Formulations of the max k-cut problem on classical and quantum computers

Recent claims on “solving” combinatorial optimization problems via quantum computers have attracted researchers to work on quantum algorithms. The max k-cut problem is a challenging combinatorial optimization problem with multiple notorious mixed integer linear optimization formulations. In this paper, we revisit the binary quadratic optimization formulation of Carlson and Nemhauser (Operations Research, 1966) and provide … Read more

Equivalences among the chi measure, Hoffman constant, and Renegar’s distance to ill-posedness

We show the equivalence among the following three condition measures of a full column rank matrix $A$: the chi measure, the signed Hoffman constant, and the signed distance to ill-posedness. The latter two measures are constructed via suitable collections of matrices obtained by flipping the signs of some rows of $A$. Our results provide a … Read more

New characterizations of Hoffman constants for systems of linear constraints

We give a characterization of the Hoffman constant of a system of linear constraints in $\R^n$ relative to a reference polyhedron $R\subseteq\R^n$. The reference polyhedron $R$ represents constraints that are easy to satisfy such as box constraints. In the special case $R = \R^n$, we obtain a novel characterization of the classical Hoffman constant. More … Read more

An algorithm to compute the Hoffman constant of a system of linear constraints

We propose a combinatorial algorithm to compute the Hoffman constant of a system of linear equations and inequalities. The algorithm is based on a characterization of the Hoffman constant as the largest of a finite canonical collection of easy-to-compute Hoffman constants. Our algorithm and characterization extend to the more general context where some of the … Read more

Completely Positive Reformulations for Polynomial Optimization

Polynomial optimization encompasses a very rich class of problems in which both the objective and constraints can be written in terms of polynomials on the decision variables. There is a well stablished body of research on quadratic polynomial optimization problems based on reformulations of the original problem as a conic program over the cone of … Read more

A distribution-free risk-reward newsvendor model: Extending Scarf’s min-max order formula

Scarf’s min-max order formula for the distribution-free risk-neutral newsvendor problem is a classical result in the field of inventory management. The min-max order formula provides, in closed-form, the order quantity that maximizes the worst-case expected profit associated with the demand of a single product when only the mean and variance of the product’s demand distribution, … Read more

Positive polynomials on unbounded equality-constrained domains

Certificates of non-negativity are fundamental tools in optimization. A “certificate” is generally understood as an expression that makes the non-negativity of the function in question evident. Some classical certificates of non-negativity are Farkas Lemma and the S-lemma. The lift-and-project procedure can be seen as a certificate of non-negativity for affine functions over the union of … Read more

Closed-form solutions to static-arbitrage upper bounds on basket options

We provide a closed-form solution to the problem of computing the sharpest static-arbitrage upper bound on the price of a European basket option, given the prices of vanilla call options in the underlying securities. Unlike previous approaches to this problem, our solution technique is entirely based on linear programming. This also allows us to obtain … Read more

Optimizing Highway Transportation at the United States Postal Service

The United States Postal Service (USPS) delivers more than 200 billion items per year. Transporting these items in a timely and cost-efficient way is a key issue if USPS is to meet its service and financial goals. The Highway Corridor Analytic Program (HCAP) is a tool that aids transportation analysts in identifying cost saving opportunities … Read more

An estimation-free, robust conditional value-at-risk portfolio allocation model

We propose a novel optimization model for risk-averse investors to obtain robust solutions for portfolio allocation problems. Unlike related models in the literature, no historical data or statistical estimation techniques are used to compute the parameters of the model. Instead, the parameters are directly obtained from current prices of options on the assets being considered. … Read more