Generating hard instances for robust combinatorial optimization

While research in robust optimization has attracted considerable interest over the last decades, its algorithmic development has been hindered by several factors. One of them is a missing set of benchmark instances that make algorithm performance better comparable, and makes reproducing instances unnecessary. Such a benchmark set should contain hard instances in particular, but so far, the standard approach to produce instances has been to sample values randomly from a uniform distribution.In this paper we introduce a new method to produce hard instances for min-max combinatorial optimization problems, which is based on an optimization model itself. Our approach does not make any assumptions on the problem structure and can thus be applied to any combinatorial problem. Using the Selection and Traveling Salesman problems as examples, we show that it is possible to produce instances which are up to 500 times harder to solve for a mixed-integer programming solver than the current state-of-the-art instances.     

Conditioning of semidefinite programs

theory. This approach also quantifies the size of permissible perturbations. We include a discussion of these results for block diagonal semidefinite programs, of which linear programming is a special case. Received November 26, 1995 / Revised version received November 1, 1998 Published online February 25, 1999     

A preliminary set of applications leading to stochastic semidefinite programs and chance-constrained semidefinite programs

Ariyawansa and Zhu have recently introduced (two-stage) stochastic semidefinite programs (with recourse) (SSDPs) [1] and chance-constrained semidefinite programs (CCSDPs) [2] as paradigms for dealing with uncertainty in applications leading to semidefinite programs. Semidefinite programs have been the subject of intense research during the past 15 years, and one of the reasons for this research activity is the novelty and variety of applications of semidefinite programs. This research activity has produced, among other things, efficient interior point algorithms for semidefinite programs. Semidefinite programs however are defined using deterministic data while uncertainty is naturally present in applications. The definitions of SSDPs and CCSDPs in [1] and [2] were formulated with the expectation that they would enhance optimization modeling in applications that lead to semidefinite programs by providing ways to handle uncertainty in data. In this paper, we present results of our attempts to create SSDP and CCSDP models in four such applications. Our results are promising and we hope that the applications presented in this paper would encourage researchers to consider SSDP and CCSDP as new paradigms for stochastic optimization when they formulate optimization models.     

Symmetry in semidefinite programs

This paper is a tutorial in a general and explicit procedure to simplify semidefinite programs which are invariant under the action of a symmetry group. The procedure is based on basic notions of representation theory of finite groups. As an example we derive the block diagonalization of the Terwilliger algebra of the binary Hamming scheme in this framework. Here its connection to the orthogonal Hahn and Krawtchouk polynomials becomes visible.     

Generating complex ontology instances from documents

This paper presents a novel Information Extraction system able to generate complex instances from free text available on the Web. The approach is based on a non-monotonical processing over ontologies, and makes use of entity recognizers and disambiguators in order to adequately extract and combine instances and their relations. Experiments conducted over the archaeology research domain provide encouraging results in both efficiency and efficacy and suggest that the tool is suitable for its application on other similar Semantic Web resources.     

A sensitivity result for semidefinite programs

We study the sensitivity of solutions of linear semidefinite programs under small arbitrary perturbations of the data. We present an elementary and self-contained proof of the differentiability of the solutions as functions of the perturbations, and we characterize the derivative as the solution of a system of linear equations.     

Solving large-scale semidefinite programs in parallel

We describe an approach to the parallel and distributed solution of large-scale, block structured semidefinite programs using the spectral bundle method. Various elements of this approach (such as data distribution, an implicitly restarted Lanczos method tailored to handle block diagonal structure, a mixed polyhedral-semidefinite subdifferential model, and other aspects related to parallelism) are combined in an implementation called LAMBDA, which delivers faster solution times than previously possible, and acceptable parallel scalability on sufficiently large problems. This work was supported in part by NSF grants DMS-0215373 and DMS-0238008.     

Constructive generation of very hard 3-colorability instances

Graph colorability (COL), is a typical constraint satisfaction problem to which phase transition phenomena (PTs), are important in the computational complexity of combinatorial search algorithms. PTs are significant and subtle because, in the PT region, extraordinarily hard problem instances are found, which may require exponential-order computational time to solve. To clarify PT mechanism, many studies have been undertaken to produce very hard instances, many of which were based on generate-and-test approaches. We propose a rather systematic or constructive algorithm that repeats the embedding of 4-critical graphs to arbitrarily generate large extraordinarily hard 3-colorability instances. We demonstrated experimentally that the computational cost to solve our generated instances is of an exponential order of the number of vertices by using a few actual coloring algorithms and constraint satisfaction algorithms.     

First and second order analysis of nonlinear semidefinite programs

In this paper we study nonlinear semidefinite programming problems. Convexity, duality and first-order optimality conditions for such problems are presented. A second-order analysis is also given. Second-order necessary and sufficient optimality conditions are derived. Finally, sensitivity analysis of such programs is discussed.     

