Carathéodory’s lemma states that if we have a linear combination of vectors in R^n, we can rewrite this combination using a linearly independent subset. This result has been successfully applied in nonlinear optimization in many contexts. In this work we present a new version of this celebrated theorem, in which we obtained new bounds for … Read more
Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Approximate KKT and Approximate Gradient Projection conditions are analyzed in this work. These conditions are not necessarily equivalent. Implications between different conditions and counter-examples will be shown. Algorithmic consequences will be discussed. ArticleDownload View PDF
We establish results for the problem of tracking a time-dependent manifold arising in online nonlinear programming by casting this as a generalized equation. We demonstrate that if points along a solution manifold are consistently strongly regular, it is possible to track the manifold approximately by solving a linear complementarity problem (LCP) at each time step. … Read more
We present two strategies for choosing a “hot” starting-point in the context of an infeasible Potential Reduction (PR) method for convex Quadratic Programming. The basic idea of both strategies is to select a preliminary point and to suitably scale it in order to obtain a starting point such that its nonnegative entries are sufficiently bounded … Read more
Parallel iterative methods are power tool for solving large system of linear equations (LQs). The existing parallel computing research results are all most concentred to sparse system or others particular structure, and all most based on parallel implementing the classical relaxation methods such as Gauss-Seidel, SOR, and AOR methods e±ciently on multiprcessor systems. In this … Read more
We present a row-by-row (RBR) method for solving semidefinite programming (SDP) problem based on solving a sequence of problems obtained by restricting the n-dimensional positive semidefinite constraint on the matrix X. By fixing any (n-1)-dimensional principal submatrix of X and using its (generalized) Schur complement, the positive semidefinite constraint is reduced to a simple second-order … Read more
In this chapter we present a primal-dual interior point algorithm for solving constrained nonlinear programming problems. Switching rules are implemented that aim at exploiting the merits and avoiding the drawbacks of three different merit functions. The penalty parameter is determined using an adaptive penalty strategy that ensures a descent property for the merit function. The … Read more
The Matlab implementation of a trust-region Gauss-Newton method for bound-constrained nonlinear least-squares problems is presented. The solver, called TRESNEI, is adequate for zero and small-residual problems and handles the solution of nonlinear systems of equalities and inequalities. The structure and the usage of the solver are described and an extensive numerical comparison with functions from … Read more
We develop a framework for a class of derivative-free algorithms for the least-squares minimization problem. These algorithms are based on polynomial interpolation models and are designed to take advantages of the problem structure. Under suitable conditions, we establish the global convergence and local quadratic convergence properties of these algorithms. Promising numerical results indicate the algorithm … Read more
The Sensor Network Localization Problem (SNLP), arising from many applied fields related with environmental monitoring, has attracted much research during the last years. Solving the SNLP deals with the reconstruction of a geometrical structure from incomplete pairwise distances between sensors. In this paper we present an heuristic multistage approach in which the solving strategy is … Read more