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 it is related to the use of the concept of proximal trajectory. Convergence to a stationary point is proved for locally Lipschitz functions.
We apply nonsmooth optimization techniques to classification problems, with particular reference to the TSVM (Transductive Support Vector Machine) approach, where the considered decision function is nonconvex and nondifferentiable and then 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.
Abstract. 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.
Abstract. 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 the 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.
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