Abstract: Recently, Slofstra proved that the set of quantum correlations is not closed. We prove that the set of synchronous quantum correlations is not closed, which implies his result, by giving an example of a synchronous game that has a perfect quantum approximate strategy but no perfect quantum strategy. We also exhibit a graph for which the quantum independence number and the quantum approximate independence number are different. We prove new characterisations of synchronous quantum approximate correlations and synchronous quantum spatial correlations. We solve the synchronous approximation problem of Dykema and the second author, which yields a new equivalence of Connes’ embedding problem in terms of synchronous correlations.

Abstract: Vector optimization is continuously needed in several science fields, particularly in economy and engineering. The evolution of these fields depends, in part, on the improvements in vector optimization in mathematical programming. © 2013 Bentham Science Publishers Ltd. All rights reserved.

Abstract: With the development of deep hashing learning, several end-to-end deep architectures have been proposed for fast image retrieval. However, learning to hash is essentially a mixed integer nonlinear optimisation problem with the non-deterministic polynomial-time (NP)-hard nature, which makes the standard back-propagation algorithm infeasible. A novel pairwise loss function with additional binary constraint via siamese network is proposed to improve the representation ability of hash codes. Compared to previous works, we force the output of each hidden node close to −1 or +1, which can produce more compact and discriminative hash codes. Extensive experimental results demonstrate that the method can generate more favourable results than existing state-of-the-art hash function learning methods with large margins.

Abstract: The alternating direction method of multipliers (ADMMs) decoding of low-density parity-check codes has received many attentions due to its excellent performance at the error floor region. In this letter, we develop a parameter-free decoder based on linear program decoding by replacing the binary constraint with the intersection of a box and an $\ell _{p}$ sphere. An efficient $\ell _{2}$ -box ADMM is designed to handle this model in a distributed fashion. Numerical experiments demonstrate that our decoder attains better adaptability to different signal-to-noise ratio and channels.

Abstract: We propose a total variation-based variational model for nonblind binary image deblurring. The binary constraint is considered using the double-well function as the penalty term. We show the existence of a minimizer for the proposed model. By using operator splitting and alternating split Bregman, we get an effective numerical algorithm for the proposed model. Different from the existing methods in which the binary values are assumed to be known, our method can estimate the binary values automatically in the iteration process. Numerical results and comparisons demonstrate that the proposed algorithm is promising.

Abstract: Cross-modal hashing is a method which projects heterogeneous multimedia data into a common low-dimensional latent space. Many methods based on hash codes try to keep the relationship between text and corresponding image, and relax the original discrete learning problem into a continuous learning problem. However, these methods may produce ineffective hash codes since they do not make full use of the relationship between different modalities and simply relax the discrete binary constraint into a continuous problem. Collective matrix factorization (CMF) has achieved impressive results in mining semantic concepts or latent topics from image/text. In this paper, we propose a new supervised learning framework which unifies CMF method that maximizes the correlation between two modalities and discrete cyclic coordinate descent (DCC) method that solves NP-hard problems, which ensures that the hash codes generated in the cross-modal are more accurate and efficient. Experiments on three benchmark data sets show the effectiveness of the proposed method.

Abstract: In many practical situations, we would like to maximize (or minimize) several different criteria, and it is not clear how much weight to assign to each of these criteria. Such situations are ubiquitous and thus, it is important to be able to solve the corresponding multi-objective optimization problems. There exist many heuristic methods for solving such problems. In this paper, we reformulate multi-objective optimization as a constraint satisfaction problem, and we show that this reformulation explains two widely use multi-objective optimization techniques: optimizing a weighted sum of the objective functions and optimizing the product of normalized values of these functions.

Abstract: We investigate a class of general combinatorial graph problems, including MAX-CUT and community detection, reformulated as quadratic objectives over nonconvex constraints and solved via the alternating direction method of multipliers (ADMM). We propose two reformulations: one using vector variables and a binary constraint, and the other further reformulating the Burer-Monteiro form for simpler subproblems. Despite the nonconvex constraint, we prove the ADMM iterates converge to a stationary point in both formulations, under mild assumptions. Additionally, recent work suggests that in this latter form, when the matrix factors are wide enough, local optimum with high probability is also the global optimum. To demonstrate the scalability of our algorithm, we include results for MAX-CUT, community detection, and image segmentation benchmark and simulated examples.

Abstract: The optimization computation is an essential transversal branch of operations research which is primordial in many technical fields: transport, finance, networks, energy, learning, etc. In fact, it aims to minimize the resource consumption and maximize the generated profits. This work provides a new method for cost optimization which can be applied either on path optimization for graphs or on binary constraint reduction for Constraint Satisfaction Problem (CSP). It is about the computing of the “transitive closure of a given binary relation with respect to a property.” Thus, this paper introduces the mathematical background for the transitive closure of binary relations. Then, it gives the algorithms for computing the closure of a binary relation according to another one. The elaborated algorithms are shown to be polynomial. Since this technique is of great interest, we show its applications in some important industrial fields.