Engineering topics

Polyak, R.

Publications and source records attributed to Polyak, R..

4 records

NASA NTRS1996

The Newton Modified Barrier Method for QP Problems

The Modified Barrier Functions (MBF) have elements of both Classical Lagrangians (CL) and Classical Barrier Functions (CBF). The MBF methods find an unconstrained minimizer of some smooth barrier function in primal space and then update the Lagrange multipliers, while the barrier parameter either remains fixed or can be updated at each step. The numerical realization of the MBF method leads to the Newton MBF method, where the primal minimizer is found by using Newton's method. This minimizer is then used to update the Lagrange multipliers. In this paper, we examine the Newton MBF method for the Quadratic Programming (QP) problem. It will be shown that under standard second-order optimality conditions, there is a ball around the primal solution and a cut cone in the dual space such that for a set of Lagrange multipliers in this cut cone, the method converges quadratically to the primal minimizer from any point in the aforementioned ball, and continues to do so after each Lagrange multiplier update. The Lagrange multipliers remain within the cut cone and converge linearly to their optimal values. Any point in this ball will be called a "hot start". Starting at such a "hot start", at most Omicron(1n 1n epsilon(exp -1)) Newton steps are sufficient to perform the primal minimization which is necessary for the Lagrange multiplier update. Here, epsilon > 0 is the desired accuracy. Because of the linear convergence of the Lagrange multipliers, this means that only Omicron(1n epsilon(exp -1))omicron(ln 1n epsilon(exp-1)) Newton steps are required to reach an epsilon-approximation to the solution from any "hot start". In order to reach the "hot start", one has to perform Omicron(square root(m) 1n C) Newton steps, where m characterizes the size of the problem and C > 0 is the condition number of the QP problem. This condition number will be characterized explicitly in terms of key parameters of the QP problem, which in turn depend on the input data and the size of the problem.

Melman, A.↗

NASA NTRS1994

A Numerical Comparison of Barrier and Modified Barrier Methods for Large-Scale Bound-Constrained Optimization

When a classical barrier method is applied to the solution of a nonlinear programming problem with inequality constraints, the Hessian matrix of the barrier function becomes increasingly ill-conditioned as the solution is approached. As a result, it may be desirable to consider alternative numerical algorithms. We compare the performance of two methods motivated by barrier functions. The first is a stabilized form of the classical barrier method, where a numerically stable approximation to the Newton direction is used when the barrier parameter is small. The second is a modified barrier method where a barrier function is applied to a shifted form of the problem, and the resulting barrier terms are scaled by estimates of the optimal Lagrange multipliers. The condition number of the Hessian matrix of the resulting modified barrier function remains bounded as the solution to the constrained optimization problem is approached. Both of these techniques can be used in the context of a truncated-Newton method, and hence can be applied to large problems, as well as on parallel computers. In this paper, both techniques are applied to problems with bound constraints and we compare their practical behavior.

Nash, Stephen G.↗

NASA NTRS1992

Application of Newton modified barrier method (NMBM) to structural optimization

This paper presents the application of the NMBM to obtain a minimum weight structure with constraints on displacements and minimum sizes. The solution to the problem is obtained via minimizing the Modified Barrier Function (MBF) at each step by using the Newton Method and updating Lagrange multipliers. The Lagrange multipliers are updated by using the value of the constraints at the minimum of the MBF. Three truss problems with a different number of design variables are solved. The convergence to the minimum weight design was found to be monotonic and the algorithm is potentially robust for solving problems with a large number of design variables.

Khot, N. S.↗

NASA NTRS1990

Newton modified barrier method in constrained optimization

In this paper, we develop and investigate the Newton method for solving constrained (non-smooth) optimization problems. This approach is based on the modified barrier functions (MBF) theory and on the global converging step-size version of the Newton method for smooth unconstrained optimization. Due to the excellent properties of the MBF near primal-dual solution, the Newton modified barrier method (NMBM) has a better rate of convergence, better complexity bound, and is much more stable in the final stage of the computational process than the methods which are based on the classical barrier functions (CBF).

Polyak, R.↗