An interior point method for solving semidefinite programs using cutting planes and weighted analytic centers

We investigate solving semidefinite programs (SDPs) with an interior point method called SDP-CUT, which utilizes weighted analytic centers and cutting plane constraints. SDP-CUT iteratively refines the feasible region to achieve the optimal solution. The algorithm uses Newton’s method to compute the weighted analytic center. We investigate different stepsize determining techniques. We found that using Newton's method with exact line search is generally the best implementation of the algorithm. We have also compared our algorithm to the SDPT3 method and found that SDP-CUT initially gets into the neighborhood of the optimal solution in less iterations on all our test problems. SDP-CUT also took less iterations to reach optimality on many of the problems. However, SDPT3 required less iterations on most of the test problems and less time on all the problems. Some theoretical properties of the convergence of SDP-CUT are also discussed

Monte Carlo algorithms for the detection of necessary linear matrix inequality constraints

We reduce the size of large semidefinite programming problems by identifying necessary linear matrix inequalities (LMI's) using Monte Carlo techniques. We describe three algorithms for detecting necessary LMI constraints that extend algorithms used in linear programming to semidefinite programming. We demonstrate that they are beneficial and could serve as tools for a semidefinite programming preprocessor. A necessary LMI is one whose removal changes the feasible region defined by all the LMI constraints. The general problem of checking whether or not a particular LMI is necessary is NP-complete. However, the methods we describe are polynomial in each iteration, and the number of iterations can be limited by stopping rules. This provides a practical method for reducing the size of some large Semidefinite Programming problems before one attempts to solve them. We demonstrate the applicability of this approach to solving instances of the Lowner ellipsoid problem. We also consider the problem of classification of all the constraints of a semidefinite programming problem as redundant or necessary

On Second-Order Cone Functions

We consider the second-order cone function (SOCF) $f: {\mathbb R}^n \to \mathbb R$ defined by $f(x)= c^T x + d -\|A x + b \|$. Every SOCF is concave. We give necessary and sufficient conditions for strict concavity of $f$. The parameters $A \in {\mathbb R}^{m \times n}$ and $b \in {\mathbb R}^m$ are not uniquely determined. We show that every SOCF can be written in the form $f(x) = c^T x + d -\sqrt{\delta^2 + (x-x_*)^TM(x-x_*)}$. We give necessary and sufficient conditions for the parameters $c$, $d$, $\delta$, $M = A^T A$, and $x_*$ to be uniquely determined. We also give necessary and sufficient conditions for $f$ to be bounded above.Comment: 21 pages, 5 figure

A probabilistic method for detecting multivariate extreme outliers

Given a data set arising from a series of observations, an outlier is a value that deviates substantially from the natural variability of the data set as to arouse suspicions that it was generated by a different mechanism. We call an observation an extreme outlier if it lies at an abnormal distance from the "center" of the data set. We introduce the Monte Carlo SCD algorithm for detecting extreme outliers. The algorithm finds extreme outliers in terms of a subset of the data set called the outer shell. Each iteration of the algorithm is polynomial. This could be reduced by preprocessing the data to reduce its size. This approach has an interesting new feature. It estimates a relative measure of the degree to which a data point on the outer shell is an outlier (its "outlierness"). This measure has potential for serendipitous discoveries in data mining where unusual or special behavior is of interest. Other applications include spatial filtering and smoothing in digital image processing. We apply this method to baseball data and identify the ten most exceptional pitchers of the 1998 American League. To illustrate another useful application, we also show that the SCD can be used to reduce the solution time of the D-optimal experimental design problem

Computing Weighted Analytic Center for Linear Matrix Inequalities Using Infeasible Newton’s Method

We study the problem of computing weighted analytic center for system of linear matrix inequality constraints. The problem can be solved using Standard Newton’s method. However, this approach requires that a starting point in the interior point of the feasible region be given or a Phase I problem be solved. We address the problem by using Infeasible Newton’s method applied to the KKT system of equations which can be started from any point. We implement the method using backtracking line search technique and also study the effect of large weights on the method. We use numerical experiments to compare Infeasible Newton’s method with Standard Newton’s method. The results show that Infeasible Newton’s method moves in the interior of the feasible regions often very quickly, starting from any point. We recommend it as a method for finding an interior point by setting each weight to be 1. It appears to work better than Standard Newton’s method in finding the weighted analytic center when none of weights is very large relative to the other weights. However, we find that Infeasible Newton’s method is more sensitive than Standard Newton’s method to large variation in the weights

Incorporating Fuzzy Logic and Stochastic Processes into Multiobjective Forest Management

From the Proceedings of the 2011 Meetings of the Hydrology Section - Arizona-Nevada Academy of Science - April 9, 2011, Glendale Community College, Glendale, ArizonaThis article is part of the Hydrology and Water Resources in Arizona and the Southwest collections. Digital access to this material is made possible by the Arizona-Nevada Academy of Science and the University of Arizona Libraries. For more information about items in this collection, contact [email protected]

THE BOUNDARY OF WEIGHTED ANALYTIC CENTERS FOR LINEAR MATRIX INEQUALITIES

ABSTRACT. We study the boundary of the region of weighted analytic centers for linear matrix inequality constraints. Let be the convex subset of Rn defined by q simultaneous linear matrix inequalities (LMIs) A (j) (x): = A (j) 0 + n� i=1 xiA (j) i ≻ 0, j = 1, 2,..., q, where A (j) i are symmetric matrices and x ∈ Rn. Given a strictly positive vector ω = (ω1, ω2,..., ωq), the weighted analytic center xac(ω) is the minimizer of the strictly convex function q� φω(x): = ωj log det[A (j) (x)] −1 j=1 over R. The region of weighted analytic centers, W, is a subset of R. We give several examples for which W has interesting topological properties. We show that every point on a central path in semidefinite programming is a weighted analytic center. We introduce the concept of the frame of W, which contains the boundary points of W which are not boundary points of R. The frame has the same dimension as the boundary of W and is therefore easier to compute than W itself. Furthermore, we develop a Newton-based algorithm that uses a Monte Carlo technique to compute the frame points of W as well as the boundary points of W that are also boundary points of R

Constraint Consensus Methods for Finding Interior Feasible Points in Second-Order Cones

Optimization problems with second-order cone constraints (SOCs) can be solved efficiently by interior point methods. In order for some of these methods to get started or to converge faster, it is important to have an initial feasible point or near-feasible point. In this paper, we study and apply Chinneck's Original constraint consensus method and DBmax constraint consensus method to find near-feasible points for systems of SOCs. We also develop and implement a new backtracking-like line search technique on these methods that attempts to increase the length of the consensus vector, at each iteration, with the goal of finding interior feasible points. Our numerical results indicate that the new methods are effective in finding interior feasible points for SOCs