A class of semidefinite programs with rank-one solutions

We show that a class of semidefinite programs (SDP) admits a solution that is a positive semidefinite matrix of rank at most r, where r is the rank of the matrix involved in the objective function of the SDP. The optimization problems of this class are semidefinite packing problems, which are the SDP analogs to vector packing problems. Of particular interest is the case in which our result guarantees the existence of a solution of rank one: we show that the computation of this solution actually reduces to a Second Order Cone Program (SOCP). We point out an application in statistics, in the optimal design of experiments.     

Easy and hard instances of arc ranking in directed graphs

In this paper we deal with the arc ranking problem of directed graphs. We give some classes of graphs for which the arc ranking problem is polynomially solvable. We prove that deciding whether , where G is an acyclic orientation of a 3-partite graph is an NP-complete problem. In this way we answer an open question stated by Kratochvil and Tuza in 1999.     

Large-scale semidefinite programs in electronic structure calculation

It has been a long-time dream in electronic structure theory in physical chemistry/chemical physics to compute ground state energies of atomic and molecular systems by employing a variational approach in which the two-body reduced density matrix (RDM) is the unknown variable. Realization of the RDM approach has benefited greatly from recent developments in semidefinite programming (SDP). We present the actual state of this new application of SDP as well as the formulation of these SDPs, which can be arbitrarily large. Numerical results using parallel computation on high performance computers are given. The RDM method has several advantages including robustness and provision of high accuracy compared to traditional electronic structure methods, although its computational time and memory consumption are still extremely large. The work of Mituhiro Fukuda was primarily conducted at the Courant Institute of Mathematical Sciences, New York University.     

Solving a class of semidefinite programs via nonlinear programming

In this paper, we introduce a transformation that converts a class of linear and nonlinear semidefinite programming (SDP) problems into nonlinear optimization problems. For those problems of interest, the transformation replaces matrix-valued constraints by vector-valued ones, hence reducing the number of constraints by an order of magnitude. The class of transformable problems includes instances of SDP relaxations of combinatorial optimization problems with binary variables as well as other important SDP problems. We also derive gradient formulas for the objective function of the resulting nonlinear optimization problem and show that both function and gradient evaluations have affordable complexities that effectively exploit the sparsity of the problem data. This transformation, together with the efficient gradient formulas, enables the solution of very large-scale SDP problems by gradient-based nonlinear optimization techniques. In particular, we propose a first-order log-barrier method designed for solving a class of large-scale linear SDP problems. This algorithm operates entirely within the space of the transformed problem while still maintaining close ties with both the primal and the dual of the original SDP problem. Global convergence of the algorithm is established under mild and reasonable assumptions. Received: January 5, 2000 / Accepted: October 2001?Published online February 14, 2002     

A trust region method for solving semidefinite programs

When using interior point methods for solving semidefinite programs (SDP), one needs to solve a system of linear equations at each iteration. For problems of large size, solving the system of linear equations can be very expensive. In this paper, we propose a trust region algorithm for solving SDP problems. At each iteration we perform a number of conjugate gradient iterations, but do not need to solve a system of linear equations. Under mild assumptions, the convergence of this algorithm is established. Numerical examples are given to illustrate the convergence results obtained.     

Dimension reduction for semidefinite programs via Jordan algebras

Mathematical Programming - We propose a new method for simplifying semidefinite programs (SDP) inspired by symmetry reduction. Specifically, we show if an orthogonal projection map satisfies...     

A refined theorem concerning the conditioning of semidefinite programs

Using a weaker version of the Newton-Kantorovich theorem [6] given by us in [3], we show how to refine the results given in [8] dealing with the analyzing of the effect of small perturbations in problem data on the solution. The new results are obtained under weaker hypotheses and the same computational cost as in [8].     

Facial reduction for symmetry reduced semidefinite and doubly nonnegative programs

We consider both facial reduction, FR, and symmetry reduction, SR, techniques for semidefinite programming, SDP. We show that the two together fit surprisingly well in an alternating direction method of multipliers, ADMM, approach. In fact, this approach allows for simply adding on nonnegativity constraints, and solving the doubly nonnegative, DNN , relaxation of many classes of hard combinatorial problems. We also show that the singularity degree remains the same after SR, and that the DNN relaxations considered here have singularity degree one, that is reduced to zero after FR. The combination of FR and SR leads to a significant improvement in both numerical stability and running time for both the ADMM and interior point approaches. We test our method on various DNN relaxations of hard combinatorial problems including quadratic assignment problems with sizes of more than \(n=500\). This translates to a semidefinite constraint of order 250, 000 and \(625\times 10^8\) nonnegative constrained variables, before applying the reduction techniques.