The inertia of a Hermitian matrix is defined to be a triplet composed by the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we give various closed-form formulas for the maximal and minimal values for the rank and inertia of the Hermitian expression $A + … Read more
Identifying and exploiting classes of nonconvex constraints whose feasible region is convex after branching can reduce the time to compute global solutions for nonlinear optimization problems. We develop techniques for identifying quadratic and nonlinear constraints whose feasible region can be represented as the union of a finite number of second-order cones, and we provide necessary … Read more
Research in optimization algorithm design is often accompanied by benchmarking a new al- gorithm. Some benchmarking is done as a proof-of-concept, by demonstrating the new algorithm works on a small number of dicult test problems. Alternately, some benchmarking is done in order to demonstrate that the new algorithm in someway out-performs previous methods. In this … Read more
We define the inclusion certificate as a measure that expresses each point in the domain of a collection of functions as a convex combination of other points in the domain. Using inclusion certificates, we extend the convex extensions theory to enable simultaneous convexification of functions. We discuss conditions under which the domain of the functions … Read more
We investigate in this paper a fixed parameter polynomial algorithm for the cardinality constrained quadratic optimization problem, which is NP-hard in general. More specifically, we prove that, given a problem of size $n$, the number of decision variables, and $s$, the cardinality, if, for some $0
We study approaches for obtaining convex relaxations of global optimization problems containing multilinear functions. Specifically, we compare the concave and convex envelopes of these functions with the relaxations that are obtained with a standard relaxation approach, due to McCormick. The standard approach reformulates the problem to contain only bilinear terms and then relaxes each term … Read more
We consider convex relaxations for the problem of minimizing a (possibly nonconvex) quadratic objective subject to linear and (possibly nonconvex) quadratic constraints. Let F denote the feasible region for the linear constraints. We first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on F is dominated by an alternative … Read more
An artificial neural network (ANN) is a computational model – implemented as a computer program – that is aimed at emulating the key features and operations of biological neural networks. ANNs are extensively used to model unknown or unspecified functional relationships between the input and output of a “black box” system. In order to apply … Read more
We have developed a new financial indicator – called the Interest Rate Differential Adjusted for Volatility (IRDAV) measure – to assist investors in currency markets. On a monthly basis, we rank currency pairs according to this measure and generate a basket of pairs with the highest IRDAV values. Under positive market conditions, an IRDAV based … Read more
A Euclidean distance matrix is one in which the $(i,j)$ entry specifies the squared distance between particle $i$ and particle $j$. Given a partially-specified symmetric matrix $A$ with zero diagonal, the Euclidean distance matrix completion problem (EDMCP) is to determine the unspecified entries to make $A$ a Euclidean distance matrix. We survey three different approaches … Read more