We present a new strategy for choosing primal and dual steplengths in a primal-dual interior-point algorithm for convex quadratic programming. Current implementations often scale steps equally to avoid increases in dual infeasibility between iterations. We propose that this method can be too conservative, while safeguarding an unequally-scaled steplength approach will often require fewer steps toward … Read more
We introduce a framework in which updating rules for the barrier parameter in primal-dual interior-point methods become dynamic. The original primal-dual system is augmented to incorporate explicitly an updating function. A Newton step for the augmented system gives a primal-dual Newton step and also a step in the barrier parameter. Based on local information and … Read more
In this paper, we are concerned with nonlinear minimization problems with second order cone constraints. A primal-dual interior point method is proposed for solving the problems. We also propose a new primal-dual merit function by combining the barrier penalty function and the potential function within the framework of the line search strategy, and show the … Read more
We propose two line search primal-dual interior-point methods that approximately solve a equence of equality constrained barrier subproblems. To solve each subproblem, our methods apply a modified Newton method and use an $\ell_2$-exact penalty function to attain feasibility. Our methods have strong global convergence properties under standard assumptions. Specifically, if the penalty parameter remains bounded, … Read more
Seismic reflection tomography is a method for determining a subsurface velocity model from the traveltimes of seismic waves reflecting on geological interfaces. From an optimization viewpoint, the problem consists in minimizing a nonlinear least-squares function measuring the mismatch between observed traveltimes and those calculated by ray tracing in this model. The introduction of a priori … Read more
A method for solving inequality constrained minimization problems is described. The algorithm is based on a primal-dual interior point approach, with a line search globalization strategy. A quasi-Newton technique (BFGS) with limited memory storage is used to approximate the second derivatives of the functions. The method is especially intended for solving problems with a large … Read more
In this paper we combine both trust-region and linesearch globalization strategies in a globally convergent hybrid algorithm to solve a continuously differentiable nonlinear equality constrained minimization problem. First, the trust-region approach is used to determine a descent direction of the augmented Lagrangian chosen as the merit function, and second, linesearch techniques are used to obtain … Read more
For constrained smooth or nonsmooth optimization problems, new continuously differentiable penalty functions are derived. They are proved exact in the sense that under some nondegeneracy assumption, local optimizers of a nonlinear program are precisely the optimizers of the associated penalty function. This is achieved by augmenting the dimension of the program by a variable that … Read more
This paper presents and analyzes a pattern search method for general constrained optimization based on filter methods for step acceptance. Roughly, a filter method accepts a step that either improves the objective function value or the value of some function that measures the constraint violation. The new algorithm does not compute or approximate any derivatives, … Read more
A slack-based feasible interior point method is described which can be derived as a modification of infeasible methods. The modification is minor for most line search methods, but trust region methods require special attention. It is shown how the Cauchy point, which is often computed in trust region methods, must be modified so that the … Read more