Newton Interior-point Method Research Articles

Applications of Primal-dual Interior Methods in Structural Optimization

AbstractWe are concerned with structural optimization problems for technological processes in material science that are described by partial differential equations. In particular, we consider the topology optimization of conductive media in high-power electronic devices described by Maxwell equations and the optimal design of composite ceramic materials by homogenization modeling. All these tasks lead to constrained nonconvex minimization problems with both equality and inequality constraints on the state variables and design parameters. After discretization by finite elements, we solve the discretized optimization problems by a primal-dual Newton interior-point method. Within a line-search approach, transforming iterations are applied with respect to the null space decomposition of the condensed primal-dual system to find the search direction. Some numerical experiments for the two applications are presented.

Primal-Dual Newton-Type Interior-Point Method for Topology Optimization

We consider the problem of minimization of energy dissipation in a conductive electromagnetic medium with a fixed geometry and a priori given lower and upper bounds for the conductivity. The nonlinear optimization problem is analyzed by using the primal-dual Newton interior-point method. The elliptic differential equation for the electric potential is considered as an equality constraint. Transforming iterations for the null space decomposition of the condensed primal-dual system are applied to find the search direction. The numerical experiments treat two-dimensional isotropic systems.

Numerical Comparisons of Path-Following Strategies for a Primal-Dual Interior-Point Method for Nonlinear Programming

An important research activity in primal-dual interior-point methods for general nonlinear programming is to determine effective path-following strategies and their implementations. The objective of this work is to present numerical comparisons of several path-following strategies for the local interior-point Newton method given by El-Bakry, Tapia, Tsuchiya, and Zhang. We conduct numerical experimentation of nine strategies using two central regions, three notions of proximity measures, and three merit functions to obtain an optimal solution. Six of these strategies are implemented for the first time. The numerical results show that the best path-following strategy is that given by Argaez and Tapia.

On the Global Convergence of a Modified Augmented Lagrangian Linesearch Interior-Point Newton Method for Nonlinear Programming

We consider a linesearch globalization of the local primal-dual interior-point Newton method for nonlinear programming introduced by El-Bakry, Tapia, Tsuchiya, and Zhang. The linesearch uses a new merit function that incorporates a modification of the standard augmented Lagrangian function and a weak notion of centrality. We establish a global convergence theory and present promising numerical experimentation.

Comparison of Two Sets of First-Order Conditions as Bases of Interior-Point Newton Methods for Optimization with Simple Bounds

In this paper, we compare the behavior of two Newton interior-point methods derived from two different first-order necessary conditions for the same nonlinear optimization problem with simple bounds. One set of conditions was proposed by Coleman and Li; the other is the standard KKT set of conditions. We discuss a perturbation of the CL conditions for problems with one-sided bounds and the difficulties involved in extending this to problems with general bounds. We study the numerical behavior of the Newton method applied to the systems of equations associated with the unperturbed and perturbed necessary conditions. Preliminary numerical results for convex quadratic objective functions indicate that, for this class of problems, the Newton method based on the perturbed KKT formulation appears to be the more robust.

On the Newton Interior-Point Method for Nonlinear Programming Problems

Interior-point methods have been developed largely for nonlinear programming problems. In this paper, we generalize the global Newton interior-point method introduced in Ref. 1 and we establish a global convergence theory for it, under the same assumptions as those stated in Ref. 1. The generalized algorithm gives the possibility of choosing different descent directions for a merit function so that difficulties due to small steplength for the perturbed Newton direction can be avoided. The particular choice of the perturbation enables us to interpret the generalized method as an inexact Newton method. Also, we suggest a more general criterion for backtracking, which is useful when the perturbed Newton system is not solved exactly. We include numerical experimentation on discrete optimal control problems.

Superlinear Convergence of Affine-Scaling Interior-Point Newton Methods for Infinite-Dimensional Nonlinear Problems with Pointwise Bounds

We develop and analyze a superlinearly convergent affine-scaling interior-point Newton method for infinite-dimensional problems with pointwise bounds in Lp-space. The problem formulation is motivated by optimal control problems with Lp-controls and pointwise control constraints. The finite-dimensional convergence theory by Coleman and Li [SIAM J. Optim., 6 (1996), pp. 418--445] makes essential use of the equivalence of norms and the exact identifiability of the active constraints close to an optimizer with strict complementarity. Since these features are not available in our infinite-dimensional framework, algorithmic changes are necessary to ensure fast local convergence. The main building block is a Newton-like iteration for an affine-scaling formulation of the KKT-condition. We demonstrate in an example that a stepsize rule to obtain an interior iterate may require very small stepsizes even arbitrarily close to a nondegenerate solution. Using a pointwise projection instead we prove superlinear convergence under a weak strict complementarity condition and convergence with \mbox{Q-rate $>$1} under a slightly stronger condition if a smoothing step is available. We discuss how the algorithm can be embedded in the class of globally convergent trust-region interior-point methods recently developed by M. Heinkenschloss and the authors. Numerical results for the control of a heating process confirm our theoretical findings.

On the Quadratic Convergence of the Simplified Mizuno--Todd--Ye Algorithm for Linear Programming

It is known that the Mizuno--Todd--Ye predictor-corrector primal-dual Newton interior-point method generates a duality-gap sequence which converges quadratically to zero, and this is accomplished with an iteration complexity of $O(\sqrt n L)$. Very recently, the present authors demonstrated that the iteration sequence generated by this method converges, and this convergence is to the analytic center of the solution set. In the current work we show that within a finite number of iterations, the Newton corrector step can be replaced with a simplified Newton corrector step, and the resulting algorithm maintains $O(\sqrt n L)$ iteration complexity, quadratic convergence of the duality-gap sequence to zero, and convergence of the iteration sequence (however, not necessarily to the analytic center). The simplified predictor-corrector algorithm requires only one linear solve per iteration in contrast to the two linear solves per iteration required by the original predictor-corrector algorithm.

On the convergence rate of Newton interior-point methods in the absence of strict complementarity

In the absence of strict complementarity, Monteiro and Wright [7] proved that the convergence rate for a class of Newton interior-point methods for linear complementarity problems is at best linear. They also established an upper bound of 1/4 for the Q 1-factor of the duality gap sequence when the steplengths converge to one. In the current paper, we prove that the Q 1 factor of the duality gap sequence is exactly 1/4. In addition, the convergence of the Tapia indicators is also discussed.

On the formulation and theory of the Newton interior-point method for nonlinear programming

In this work, we first study in detail the formulation of the primal-dual interior-point method for linear programming. We show that, contrary to popular belief, it cannot be viewed as a damped Newton method applied to the Karush-Kuhn-Tucker conditions for the logarithmic barrier function problem. Next, we extend the formulation to general nonlinear programming, and then validate this extension by demonstrating that this algorithm can be implemented so that it is locally and Q-quadratically convergent under only the standard Newton method assumptions. We also establish a global convergence theory for this algorithm and include promising numerical experimentation.

Higher-Order Predictor-Corrector Interior Point Methods with Application to Quadratic Objectives

In this paper, the authors explore the full utility of Mehrotra’s predictor-corrector method in the context of linear and convex quadratic programs. They describe a procedure for doing multiple corrections at each iteration and implement it within the framework of OB1. Computational results are provided for the multiple correcting procedure using several strategies for determining the number of corrections in a given iteration. The results indicate that iteration counts can be significantly reduced by allowing higher-order corrections but at the the cost of extra work per iteration. The procedure is shown to be a level-m composite Newton interior point method, where m is the number of corrections performed in an iteration.