Axioma

About: Axioma is a based out in . It is known for research contribution in the topics: Portfolio & Investment (macroeconomics). The organization has 22 authors who have published 45 publications receiving 1693 citations.

Routing for Relief Efforts

Abstract: In the aftermath of a large disaster, the routing of vehicles carrying critical supplies can greatly impact the arrival times to those in need. Because it is critical that the deliveries are both fast and fair to those being served, it is not clear that the classic cost-minimizing routing problems properly reflect the relevant priorities in disaster relief. In this paper, we take the first steps toward developing new methodologies for these problems. We focus specifically on two alternative objective functions for the traveling salesman problem (TSP) and the vehicle routing problem (VRP): one that minimizes the maximum arrival time (minmax) and one that minimizes the average arrival time (minavg). To demonstrate the potential impact of using these new objective functions, we bound the worst-case performance of optimal TSP solutions with respect to these new variants and extend these bounds to include multiple vehicles and vehicle capacity. Similarly, we examine the potential increase in routing costs that results from using these alternate objectives. We present solution approaches for these two variants of the TSP and VRP, which are based on well-known insertion and local search techniques. These are used in a series of computational experiments to help identify the types of instances in which TSP and VRP solutions can be significantly different from optimal minmax and minavg solutions.

Extension of primal-dual interior point algorithms to symmetric cones

Abstract: In this paper we show that the so-called commutative class of primal-dual interior point algorithms which were designed by Monteiro and Zhang for semidefinite programming extends word-for-word to optimization problems over all symmetric cones. The machinery of Euclidean Jordan algebras is used to carry out this extension. Unlike some non-commutative algorithms such as the XS+SX method, this class of extensions does not use concepts outside of the Euclidean Jordan algebras. In particular no assumption is made about representability of the underlying Jordan algebra. As a special case, we prove polynomial iteration complexities for variants of the short-, semi-long-, and long-step path-following algorithms using the Nesterov-Todd, XS, or SX directions.

Incorporating estimation errors into portfolio selection: Robust portfolio construction

Abstract: The authors explore the negative effect that estimation error has on mean-variance optimal portfolios. It is shown that asset weights in mean-variance optimal portfolios are very sensitive to slight changes in input parameters. This instability is magnified by the presence of constrains that asset managers typically impose on their portfolios. The authors propose to use robust mean variance, a new technique which is based on robust optimisation, a deterministic framework designed to explicitly consider parameter uncertainty in optimisation problems. Alternative uncertainty regions that create a less conservative robust problem are introduced. In fact, the authors' proposed approach does not assume that all estimation errors will negatively affect the portfolios, as is the case in traditional robust optimisation, but rather that there are as many errors with negative consequences as there are errors with positive consequences. The authors demonstrate through extensive computational experiments that portfolios generated with their proposed robust mean variance methodolgy typically outperform traditional mean variance portfolios in a variety of investment scenarios. Additionally, robust mean variance portfolios are usually less sensitive to input parameters.

A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations

Abstract: Existing global optimization techniques for nonconvex quadratic programming (QP) branch by recursively partitioning the convex feasible set and thus generate an infinite number of branch-and-bound nodes. An open question of theoretical interest is how to develop a finite branch-and-bound algorithm for nonconvex QP. One idea, which guarantees a finite number of branching decisions, is to enforce the first-order Karush-Kuhn-Tucker (KKT) conditions through branching. In addition, such an approach naturally yields linear programming (LP) relaxations at each node. However, the LP relaxations are unbounded, a fact that precludes their use. In this paper, we propose and study semidefinite programming relaxations, which are bounded and hence suitable for use with finite KKT-branching. Computational results demonstrate the practical effectiveness of the method, with a particular highlight being that only a small number of nodes are required.

Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations

Abstract: This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non- convexities: integer variables and non-convex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology. In particular, we propose new methods for generating valid inequalities from the equation Y = x x T . We use the non-convex constraint $${ Y - x x^T \preccurlyeq 0}$$ to derive disjunctions of two types. The first ones are directly derived from the eigenvectors of the matrix Y − x x T with positive eigenvalues, the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction. We also use the convex SDP constraint $${ Y - x x^T \succcurlyeq 0}$$to derive convex quadratic cuts, and we combine both approaches in a cutting plane algorithm. We present computational results to illustrate our findings.