This paper uses certain conditions for the global and superlinear convergence of the two-parameter self-scaling Broyden family of quasi-Newton algorithms for unconstraiend optimization to derive a wide interval for self-scaling updates. Numerical testing shows that such algorithms not only accelerate the convergence of the (unscaled) methods from the so-called convex class, but increase their chances … Read more
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 … Read more
A new family of numerically efficient variable metric or quasi-Newton methods for unconstrained minimization are given, which give simple possibility of adaptation for large-scale optimization. Global convergence of the methods can be established for convex sufficiently smooth functions. Some encouraging numerical experience is reported. Citation Report V876, Institute of Computer Science, AV CR, Pod Vodarenskou … Read more
The trust region subproblem (the minimization of a quadratic objective subject to one quadratic constraint and denoted TRS) has many applications in diverse areas, e.g. function minimization, sequential quadratic programming, regularization, ridge regression, and discrete optimization. In particular, it determines the step in trust region algorithms for function minimization. Trust region algorithms are popular for … Read more
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 … Read more
This paper deals with the application of pattern search methods to the numerical solution of a class of molecular geometry problems with important applications in molecular physics and chemistry. The goal is to find a configuration of a cluster or a molecule with minimum total energy. The minimization problems in this class of geometry molecular … Read more
In recent years, much work has been done on implementing a variety of algorithms in nonlinear programming software. In this paper, we analyze the performance of several state-of-the-art optimization codes on large-scale nonlinear optimization problems. Extensive numerical results are presented on different classes of problems, and features of each code that make it efficient or … Read more
We study nonsmooth unconstrained optimization problem, which includes sum of pairwise maxima of smooth functions. Minimum $l_1$-norm approximation is a particular case of this problem. Combining ideas Lagrange multipliers with smooth approximation of max-type function, we obtain a new kind of nonquadratic augmented Lagrangian. Our approach does not require artificial variables, and preserves sparse structure … Read more
The authors of this paper recently introduced a transformation \cite{BuMoZh99-1} that converts a class of semidefinite programs (SDPs) into nonlinear optimization problems free of matrix-valued constraints and variables. This transformation enables the application of nonlinear optimization techniques to the solution of certain SDPs that are too large for conventional interior-point methods to handle efficiently. Based … Read more
We study conditions under which line search Newton methods for nonlinear systems of equations and optimization fail due to the presence of singular non-stationary points. These points are not solutions of the problem and are characterized by the fact that Jacobian or Hessian matrices are singular. It is shown that, for systems of nonlinear equations, … Read more