Quadratic Zero-one Programming Research Articles

Hash Bit Selection Based on Collaborative Neurodynamic Optimization.

Hash bit selection determines an optimal subset of hash bits from a candidate bit pool. It is formulated as a zero-one quadratic programming problem subject to binary and cardinality constraints. In this article, the problem is equivalently reformulated as a global optimization problem. A collaborative neurodynamic optimization (CNO) approach is applied to solve the problem by using a group of neurodynamic models initialized with particle swarm optimization iteratively in the CNO. Lévy mutation is used in the CNO to avoid premature convergence by ensuring initial state diversity. A theoretical proof is given to show that the CNO with the Lévy mutation operator is almost surely convergent to global optima. Experimental results are discussed to substantiate the efficacy and superiority of the CNO-based hash bit selection method to the existing methods on three benchmarks.

A neurodynamic approach to zero-one quadratic programming

This paper proposes a neurodynamic approach to zero-one quadratic programming with linear constraints. Compared with one existing neurodynamic approach to such problems, the proposed one has lower dimensions of state variables and less computational cost, which makes the implementation easier. Under some suitable conditions, the stability and convergence properties of the proposed approach are established. Numerical simulation results and related comparisons show the efficiency of this proposed method in practical computation.

The Basic Algorithm for the Constrained Zero-One Quadratic Programming Problem with k-diagonal Matrix and Its Application in the Power System

Zero-one quadratic programming is a classical combinatorial optimization problem that has many real-world applications. However, it is well known that zero-one quadratic programming is non-deterministic polynomial-hard (NP-hard) in general. On one hand, the exact solution algorithms that can guarantee the global optimum are very time consuming. And on the other hand, the heuristic algorithms that generate the solution quickly can only provide local optimum. Due to this reason, identifying polynomially solvable subclasses of zero-one quadratic programming problems and their corresponding algorithms is a promising way to not only compromise these two sides but also offer theoretical insight into the complicated nature of the problem. By combining the basic algorithm and dynamic programming method, we propose an effective algorithm in this paper to solve the general linearly constrained zero-one quadratic programming problem with a k-diagonal matrix. In our algorithm, the value of k is changeable that covers different subclasses of the problem. The theoretical analysis and experimental results reveal that our proposed algorithm is reasonably effective and efficient. In addition, the placement of the phasor measurement units problem in the power system is adopted as an example to illustrate the potential real-world applications of this algorithm.

An artificial neural network for solving quadratic zero-one programming problems

This paper presents an artificial neural network to solve the quadratic zero-one programming problems under linear constraints. In this paper, by using the connection between integer and nonlinear programming, the quadratic zero-one programming problem is transformed into the quadratic programming problem with nonlinear constraints. Then, by using the nonlinear complementarity problem (NCP) function and penalty method this problem is transformed into an unconstrained optimization problem. It is shown that the Hessian matrix of the associated function in the unconstrained optimization problem is positive definite in the optimal point. To solve the unconstrained optimization problem an artificial neural network is used. The proposed neural network has a simple structure and a low complexity of implementation. It is shown here that the proposed artificial neural network is stable in the sense of Lyapunov. Finally, some numerical examples are given to show that the proposed model finds the optimal solution of this problem in the low convergence time.

Linear and quadratic programming approaches for the general graph partitioning problem

The graph partitioning problem is to partition the vertex set of a graph into a number of nonempty subsets so that the total weight of edges connecting distinct subsets is minimized. Previous research requires the input of cardinalities of subsets or the number of subsets for equipartition. In this paper, the problem is formulated as a zero-one quadratic programming problem without the input of cardinalities. We also present three equivalent zero-one linear integer programming reformulations. Because of its importance in data biclustering, the bipartite graph partitioning is also studied. Several new methods to determine the number of subsets and the cardinalities are presented for practical applications. In addition, hierarchy partitioning and partitioning of bipartite graphs without reordering one vertex set, are studied.

