Most real-world optimization problems in the areas of applied sciences, engineering and economics involve multiple, often conflicting, goals. In the mathematical model of these problems, under the necessity of reflecting discrete quantities, logical relationships or decisions, integer and 0-1 variables need to be considered. We are then in the context of MultiObjective Integer Programming (MOIP).
MOIP problems combine all the difficulties of both MultiObjective Problems (MOPs) and Mixed Integer Nonlinear Programming problems (MINLPs), which are among the class of theoretically difficult problems (NP-complete). These problems are intrinsically nonconvex and thus require global optimization techniques. Therefore, the design of efficient solution methods is a big challenge for people working in optimization and operations research.
The research activity will be devoted to the study and the definition of new optimization methodologies to deal with specific classes of MOIP problems.
The challenging nature of multiobjective integer programming problems increasingly calls for
* the definition of new ways to detect nondominated points,
* the development of efficient and easy-to-use solution methods.
From the methodological point of view, the research activity will be carried out along the following new approaches:
*Bi-objective integer nonlinear programming problems.
A key feature in the definition of criterion space search methods for this class of problems is to solve appropriate single-objective optimization problems in order to detect efficient solutions.
We plan to study specific nonlinear techniques (such as scalarization or GOAL programming) leading to single-objective/scalarized problems that should be efficiently solved to global optimality. The smart combination of the solution of these single-objective subproblems with a suitably defined enumeration strategy should lead to the definition of effective criterion space search algorithms.
*Cutting plane based criterion space search algorithms.
Starting from a specific class of multiobjective integer programming problems, we plan to study and define new valid inequalities to exclude dominated parts of the criterion space. By implementing these cuts in a criterion space search algorithm and comparing the performance of the resulting method with state-of-the art solvers (such as the balanced box method [2] for biobjective integer linear programming), we will have an insight on the efficiency of the cuts proposed.
The research activity will benefit from the expertise of the group in the context of mixed integer nonlinear programming: all members of the group have been involved in the development of exact algorithms for solving mixed integer problems. We plan to consider the succesful ideas used in the design of methods for mixed integer programming problems (see [7, 8, 5, 10, 6]), and extend them in the context of multiobjective integer programming.
For the references: See "Descrizione delle attività e dei compiti dei partecipanti"