Numerical examples of the behavior of REQP on nonlinear programming problems involving linear dependence among the constraint normals

This report illustrates, by means of numerical examples, the behavior of the constrained minimization algorithm REQP in situations where the active constraint normals are not linearly independent. The examples are intended to demonstrate that the presence of the penalty parameter in the equations for calculating the Lagrange multiplier estimates enables a useful search direction to be computed. This is shown to be true, whether the dependence among the constraint normals occurs at the solution or in some other region.     

Relaxation techniques for strictly convex network problems

Gauss—Seidel type relaxation techniques are applied in the context of strictly convex pure networks with separable cost functions. The algorithm is an extension of the Bertsekas—Tseng approach for solving the linear network problem and its dual as a pair of monotropic programming problems. The method is extended to cover the class of generalized network problems. Alternative internal tactics for the dual problem are examined. Computational experiments — aimed at the improved efficiency of the algorithm — are presented.This research was supported in part by National Science Foundation Grant No. DCR-8401098-A01.     

A method for minimizing the sum of a convex function and a continuously differentiable function

This paper presents a method for minimizing the sum of a possibly nonsmooth convex function and a continuously differentiable function. As in the convex case developed by the author, the algorithm is a descent method which generates successive search directions by solving quadratic programming subproblems. An inexact line search ensures global convergence of the method to stationary points.     

On Extending the LP Computable Risk Measures to Account Downside Risk

A mathematical model of portfolio optimization is usually quantified with mean-risk models offering a lucid form of two criteria with possible trade-off analysis. In the classical Markowitz model the risk is measured by a variance, thus resulting in a quadratic programming model. Following Sharpe’s work on linear approximation to the mean-variance model, many attempts have been made to linearize the portfolio optimization problem. There were introduced several alternative risk measures which are computationally attractive as (for discrete random variables) they result in solving linear programming (LP) problems. Typical LP computable risk measures, like the mean absolute deviation (MAD) or the Gini’s mean absolute difference (GMD) are symmetric with respect to the below-mean and over-mean performances. The paper shows how the measures can be further combined to extend their modeling capabilities with respect to enhancement of the below-mean downside risk aversion. The relations of the below-mean downside stochastic dominance are formally introduced and the corresponding techniques to enhance risk measures are derived.The resulting mean-risk models generate efficient solutions with respect to second degree stochastic dominance, while at the same time preserving simplicity and LP computability of the original models. The models are tested on real-life historical data.The research was supported by the grant PBZ-KBN-016/P03/99 from The State Committee for Scientific Research.     

Geometric Condition Measures and Smoothness Condition Measures for Closed Convex Sets and Linear Regularity of Infinitely Many Closed Convex Sets

In this paper, we study geometric condition measures and smoothness condition measures of closed convex sets, bounded linear regularity, and linear regularity. We show that, under certain conditions, the constant for the linear regularity of infinitely many closed convex sets can be characterized by the geometric condition measure of the intersection or by the smoothness condition measure of the intersection. We study also the bounded linear regularity and present some interesting properties of the general linear regularity problem.The author is grateful to the referees for valuable and constructive suggestions. In particular, she thanks a referee for drawing her attention to Corollary 5.14 of Ref. 3, which inspired her to derive Theorem 4.2 and Corollary 4.2 in the revision of this paper.     

Pattern search ranking and selection algorithms for mixed variable simulation-based optimization

The class of generalized pattern search (GPS) algorithms for mixed variable optimization is extended to problems with stochastic objective functions. Because random noise in the objective function makes it more difficult to compare trial points and ascertain which points are truly better than others, replications are needed to generate sufficient statistical power to draw conclusions. Rather than comparing pairs of points, the approach taken here augments pattern search with a ranking and selection (R&S) procedure, which allows for comparing many function values simultaneously. Asymptotic convergence for the algorithm is established, numerical issues are discussed, and performance of the algorithm is studied on a set of test problems.     

Continuity and Uniqueness of Equilibria for Linear Exchange Economies

The purpose of this paper is to study the continuity and uniqueness properties of equilibria for linear exchange economies. We characterize the sets of utility vectors and initial endowments for which the equilibrium price is unique and respectively the set for which the equilibrium allocation is unique. We show that the equilibrium allocation correspondence is continuous with respect to the initial endowments and we characterize the set of full measure where the equilibrium allocation correspondence with respect to the initial endowments and utility vectors is continuous.     

All Admissible Linear Estimators under Quadratic Loss in Multivariate Model

§ 1.Introduction and Notations　In this paper,for any given matrices A and B,A B denotes the Kronecker productof A and B,A is a vector formed by stacking the columns of A under each other,μ(A)is a space generated by the columns of A,and PA=A(A′A) - A′. Fourthmore,if A andB are square matrices,then A>B and A≥ B mean that A-B is a symmetrical positiveand nonnegative matrix,respectively,andλi(A) is the i-th largest eigenvalue of A.　Consider general multivariate linear modelY …     

Robust supergain beamforming for circular array via second-order cone programming

The phenomenon of supergain for a circular array and its robust beamforming are presented. The coplanar superdirective array gain of the circular array, although it is not so extreme as an endfire line array, outperforms a lot over that of a conventional delay-and-sum beamformer in isotropic noise fields when the inter-element spacings are much smaller than one-half wavelength. However, optimum beamforming algorithms can be extremely sensitive to slight errors in array characteristics. The performance are known to degrade significantly if some of underlying assumptions on the sensor array is violated. Therefore, white noise gain constraint is used to improve the robustness of the supergain beamformer against random errors. We show that the design of the weight vector of robust supergain beamformer can be reformulated as a form of second-order cone programming and resolved efficiently via the well-established interior point method. Results of computer simulation for a 24-element circular array confirm satisfactory performance of the approach proposed in this paper.     

Exact solution of a class of nonlinear knapsack problems

We consider a class of nonlinear knapsack problems with applications in service systems design and facility location problems with congestion. We provide two linearizations and their respective solution approaches. The first is solved directly using a commercial solver. The second is a piecewise linearization that is solved by a cutting plane method.