimization problem. Then this simpler problem will be tackled by combining the approach prosed in [De Santis, Di Pillo, Lucidi, COAP (2012)] with the use of efficient Newto-type directions.
The more application-oriented research activity will consider difficult real world problems deriving from the of the following fields:
-optimal designs of electrical motors and electrical magnetic apparatus;
-optimal ship design problems;
-managements of healthcare services;
-optimal trading strategies;
-workforce management;
-machine learning techniques;
- data mining techniques.
In this contest, the common approach could be to study in depth the considered real problem, to model it as a suitable optimization problem.
The use of efficient optimization methods in the above described could produce a significant impact. For example, an increase of electric motors efficiency would lead
- to reduce expenses resulting from the use of electric motors from the industry (according to 2014 data, using more efficient electric
motors could lead to savings of up to 4 million euro/ year):
- to save energy consumption, and, consequently, a decrease of the demand for electricity and less emissions related to power
generation, in agreement with EU directives.
This research activity should provide the scientific and engineering community new methods by producing codes freely available for download from the url
http://www.dis.uniroma1.it/~lucidi