Cuts for mixed 0-1 conic programming

In this we paper we study techniques for generating valid convex constraints for mixed 0-1 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 0-1 linear programs, such as the Gomory cuts, the lift-and-project cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 0-1 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 0-1 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank-1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.Research partially supported by NSF grants CCR-00-09972, DMS-01-04282 and ONR grant N000140310514.     

Ambiguous chance constrained problems and robust optimization

In this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain. We focus primarily on the special case where the uncertainty set of the distributions is of the form where ρ_p denotes the Prohorov metric. The ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure The main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability. This result is established using the Strassen-Dudley Representation Theorem that states that when the distributions of two random variables are close in the Prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability. We also show that the robust sampled problem can be solved efficiently both in theory and in practice. Research partially supported by NSF grant CCR-00-09972. Research partially supported by NSF grants CCR-00-09972, DMS-01-04282, and ONR grant N000140310514.     

On two-stage convex chance constrained problems

In this paper we develop approximation algorithms for two-stage convex chance constrained problems. Nemirovski and Shapiro (Probab Randomized Methods Des Uncertain 2004) formulated this class of problems and proposed an ellipsoid-like iterative algorithm for the special case where the impact function f (x, h) is bi-affine. We show that this algorithm extends to bi-convex f (x, h) in a fairly straightforward fashion. The complexity of the solution algorithm as well as the quality of its output are functions of the radius r of the largest Euclidean ball that can be inscribed in the polytope defined by a random set of linear inequalities generated by the algorithm (Nemirovski and Shapiro in Probab Randomized Methods Des Uncertain 2004). Since the polytope determining r is random, computing r is difficult. Yet, the solution algorithm requires r as an input. In this paper we provide some guidance for selecting r. We show that the largest value of r is determined by the degree of robust feasibility of the two-stage chance constrained problem—the more robust the problem, the higher one can set the parameter r. Next, we formulate ambiguous two-stage chance constrained problems. In this formulation, the random variables defining the chance constraint are known to have a fixed distribution; however, the decision maker is only able to estimate this distribution to within some error. We construct an algorithm that solves the ambiguous two-stage chance constrained problem when the impact function f (x, h) is bi-affine and the extreme points of a certain “dual” polytope are known explicitly. Research partially supported by NSF grants CCR-00-09972, DMS-01-04282 and ONR grant N000140310514.     

Product-form Cholesky factorization in interior point methods for second-order cone programming

Second-order cone programming (SOCP) problems are typically solved by interior point methods. As in linear programming (LP), interior point methods can, in theory, solve SOCPs in polynomial time and can, in practice, exploit sparsity in the problem data. Specifically, when cones of large dimension are present, the density that results in the normal equations that are solved at each iteration can be remedied in a manner similar to the treatment of dense columns in an LP. Here we propose a product-form Cholesky factorization (PFCF) approach, and show that it is more numerically stable than the alternative Sherman-Morrison-Woodbury approach. We derive several PFCF variants and compare their theoretical perfomance. Finally, we prove that the elements of L in the Cholesky factorizations LDL^T that arise in interior point methods for SOCP are uniformly bounded as the duality gap tends to zero as long as the iterates remain is some conic neighborhood of the cental path.Mathematics Subject Classification (1991): 90C25, 90C51, 15A23Research supported in part by NSF Grants CDA 97-26385, DMS 01-04282, ONR Grant N000140310514 and DOE Grant GE-FG01-92ER-25126     

The Lévy-Hinčin representation for random compact convex subsets which are infinitely divisible under Minkowski addition

Summary This paper establishes the Lévy-Hinčin representation for all random compact convex subsets of ℝ which are infinitely divisible for Minkowski addition. Research partially supported by NSF grants No. MCS 8100728 and DMS-8318610 Research partially supported by NSF grant No. MCS 8301326     

Perfect matchings in balanced hypergraphs

We generalize Hall's condition for the existence of a perfect matching in a bipartite graph, to balanced hypergraphs.This work was partially supported in part by NSF grants DMI-9424348, DMS-9509581 and ONR grant N00014-89-J-1063. Ajai Kapoor was also supported by a grant from Gruppo Nazionale Delle Riccerche-CNR. Finally, we acknowledge the support of Laboratiore ARTEMIS, Université Joseph Fourier, Grenoble.     

