Antonio Fuduli

Minimizing Nonconvex Nonsmooth Functions via Cutting Planes and Proximity Control

We describe an extension of the classical cutting plane algorithm to tackle the unconstrained minimization of a nonconvex, not necessarily differentiable function of several variables. The method is based on the construction of both a lower and an upper polyhedral approximation to the objective function and is related to the use of the concept of proximal trajectory. Convergence to a stationary point is proved for weakly semismooth functions.

Nonsmooth Optimization Techniques for Semisupervised Classification

We apply nonsmooth optimization techniques to classification problems, with particular reference to the transductive support vector machine (TSVM) approach, where the considered decision function is nonconvex and nondifferentiable, hence difficult to minimize. We present some numerical results obtained by running the proposed method on some standard test problems drawn from the binary classification literature.

A DC piecewise affine model and a bundling technique in nonconvex nonsmooth minimization

We introduce an algorithm to minimize a function of several variables with no convexity nor smoothness assumptions. The main peculiarity of our approach is the use of an objective function model which is the difference of two piecewise affine convex functions. Bundling and trust region concepts are embedded into the algorithm. Convergence of the algorithm to a stationary point is proved and some numerical results are reported.

Non-smoothness in classification problems

We review the role played by non-smooth optimization techniques in many recent applications in classification area. Starting from the classical concept of linear separability in binary classification, we recall the more general concepts of polyhedral, ellipsoidal and max–min separability. Finally we focus our attention on the support vector machine (SVM) approach and on the more recent transductive SVM technique.

Integrated Shipment Dispatching and Packing Problems: a Case Study

In this paper we examine a consolidation and dispatching problem motivated by a multinational chemical company which has to decide routinely the best way of delivering a set of orders to its customers over a multi-day planning horizon. Every day the decision to be made includes order consolidation, vehicle dispatching as well as load packing into the vehicles. We develop a heuristic based on a cutting plane framework, in which a simplified Integer Linear Program (ILP) is solved to optimality. Since the ILP solution may correspond to a infeasible loading plan, a feasibility check is performed through a tailored heuristic for a three-dimensional bin packing problem with side constraints. If this test fails, a cut able to remove the infeasible solution is generated and added to the simplified ILP. Then the procedure is iterated. Computational results show that our procedure allows achieving remarkable cost savings.

Support Vector Machine Polyhedral Separability in Semisupervised Learning

We introduce separation margin maximization, a characteristic of the Support Vector Machine technique, into the approach to binary classification based on polyhedral separability and we adopt a semisupervised classification framework.In particular, our model aims at separating two finite and disjoint sets of points by means of a polyhedral surface in the semisupervised case, that is, by exploiting information coming from both labeled and unlabeled samples. Our formulation requires the minimization of a nonconvex nondifferentiable error function. Numerical results are presented on several data sets drawn from the literature.

A Nonmonotone Proximal Bundle Method with (Potentially) Continuous Step Decisions

We present a convex nondifferentiable minimization algorithm of proximal bundle type that does not rely on measuring descent of the objective function to declare the so-called serious steps; rather, a merit function is defined which is decreased at each iteration, leading to a (potentially) continuous choice of the stepsize between zero (the null step) and one (the serious step). By avoiding the discrete choice the convergence analysis is simplified, and we can more easily obtain efficiency estimates for the method. Some choices for the step selection actually reproduce the dichotomic behavior of standard proximal bundle methods but shed new light on the rationale behind the process, and ultimately with different rules; furthermore, using nonlinear upper models of the function in the step selection process can lead to actual fractional steps.

A bundle modification strategy for convex minimization

We present a new bundle algorithm for minimizing convex not necessarily smooth functions. The novelty of our approach is based on a bundle modification strategy that we apply whenever the stability center is updated and which is aimed at substituting the points of the bundle by new points characterized by possibly better values of the objective function. Convergence of the algorithm is proved and numerical results are presented.

ON THE PERFORMANCE OF SWITCHING BFGS/SR1 ALGORITHMS FOR UNCONSTRAINED OPTIMIZATION

This paper studies some possible combinations of the best features of the quasi-Newton symmetric rank-one (SR1), BFGS and extra updating BFGS algorithms for solving nonlinear unconstrained optimization problems. These combinations depend on switching between the BFGS and SR1 updates so that certain desirable properties are imposed. The presented numerical results show that the proposed switching algorithm outperforms the robust BFGS method. *E-mail: albaali@squ.edu.om †E-mail: antonio.fuduli@unile.it

The Proximal Trajectory Algorithm in SVM Cross Validation

We propose a bilevel cross-validation scheme for support vector machine (SVM) model selection based on the construction of the entire regularization path. Since such path is a particular case of the more general proximal trajectory concept from nonsmooth optimization, we propose for its construction an algorithm based on solving a finite number of structured linear programs. Our methodology, differently from other approaches, works directly on the primal form of SVM. Numerical results are presented on binary data sets drawn from literature.