Exact Solution Methods for the kk-item Quadratic Knapsack Problem

The purpose of this paper is to solve the 0-1 $k$-item quadratic knapsack problem $(kQKP)$, a problem of maximizing a quadratic function subject to two linear constraints. We propose an exact method based on semidefinite optimization. The semidefinite relaxation used in our approach includes simple rank one constraints, which can be handled efficiently by interior point methods. Furthermore, we strengthen the relaxation by polyhedral constraints and obtain approximate solutions to this semidefinite problem by applying a bundle method. We review other exact solution methods and compare all these approaches by experimenting with instances of various sizes and densities.Comment: 12 page

A hybrid constraint programming and semidefinite programming approach for the stable set problem

This work presents a hybrid approach to solve the maximum stable set problem, using constraint and semidefinite programming. The approach consists of two steps: subproblem generation and subproblem solution. First we rank the variable domain values, based on the solution of a semidefinite relaxation. Using this ranking, we generate the most promising subproblems first, by exploring a search tree using a limited discrepancy strategy. Then the subproblems are being solved using a constraint programming solver. To strengthen the semidefinite relaxation, we propose to infer additional constraints from the discrepancy structure. Computational results show that the semidefinite relaxation is very informative, since solutions of good quality are found in the first subproblems, or optimality is proven immediately.Comment: 14 page

Credible Autocoding of Convex Optimization Algorithms

International audienceThe efficiency of modern optimization methods, coupled with increasing computational resources, has led to the possibility of real-time optimization algorithms acting in safety critical roles. There is a considerable body of mathematical proofs on on-line optimization programs which can be leveraged to assist in the development and verification of their implementation. In this paper, we demonstrate how theoretical proofs of real-time optimization algorithms can be used to describe functional properties at the level of the code, thereby making it accessible for the formal methods community. The running example used in this paper is a generic semi-definite programming (SDP) solver. Semi-definite programs can encode a wide variety of optimization problems and can be solved in polynomial time at a given accuracy. We describe a top-to-down approach that transforms a high-level analysis of the algorithm into useful code annotations. We formulate some general remarks about how such a task can be incorporated into a convex programming autocoder. We then take a first step towards the automatic verification of the optimization program by identifying key issues to be adressed in future work

An inexact dual logarithmic barrier method for solving sparse semidefinite programs

A dual logarithmic barrier method for solving large, sparse semidefinite programs is proposed in this paper. The method avoids any explicit use of the primal variable X and therefore is well-suited to problems with a sparse dual matrix S. It relies on inexact Newton steps in dual space which are computed by the conjugate gradient method applied to the Schur complement of the reduced KKT system. The method may take advantage of low-rank representations of matrices Ai to perform implicit matrix-vector products with the Schur complement matrix and to compute only specific parts of this matrix. This allows the construction of the partial Cholesky factorization of the Schur complement matrix which serves as a good preconditioner for it and permits the method to be run in a matrix-free scheme. Convergence properties of the method are studied and a polynomial complexity result is extended to the case when inexact Newton steps are employed. A Matlab-based implementation is developed and preliminary computational results of applying the method to maximum cut and matrix completion problems are reported

Online Optimization of Complex Transportation Systems

This paper discusses online optimization of real-world transportation systems. We concentrate on transportation problems arising in production and manufacturing processes, in particular in company internal logistics. We describe basic techniques to design online optimization algorithms for such systems, but our main focus is decision support for the planner: which online algorithm is the most appropriate one in a particular setting? We show by means of several examples that traditional methods for the evaluation of online algorithms often do not suffice to judge the strengths and weaknesses of online algorithms. We present modifications of well-known evaluation techniques and some new methods, and we argue that the selection of an online algorithm to be employed in practice should be based on a sound combination of several theoretical and practical evaluation criteria, including simulation

Elementary landscape decomposition of the 0-1 unconstrained quadratic optimization

Journal of Heuristics, 19(4), pp.711-728Landscapes’ theory provides a formal framework in which combinatorial optimization problems can be theoretically characterized as a sum of an especial kind of landscape called elementary landscape. The elementary landscape decomposition of a combinatorial optimization problem is a useful tool for understanding the problem. Such decomposition provides an additional knowledge on the problem that can be exploited to explain the behavior of some existing algorithms when they are applied to the problem or to create new search methods for the problem. In this paper we analyze the 0-1 Unconstrained Quadratic Optimization from the point of view of landscapes’ theory. We prove that the problem can be written as the sum of two elementary components and we give the exact expressions for these components. We use the landscape decomposition to compute autocorrelation measures of the problem, and show some practical applications of the decomposition.Spanish Ministry of Sci- ence and Innovation and FEDER under contract TIN2008-06491-C04-01 (the M∗ project). Andalusian Government under contract P07-TIC-03044 (DIRICOM project)

HLA genotyping in the international Type 1 Diabetes Genetics Consortium

Background Although human leukocyte antigen (HLA) DQ and DR loci appear to confer the strongest genetic risk for type 1 diabetes, more detailed information is required for other loci within the HLA region to understand causality and stratify additional risk factors. The Type 1 Diabetes Genetics Consortium (T1DGC) study design included high-resolution genotyping of HLA-A, B, C, DRB1, DQ, and DP loci in all affected sibling pair and trio families, and cases and controls, recruited from four networks worldwide, for analysis with clinical phenotypes and immunological markers

KIR and HLA Loci Are Associated with Hepatocellular Carcinoma Development in Patients with Hepatitis B Virus Infection: A Case-Control Study

BACKGROUND: Natural killer (NK) cells activation has been reported to contribute to inflammation and liver injury during hepatitis B virus (HBV) infection both in transgenic mice and in patients. However, the role of NK cells in the process of HBV-associated hepatocellular carcinoma (HCC) development has not been addressed. Killer cell immunoglobulin-like receptors (KIRs) are involved in regulating NK cell activation through recognition of specific human leukocyte antigen (HLA) class I allotypes. METHODOLOGY/PRINCIPAL FINDINGS: To investigate whether KIR and HLA genes could influence the risk of HBV-associated HCC development, 144 HBV-infected patients with HCC and 189 well-matched HBV infectors with chronic hepatitis or cirrhosis as non-HCC controls were enrolled in this study. The presence of 12 loci of KIR was detected individually. HLA-A, -B, -C loci were genotyped with high-resolution. HLA-C group 1 homozygote (OR = 2.02; p = 0.005), HLA-Bw4-80I (OR = 2.67; p = 2.0E-04) and combination of full-length form and 22 bp-deleted form of KIR2DS4 (KIR2DS4/1D) (OR = 1.89; p = 0.017) were found associated with HCC incidence. When the combined effects of these three genetic factors were evaluated, more risk factors were observed correlating with higher odds ratios for HCC incidence (P trend = 7.4E-05). Because all the risk factors we found have been reported to result in high NK cell functional potential by previous studies, our observations suggest that NK cell activation may contribute to HBV-associated HCC development. CONCLUSIONS/SIGNIFICANCE: In conclusion, this study has identified significant associations that suggest an important role for NK cells in HCC incidence in HBV-infected patients. Our study is useful for HCC surveillance and has implications for novel personalized therapy strategy development aiming at HCC prevention in HBV-infected patients