Interior-point methods for nonconvex nonlinear programming: Jamming and numerical testing

The paper considers an example of Wächter and Biegler which is shown to converge to a nonstationary point for the standard primal–dual interior-point method for nonlinear programming. The reason for this failure is analyzed and a heuristic resolution is discussed. The paper then characterizes the performance of LOQO, a line-search interior-point code, on a large test set of nonlinear programming problems. Specific types of problems which can cause LOQO to fail are identified.Research of the first and third authors supported by NSF grant DMS-9870317, ONR grant N00014-98-1-0036.Research of the second author supported by NSF grant DMS-9805495.     

Packing directed circuits

We prove a conjecture of Younger, that for every integern0 there exists an integert0 such that for every digraphG, eitherG hasn vertex-disjoint directed circuits, orG can be made acyclic by deleting at mostt vertices.Research partially supported by DONET ECHM contract CHRXCT930090.Research partially supported by DIMACS, by NSF grant DMS-9401981 and by ONR grant N00014-92-J-1965, and partially performed under a consulting agreement with Bellcore.Research partially supported by DIMACS, by Université de Paris VI, by NSF grant DMS-9303761 and by ONR grant N00014-93-1-0325, and partially performed under a consulting agreement with Bellcore.     

Shortest paths algorithms: Theory and experimental evaluation

We conduct an extensive computational study of shortest paths algorithms, including some very recent algorithms. We also suggest new algorithms motivated by the experimental results and prove interesting theoretical results suggested by the experimental data. Our computational study is based on several natural problem classes which identify strengths and weaknesses of various algorithms. These problem classes and algorithm implementations form an environment for testing the performance of shortest paths algorithms. The interaction between the experimental evaluation of algorithm behavior and the theoretical analysis of algorithm performance plays an important role in our research. This work was done while Boris V. Cherkassky was visiting Stanford University Computer Science Department and supported by the NSF and Powell Foundation grants mentioned below. Andrew V. Goldberg was supported in part by ONR Young Investigator Award N00014-91-J-1855, NSF Presidential Young Investigator Grant CCR-8858097 with matching funds from AT&T, DEC, and 3M, and a grant from Powell Foundation. Corresponding author. This work was done while Tomasz Radzik was a Postdoctoral Fellow at SORIE, Cornell University, and supported by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550, and by the Packard Fellowship of éva Tardos.     

A Feasibility Pump for mixed integer nonlinear programs

We present an algorithm for finding a feasible solution to a convex mixed integer nonlinear program. This algorithm, called Feasibility Pump, alternates between solving nonlinear programs and mixed integer linear programs. We also discuss how the algorithm can be iterated so as to improve the first solution it finds, as well as its integration within an outer approximation scheme. We report computational results. P. Bonami is supported in part by a grant from IBM and by ANR grant BLAN06-1-138894. G. Cornuéjols is supported in part by NSF grant CMMI-0653419, ANR grant BLAN06-1-138894 and ONR grant N00014-03-1-0188. Part of this research was carried out when Andrea Lodi was Herman Goldstine Fellow of the IBM T.J. Watson Research Center whose support is gratefully acknowledged. F. Margot is supported in part by a grant from IBM and by ONR grant N00014-03-1-0188.     

Solving combinatorial optimization problems using Karmarkar's algorithm

We describe a cutting plane algorithm for solving combinatorial optimization problems. The primal projective standard-form variant of Karmarkar's algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem.Computational facilities provided by the Cornell Computational Optimization Project supported by NSF Grant DMS-8706133 and by the Cornell National Supercomputer Facility. The Cornell National Supercomputer Facility is a resource of the Center for Theory and Simulation in Science and Engineering at Cornell Unversity, which is funded in part by the National Science Foundation, New York State, and the IBM Corporation. The research of both authors was partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research partially supported by ONR Grant N00014-90-J-1714.Research partially supported by NSF Grant ECS-8602534 and by ONR Contract N00014-87-K-0212.     

On computing the center of a convex quadratically constrained set

We propose a method for finding analytic center of a convex feasible region whose boundaries are defined by quadratic functions. The algorithm starts from an arbitrary initial point and approaches to the desired center by simultaneously reducing infeasibility or slackness of all constraints. A partial Newton step is taken at each iteration.Research supported in part by the ONR under grant N00014-87-K-0214 and by the NSF under grant CCR-8810107.Research supported in part by the NSF under grant ECS-8721709.     

