In a number of application areas, it is desirable to obtain sparse solutions. Minimizing the number of nonzeroes of the solution (its l0-norm) is a difficult nonconvex optimization problem, and is often approximated by the convex problem of minimizing the l1-norm. In contrast, we consider exact formulations as mathematical programs with complementarity constraints and their … Read more
This paper considers a bilevel nonlinear program (NLP) whose lower-level problem satisfies a linear independence constraint qualification (LICQ) and a strong second-order condition (SSOC). One would expect the resulting mathematical program with complementarity constraints (MPCC), whose constraints are the first-order optimality conditions of the lower-level NLP, to satisfy an MPEC-LICQ. We provide an example which … Read more
We consider an equilibrium problem defined on a convex set, whose cost bifunction may not be monotone. We show that this problem can be solved by the inexact partial proximal method with quasi distance. As an application, at the psychological level of behavioral dynamics, this paper shows two points: i) how a dual equilibrium problem … Read more
We present a filter linesearch method for solving general nonlinear and nonconvex optimization problems. The method is of the filter variety, but uses a robust (always feasible) subproblem based on an exact penalty function to compute a search direction. This contrasts traditional filter methods that use a (separate) restoration phase designed to reduce infeasibility until … Read more
We investigate constrained first order techniques for training Support Vector Machines (SVM) for online classification tasks. The methods exploit the structure of the SVM training problem and combine ideas of incremental gradient technique, gradient acceleration and successive simple calculations of Lagrange multipliers. Both primal and dual formulations are studied and compared. Experiments show that the … Read more
The industrial treatment of waste paper in order to regain valuable fibers from which recovered paper can be produced, involves several steps of preparation. One important step is the separation of stickies that are normally attached to the paper. If not properly separated, remaining stickies reduce the quality of the recovered paper or even disrupt … Read more
We present an approach for nonlinear programming (NLP) based on the direct minimization of an exact di erentiable penalty function using trust-region Newton techniques. As opposed to existing algorithmic approaches to NLP, the approach provides all the features required for scalability: it can eciently detect and exploit directions of negative curvature, it is superlinearly convergent, and … Read more
In recent years, many traditional optimization methods have been successfully generalized to minimize objective functions on manifolds. In this paper, we first extend the general traditional constrained optimization problem to a nonlinear programming problem built upon a general Riemannian manifold $\mathcal{M}$, and discuss the first-order and the second-order optimality conditions. By exploiting the differential geometry … Read more
Sequential quadratic programming (SQP) methods are a popular class of methods for nonlinearly constrained optimization. They are particularly effective for solving a sequence of related problems, such as those arising in mixed-integer nonlinear programming and the optimization of functions subject to differential equation constraints. Recently, there has been considerable interest in the formulation of \emph{stabilized} … Read more
In this paper, we present a primal-dual interior-point method for solving nonlinear programming problems. It employs a Levenberg-Marquardt (LM) perturbation to the Karush-Kuhn-Tucker (KKT) matrix to handle indefinite Hessians and a line search to obtain sufficient descent at each iteration. We show that the LM perturbation is equivalent to replacing the Newton step by a … Read more