Robert J. Vanderbei

An Interior-Point Method for Semidefinite Programming

We propose a new interior-point-based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices....

An Interior-Point Algorithm for Nonconvex Nonlinear Programming

The paper describes an interior-point algorithm for nonconvex nonlinear programming which is a direct extension of interior-point methods for linear and quadratic programming. Major modifications include a merit function and an altered search direction to ensure that a descent direction for the merit function is obtained. Preliminary numerical testing indicates that the method is robust. Further, numerical comparisons with MINOS and LANCELOT show that the method is efficient, and has the promise of greatly reducing solution times on at least some classes of models.

LOQO:an interior point code for quadratic programming

This paper describes a software package, called LOQO, which implements a primal-dual interior-point method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex and general nonlinear programming, since a detailed paper describing these extensions was published recently elsewhere. In particular, we emphasize the importance of establishing and maintaining symmetric quasidefiniteness of the reduced KKT system. We show that the industry standard MPS format can be nicely formulated in such a way to provide quasidefiniteness. Computational results are included for a variety of linear and quadratic programming problems.

A modification of karmarkar's linear programming algorithm

We present a modification of Karmarkars linear programming algorithm. Our algorithm uses a recentered projected gradient approach thereby obviatinga priori knowledge of the optimal objective function value. Assuming primal and dual nondegeneracy, we prove that our algorithm converges. We present computational comparisons between our algorithm and the revised simplex method. For small, dense constraint matrices we saw little difference between the two methods.

Response surface modeling of Pb(II) removal from aqueous solution by Pistacia vera L.: Box-Behnken experimental design.

A three factor, three-level Box-Behnken experimental design combining with response surface modeling (RSM) and quadratic programming (QP) was employed for maximizing Pb(II) removal from aqueous solution by Antep pistachio (Pistacia vera L.) shells based on 17 different experimental data obtained in a lab-scale batch study. Three independent variables (initial pH of solution (pH(0)) ranging from 2.0 to 5.5, initial concentration of Pb(II) ions (C(0)) ranging from 5 to 50 ppm, and contact time (t(C)) ranging from 5 to 120 min) were consecutively coded as x(1), x(2) and x(3) at three levels (-1, 0 and 1), and a second-order polynomial regression equation was then derived to predict responses. The significance of independent variables and their interactions were tested by means of the analysis of variance (ANOVA) with 95% confidence limits (alpha=0.05). The standardized effects of the independent variables and their interactions on the dependent variable were also investigated by preparing a Pareto chart. The optimum values of the selected variables were obtained by solving the quadratic regression model, as well as by analysing the response surface contour plots. The optimum coded values of three test variables were computed as x(1)=0.125, x(2)=0.707, and x(3)=0.107 by using a LOQO/AMPL optimization algorithm. The experimental conditions at this global point were determined to be pH(0)=3.97, C(0)=43.4 ppm, and t(C)=68.7 min, and the corresponding Pb(II) removal efficiency was found to be about 100%.

EXTRASOLAR PLANET FINDING VIA OPTIMAL APODIZED-PUPIL AND SHAPED-PUPIL CORONAGRAPHS

In this paper we examine several different apodization approaches to achieving high-contrast imaging of extrasolar planets and compare different designs on a selection of performance metrics. These approaches are characterized by their use of the pupils transmission function to focus the starlight rather than by masking the star in the image plane as in a classical coronagraph. There are two broad classes of pupil coronagraphs examined in this paper: apodized pupils with spatially varying transmission functions and shaped pupils, whose transmission values are either 0 or 1. The latter are much easier to manufacture to the needed tolerances. In addition to comparing existing approaches, numerical optimization is used to design new pupil shapes. These new designs can achieve nearly as high a throughput as the best apodized pupils and perform significantly better than the apodized square aperture design. The new shaped pupils enable searches of 50%-100% of the detectable region, suppress the stars light to below 10-10 of its peak value, and have inner working distances as small as 2.8λ/D. Pupils are shown for terrestrial planet discovery using square, rectangular, circular, and elliptical apertures. A mask targeted at Jovian planet discovery is also presented, in which contrast is given up to yield greater throughput.

LOQO user's manual — version 3.10

LOQO is a system for solving smooth constrained optimization problems. The problems can be linear or nonlinear, convex or nonconvex, constrained or unconstrained. The only real restriction is that the functions defining the problem be smooth (at the points evaluated by the algorithm). If the problem is convex, LOQO finds a globally optimal solution. Otherwise, it finds a locally optimal solution near to a given starting point. This manual describes 1. how to install LOQO on your hardware. 2. how to use AMPL together with LOQO to solve general optimization problems, 3. how to use the subroutine library to formulate and solve optimization problems, and 4. how to formulate and solve linear and quadratic programs in MPS format.

Symmetric Quasidefinite Matrices

It is stated here that a symmetric matrix K is quasidefinite if it has the form \[ K = \begin{bmatrix} - E & A^T \\ A & F \end{bmatrix}, \] where E and F are symmetric positive definite matrices. Although such matrices are indefinite, it is shown that any symmetric permutation of a quasidefinite matrix yields a factorization

Interior-Point Methods for Nonconvex Nonlinear Programming: Filter Methods and Merit Functions

LDL^T

Symmetric indefinite systems for interior point methods

.This result is applied to obtain a new approach for solving the symmetric indefinite systems arising in interior-point methods for linear and quadratic programming. These systems are typically solved either by reducing to a positive definite system or by performing a Bunch–Parlett factorization of the full indefinite system at every iteration. This is an intermediate approach based on reducing to a quasidefinite system. This approach entails less fill-in than further reducing to a positive definite system, but is based on a static ordering and is therefore more efficient than performing Bunch–Parlett factorizations of the original indefinite system.