Portfolio optimization in the presence of estimation errors on the expected asset returns

It is well known that the classical Markowitz model for portfolio optimization is extremely sensitive to estimation errors on the expected asset returns. Robust optimization mitigates this issue. We focus on ellipsoidal uncertainty sets around the point estimates of the expected asset returns. We investigate the performance of diagonal estimation-error matrices in the description of … Read more

On the Rational Polytopes with Chvatal Rank 1

We study the following problem: given a rational polytope with Chvatal rank 1, does it contain an integer point? Boyd and Pulleyblank observed that this problem is in the complexity class NP ∩ co-NP, an indication that it is probably not NP-complete. It is open whether there is a polynomial time algorithm to solve the … Read more

On Some Polytopes Contained in the 0,1 Hypercube that Have a Small Chvatal Rank

In this paper, we consider polytopes P that are contained in the unit hypercube. We provide conditions on the set of 0,1 vectors not contained in P that guarantee that P has a small Chvatal rank. Our conditions are in terms of the subgraph induced by these infeasible 0,1 vertices in the skeleton graph of … Read more

Computational Aspects of Bayesian Solution Estimators in Stochastic Optimization

We study a class of stochastic programs where some of the elements in the objective function are random, and their probability distribution has unknown parameters. The goal is to find a good estimate for the optimal solution of the stochastic program using data sampled from the distribution of the random elements. We investigate two common … Read more

From Estimation to Optimization via Shrinkage

We study a class of quadratic stochastic programs where the distribution of random variables has unknown parameters. A traditional approach is to estimate the parameters using a maximum likelihood estimator (MLE) and to use this as input in the optimization problem. For the unconstrained case, we show that an estimator that “shrinks” the MLE towards … Read more

Incorporating Black-Litterman Views in Portfolio Construction when Stock Returns are a Mixture of Normals

In this paper, we consider the basic problem of portfolio construction in financial engineering, and analyze how market-based and analytical approaches can be combined to obtain efficient portfolios. As a first step in our analysis, we model the asset returns as a random variable distributed according to a mixture of normal random variables. We then … Read more

Disjunctive Cuts for Cross-Sections of the Second-Order Cone

In this paper we provide a unified treatment of general two-term disjunctions on cross-sections of the second-order cone. We derive a closed-form expression for a convex inequality that is valid for such a disjunctive set and show that this inequality is sufficient to characterize the closed convex hull of all two-term disjunctions on ellipsoids and … Read more

On the relative strength of families of intersection cuts arising from pairs of tableau constraints in mixed integer programs

We compare the relative strength of valid inequalities for the integer hull of the feasible region of mixed integer linear programs with two equality constraints, two unrestricted integer variables and any number of nonnegative continuous variables. In particular, we prove that the closure of Type~2 triangle (resp. Type~3 triangle; quadrilateral) inequalities, are all within a … Read more

How tight is the corner relaxation? Insights gained from the stable set problem

The corner relaxation of a mixed-integer linear program is a central concept in cutting plane theory. In a recent paper Fischetti and Monaci provide an empirical assessment of the strength of the corner and other related relaxations on benchmark problems. In this paper we give a precise characterization of the bounds given by these relaxations … Read more

A probabilistic analysis of the strength of the split and triangle closures

In this paper we consider a relaxation of the corner polyhedron introduced by Andersen et al., which we denote by RCP. We study the relative strength of the split and triangle cuts of RCP’s. Basu et al. showed examples where the split closure can be arbitrarily worse than the triangle closure under a `worst-cost’ type … Read more