Laura Carosi

Duality in Multiobjective Optimization Problems with Set Constraints

We propose four different duality problems for a vector optimization program with a set constraint, equality and inequality constraints. For all dual problems we state weak and strong duality theorems based on different generalized concavity assumptions. The proposed dual problems provide a unified framework generalizing Wolfe and Mond-Weir results.

On Generalized Linearity of Quadratic Fractional Functions

Quadratic fractional functions are proved to be quasilinear if and only if they are pseudo-linear. For these classes of functions, some characterizations are provided by means of the inertia of the quadratic form and the behavior of the gradient of the function itself. The study is then developed showing that generalized linear quadratic fractional functions share a particular structure. Therefore it is possible to suggest a sort of “canonical form” for those functions. A wider class of functions given by the sum of a quadratic fractional function and a linear one is also studied. In this case generalized linearity is characterized by means of simple conditions. Finally, it is deepened on the role played by generalized linear quadratic fractional functions in optimization problems.

A sequential method for a class of pseudoconcave fractional problems

The aim of the paper is to maximize a pseudoconcave function which is the sum of a linear and a linear fractional function subject to linear constraints. Theoretical properties of the problem are first established and then a sequential method based on a simplex-like procedure is suggested.

Cost Savings in Wastewater Treatment Processes: the Role of Environmental and Operational Drivers

In this study, 139 Tuscan Wastewater Treatment Plants (WWTPs) were analyzed with the aim of evaluating their efficiency and highlighting the main efficiency drivers, as well as distinguishing among wastewater features, WWTP technology, other features of WWTPs, output variables, and sludge disposal. From a methodological point of view, the proposed method includes an ordinary least squares analysis of total plant costs regressed on a set of 28 exogenous variables and a two-stage Data Envelopment Analysis model, where efficiency scores are obtained through weight restrictions. Moreover, the results of this study demonstrate that, with the exception of the “other features of WWTPs”, all other clusters of variables exert a negative effect on cost savings; in other words, larger scale of operations and higher usage of the productive capacity (grouped as “other features of WWTPs”) can improve cost efficiency.

Duality in Fractional Programming Problems with Set Constraints

Duality is studied for a minimization problem with finitely many inequality and equality constraints and a set constraint where the constraining convex set is not necessarily open or closed. Under suitable generalized convexity assumptions we derive a weak, strong and strict converse duality theorem. By means of a suitable transformation of variables these results are then applied to a class of fractional programs involving a ratio of a convex and an affine function with a set constraint in addition to inequality and equality constraints. The results extend classical fractional programming duality by allowing for a set constraint involving a convex set that is not necessarily open or closed.

Simplex-like sequential methods for a class of generalized fractional programs

A sequential method for a class of generalized fractional programming problems is proposed. The considered objective function is the ratio of powers of affine functions and the feasible region is a polyhedron, not necessarily bounded. Theoretical properties of the optimization problem are first established and the maximal domains of pseudoconcavity are characterized. When the objective function is pseudoconcave in the feasible region, the proposed algorithm takes advantage of the nice optimization properties of pseudoconcave functions; the particular structure of the objective function allows to provide a simplex-like algorithm even when the objective function is not pseudoconcave. Computational results validate the nice performance of the proposed algorithm.

On the pseudoconvexity and pseudolinearity of some classes of fractional functions

The aim of the article is to study the pseudoconvexity (pseudoconcavity) of the ratio between a quadratic function and the square of an affine function. Applying the Charnes–Cooper transformation of variables the function is transformed in a quadratic one defined on a suitable halfspace. The characterization of the pseudoconvexity of such a quadratic function allows to give necessary and sufficient conditions for the pseudoconvexity and the pseudolinearity of the ratio in terms of the initial data.

ON THE CONNECTIONS BETWEEN SEMIDEFINITE OPTIMIZATION AND VECTOR OPTIMIZATION

Abstract This paper works out connections between semidefinite optimization and vector optimization. It is shown that well-known semidefinite optimization problems are scalarized versions of a general vector optimization problem. This scalarization leads to the minimization of the trace or the maximal eigenvalue.

Water pollution in wastewater treatment plants: An efficiency analysis with undesirable output

Abstract The environmental efficiency of 96 Tuscan (Italian) wastewater treatment plants (WWTPs) is investigated taking into account the quality of the outgoing water in terms of pollutant. In this regard, the presence of the residual nitrogen in the outgoing treated water is considered as undesirable output. The efficiency analysis is performed by applying a novel integrated Analytic Hierarchy Process/Non-radial Directional Distance Function (AHP/NDDF) approach, combining the benefits of the two techniques. Similarly to the standard NDDF approach, the suggested model allows to include simultaneously inputs, desirable and undesirable outputs and not to overestimate the efficiency scores. At the same time, the AHP inclusion gives the possibility to directly take into account the decision maker preferences in the weighting system and to encompass some existing directional distance function models as special cases. The obtained results are then used to identify the efficiency explanatory variables: among them, the facilities’ capacity, the percentage of wastewater discharged by the industrial and agricultural activities and the level of compliance with the pollutant concentration threshold set by the legislator have a significant impact on the WWTP performance. The integrated performance assessment allows the water authorities to combine the WWTP efficiency together with the environmental sustainability issue and it has the potential for further promising environmental inspections.

Some Classes of Pseudoconvex Fractional Functions via the Charnes-Cooper Transformation

Using a very recent approach based on the Charnes-Cooper trasformation we characterize the pseudoconvexity of the sum between a quadratic fractional function and a linear one. Furthemore we prove that the ratio between a quadratic fractional function and the cube of an affine one is pseudoconvex if and only if the product between a quadratic fractional function and an affine one is pseudoconvex and we provide a sort of canonical form for this latter class of functions. Benefiting by the new results we are able to characterize the pseudoconvexity of the ratio between a quadratic fractional function and the cube of an affine one.