An exact primal–dual penalty method approach to warmstarting interior-point methods for linear programming

One perceived deficiency of interior-point methods in comparison to active set methods is their inability to efficiently re-optimize by solving closely related problems after a warmstart. In this paper, we investigate the use of a primal–dual penalty approach to overcome this problem. We prove exactness and convergence and show encouraging numerical results on a set of linear and mixed integer programming problems. Research of the first author is sponsored by ONR grant N00014-04-1-0145. Research of the second author is supported by NSF grant DMS-0107450.     

Interior-point methods for nonconvex nonlinear programming: regularization and warmstarts

In this paper, we investigate the use of an exact primal-dual penalty approach within the framework of an interior-point method for nonconvex nonlinear programming. This approach provides regularization and relaxation, which can aid in solving ill-behaved problems and in warmstarting the algorithm. We present details of our implementation within the loqo algorithm and provide extensive numerical results on the CUTEr test set and on warmstarting in the context of quadratic, nonlinear, mixed integer nonlinear, and goal programming. Research of the first author is sponsored by ONR grant N00014-04-1-0145. Research of the second author is supported by NSF grant DMS-0107450.     

A partition relation for successors of Large Cardinals

We address partition problems of Erd?s and Hajnal by showing that for all , if and carries a -dense ideal. If is measurable we show that for where is a very large ordinal less than that is closed under all primitive recursive ordinal operations. Received: 27 June 2001 / Revised version: 5 December 2001 / Published online: 4 February 2003 The first author was partially supported by NSF grant DMS-0101155 and the Equipe d'Analyse Univ. of Paris 6. The second author was partially supported by NSF grants DMS-0072560 and DMS-9704477.     

Rank two vector bundles over regular elliptic surfaces

Research partially supported by NSF grants DMS-85-03743 and DMS-87-03569 and the Alfred P. Sloan Foundation     

Morera and mean-value type theorems in the hyperbolic disk

Some Morera and mean-value type theorems are proved in the hyperbolic disk. Partially supported by NSF grants DMS-9000619 and CDR-8803012. Partially supported by NSF grants DMS-8703072 and DMS-9000619, and DGI-CYT grant PB 89-0311.     

Gradient estimates,Harnack inequalities and estimates for heat kernels of the sum of squares of vector fields

Research partially supported by NSF grant no. DMS-87-04209 and DMS-90-04062     

Polynomial dual network simplex algorithms

We show how to use polynomial and strongly polynomial capacity scaling algorithms for the transshipment problem to design a polynomial dual network simplex pivot rule. Our best pivoting strategy leads to an O(m ² logn) bound on the number of pivots, wheren andm denotes the number of nodes and arcs in the input network. If the demands are integral and at mostB, we also give an O(m(m+n logn) min(lognB, m logn))-time implementation of a strategy that requires somewhat more pivots.Research supported by AFOSR-88-0088 through the Air Force Office of Scientific Research, by NSF grant DOM-8921835 and by grants from Prime Computer Corporation and UPS.Research supported by NSF Research Initiation Award CCR-900-8226, by U.S. Army Research Office Grant DAAL-03-91-G-0102, and by ONR Contract N00014-88-K-0166.Research supported in part by a Packard Fellowship, an NSF PYI award, a Sloan Fellowship, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.     

A fourth-order numerical method for the planetary geostrophic equations with inviscid geostrophic balance

The planetary geostrophic equations with inviscid balance equation are reformulated in an alternate form, and a fourth-order finite difference numerical method of solution is proposed and analyzed in this article. In the reformulation, there is only one prognostic equation for the temperature field and the velocity field is statically determined by the planetary geostrophic balance combined with the incompressibility condition. The key observation is that all the velocity profiles can be explicitly determined by the temperature gradient, by utilizing the special form of the Coriolis parameter. This brings convenience and efficiency in the numerical study. In the fourth-order scheme, the temperature is dynamically updated at the regular numerical grid by long-stencil approximation, along with a one-sided extrapolation near the boundary. The velocity variables are recovered by special solvers on the 3-D staggered grid. Furthermore, it is shown that the numerical velocity field is divergence-free at the discrete level in a suitable sense. Fourth order convergence is proven under mild regularity requirements. R. Samelson was supported by NSF grant OCE04-24516 and Navy ONR grant N00014-05-1-0891. R. Temam was supported by NSF grant DMS-0604235 and the research fund of Indiana University. S. Wang was supported by NSF grant DMS-0605067 and Navy ONR grant N00014-05-1-0218.