On the irreducibility, Lyapunov rank, and automorphisms of speical Bishop-Phelps cones

Motivated by optimization considerations, we consider special Bishop-Phelps cones in R^n which are of the form {(t,x): t \geq ||x||} for some norm on R^(n-1). We show that for n bigger than 2, such cones are always irreducible. De fining the Lyapunov rank of a proper cone K as the dimension of the Lie algebra of … Read more

Strongly Agree or Strongly Disagree?: Rating Features in Support Vector Machines

In linear classifiers, such as the Support Vector Machine (SVM), a score is associated with each feature and objects are assigned to classes based on the linear combination of the scores and the values of the features. Inspired by discrete psychometric scales, which measure the extent to which a factor is in agreement with a … Read more

Information Relaxations, Duality, and Convex Dynamic Programs

We consider the information relaxation approach for calculating performance bounds for stochastic dynamic programs (DPs), following Brown, Smith, and Sun (2010). This approach generates performance bounds by solving problems with relaxed nonanticipativity constraints and a penalty that punishes violations of these nonanticipativity constraints. In this paper, we study DPs that have a convex structure and … Read more

Mixed-Integer Rounding Enhanced Benders Decomposition for Multiclass Service System Staffing and Scheduling with Arrival Rate Uncertainty

We study server scheduling in multiclass service systems under stochastic uncertainty in the customer arrival volumes. Common practice in such systems is to first identify staffing levels, and then determine schedules for the servers that cover these targets. We propose a new stochastic integer programming model that integrates these two decisions, which can yield lower … Read more

Conic separation of finite sets: The non-homogeneous case

We address the issue of separating two finite sets in $\mathbb{R}^n $ by means of a suitable revolution cone $$ \Gamma (z,y,s)= \{x \in \mathbb{R}^n :\, s\,\Vert x-z\Vert – y^T(x-z)=0\}.$$ One has to select the aperture coefficient $s$, the axis $y$, and the apex $z$ in such a way as to meet certain optimal separation … Read more

Conic separation of finite sets:The homogeneous case

This work addresses the issue of separating two finite sets in $\mathbb{R}^n $ by means of a suitable revolution cone $$ \Gamma (z,y,s)= \{x \in \mathbb{R}^n : s\,\Vert x-z\Vert – y^T(x-z)=0\}.$$ The specific challenge at hand is to determine the aperture coefficient $s$, the axis $y$, and the apex $z$ of the cone. These parameters … Read more

PEBBL: An Object-Oriented Framework for Scalable Parallel Branch and Bound

PEBBL is a C++ class library implementing the underlying operations needed to support a wide variety of branch-and-bound algorithms in a message-passing parallel computing environment. Deriving application-speci c classes from PEBBL, one may create parallel branch-and-bound applications through a process focused on the unique aspects of the application, while relying on PEBBL for generic aspects of … Read more

Gauge optimization, duality, and applications

Gauge functions significantly generalize the notion of a norm, and gauge optimization, as defined by Freund (1987), seeks the element of a convex set that is minimal with respect to a gauge function. This conceptually simple problem can be used to model a remarkable array of useful problems, including a special case of conic optimization, … Read more

The divergence of the BFGS and the Gauss Newton Methods

We present examples of divergence for the BFGS and Gauss Newton methods. These examples have objective functions with bounded level sets and other properties concerning the examples published recently in this journal, like unit steps and convexity along the search lines. As these other examples, the iterates, function values and gradients in the new examples … Read more

Minimizing Value-at-Risk in Single-Machine Scheduling

The vast majority of the machine scheduling literature focuses on deterministic problems in which all data is known with certainty a priori. In practice, this assumption implies that the random parameters in the problem are represented by their point estimates in the scheduling model. The resulting schedules may perform well if the variability in the … Read more