43 research outputs found
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 optimization criteria for evaluating the quality of a solution estimator, one based on the difference in objective values, and the other based on the Euclidean distance between solutions. We use risk as the expected value of such criteria over the sample space. Under a Bayesian framework, where a prior distribution is assumed for the unknown parameters, two natural estimation-optimization strategies arise. A separate scheme first finds an estimator for the unknown parameters, and then uses this estimator in the optimization problem. A joint scheme combines the estimation and optimization steps by directly adjusting the distribution in the stochastic program. We analyze the risk difference between the solutions obtained from these two schemes for several classes of stochastic programs, while providing insight on the computational effort to solve these problems
Intersecting restrictions in clutters
A clutter is intersecting if the members do not have a common element yet every two members intersect. It has been conjectured that for clutters without an intersecting minor, total primal integrality and total dual integrality of the corresponding set covering linear system must be equivalent. In this paper, we provide a polynomial characterization of clutters without an intersecting minor. One important class of intersecting clutters comes from projective planes, namely the deltas, while another comes from graphs, namely the blockers of extended odd holes. Using similar techniques, we provide a poly- nomial algorithm for finding a delta or the blocker of an extended odd hole minor in a given clutter. This result is quite surprising as the same problem is NP-hard if the input were the blocker instead of the clutter
Revival of the Gomory Cuts in the 1990’s
In the early 90’s, the research community was unanimous: In order to solve integer programs
of meaningful sizes, one had to exploit the structure of the underlying combinatorial problem;
Gomory cuts (Gomory, 1960, 1963) made elegant theory (because they did not require
knowledge of the underlying structure) but were utterly useless in practice (because they did
not use the underlying structure!)
The max-flow min-cut property and ±1-resistant sets
A subset of the unit hypercube {0, 1}n is cube-ideal if its convex hull is described by hypercube and generalized set covering inequalities. In this paper, we provide a structure theorem for cube-ideal sets S ⊆ {0, 1}n such that, for any point x ∈ {0, 1}n , S − {x} and S ∪ {x} are cube-ideal. As a consequence of the structure theorem, we see that cuboids of such sets have the max-flow min-cut property
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 discuss how to construct portfolios that minimize the Conditional Value-at-Risk (CVaR) under this probabilistic model via a convex program. We also construct a second-order cone representable approximation of the CVaR under the mixture model, and demonstrate its theoretical and empirical accuracy. Furthermore, we incorporate the market equilibrium information into this procedure through the well-known Black-Litterman approach via an inverse optimization framework by utilizing
the proposed approximation. Our computational experiments on a real dataset show that this approach with an emphasis on the market equilibrium typically yields less risky portfolios than a purely market-based portfolio while producing similar returns on average
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 discuss how to construct portfolios that minimize the Conditional Value-at-Risk (CVaR) under this probabilistic model via a convex program. We also construct a second-order cone representable approximation of the CVaR under the mixture model, and demonstrate its theoretical and empirical accuracy. Furthermore, we incorporate the market equilibrium information into this procedure through the well-known Black-Litterman approach via an inverse optimization framework by utilizing
the proposed approximation. Our computational experiments on a real dataset show that this approach with an emphasis on the market equilibrium typically yields less risky portfolios than a purely market-based portfolio while producing similar returns on average
Branching on General Disjunctions
This paper considers a modification of the branch-and-cut algorithm for
Mixed Integer Linear Programming where branching is performed on general disjunctions
rather than on variables. We select promising branching disjunctions based
on a heuristic measure of disjunction quality. This measure exploits the relation between
branching disjunctions and intersection cuts. In this work, we focus on disjunctions
defining the mixed integer Gomory cuts at an optimal basis of the linear
programming relaxation. The procedure is tested on instances from the literature.
Experiments show that, for a majority of the instances, the enumeration tree obtained
by branching on these general disjunctions has a smaller size than the tree obtained by
branching on variables, even when variable branching is performed using full strong
branching
A Connection between Cutting Plane Theory and the Geometry of Numbers
In this paper, we relate several questions about cutting planes to a fundamental problem in the
geometry of numbers, namely, the closest vector problem. Using this connection we show that the dominance,
membership and validity problems are NP-complete for Chvátal and split cuts