An improved linearization strategy for zero-one quadratic programming problems

We present a new linearized model for the zero-one quadratic programming problem, whose size is linear in terms of the number of variables in the original nonlinear problem. Our derivation yields three alternative reformulations, each varying in model size and tightness. We show that our models are at least as tight as the one recently proposed in [7], and examine the theoretical relationship of our models to a standard linearization of the zero-one quadratic programming problem. Finally, we demonstrate the efficacy of solving each of these models on a set of randomly generated test instances.

Long-term prospective on-line real-time seizure prediction

Objective Epilepsy, one of the most common neurological disorders, constitutes a unique opportunity to study the dynamics of spatiotemporal state transitions in real, complex, nonlinear dynamical systems. In this study, we evaluate the performance of a prospective on-line real-time seizure prediction algorithm in two patients from a common database. Methods We previously demonstrated that measures of chaos and angular frequency, estimated from electroencephalographic (EEG) signals recorded at critical sites in the cerebral cortex, progressively converge (i.e. become dynamically entrained) as the epileptic brain transits from the asymptomatic interictal state to the ictal state (seizure) (Iasemidis et al., 2001, 2002a, 2003a). This observation suggested the possibility of developing algorithms to predict seizures well ahead of their occurrences. One of the central points in those investigations was the application of optimization theory, specifically quadratic zero-one programming, for the selection of the critical cortical sites. This current study combines that observation with a dynamical entrainment detection method to prospectively predict epileptic seizures. The algorithm was tested in two patients with long-term (107.54 h) and multi-seizure EEG data B and C (Lehnertz and Litt, 2004). Results Analysis from the 2 test patients resulted in the prediction of up to 91.3% of the impending 23 seizures, about 89±15 min prior to seizure onset, with an average false warning rate of one every 8.27 h and an allowable prediction horizon of 3 h. Conclusions The algorithm provides warning of impending seizures prospectively and in real time, that is, it constitutes an on-line and real-time seizure prediction scheme. Significance These results suggest that the proposed seizure prediction algorithm could be used in novel diagnostic and therapeutic applications in epileptic patients.

Prediction of Human Epileptic Seizures based on Optimization and Phase Changes of Brain Electrical Activity

The phenomenon of epilepsy, one of the most common neurological disorders, constitutes a unique opportunity to study the dynamics of spatiotemporal state transitions in real, complex, nonlinear dynamical systems. We previously demonstrated that measures of chaos and angular frequency obtained from electroencephalographic (EEG) signals generated by critical sites in the cerebral cortex converge progressively (dynamical entrainment) from the asymptomatic interictal state to the ictal state (seizure) [L.D. Iasemidis, P. Pardalos, J.C. Sackellares and D.-S. Shiau (2001). Quadratic binary programming and dynamical system approach to determine the predictability of epileptic seizures. J. Combinatorial Optimization, 5, 9–26; L.D. Iasemidis, D.-S. Shiau, P.M. Pardalos and J.C. Sackellares (2002). Phase entrainment and predictability of epileptic seizures. In: P.M. Pardalos and J. Principe (Eds.), Biocomputing, pp. 59–84. Kluwer Academic Publishers]. This observation suggests the possibility of developing algorithms to predict seizures. One of the central points of those investigations was the application of optimization theory, specifically quadratic zero-one programming, for the selection of the cortical sites that exhibit preictal dynamical entrainment. In this study we present results from the application of this methodology to the prediction of epileptic seizures. Analysis of continuous, long-term (18–140 h), multielectrode EEG recordings from 5 patients resulted in the prediction of 88% of the impending 50 seizures, on average about 83 min prior to seizure onset, with an average false warning rate of one every 5.26 h. These results suggest that this seizure prediction algorithm performs well enough to be used in diagnostic and therapeutic applications in epileptic patients. Similar algorithms may be useful for certain spatiotemporal state transitions in other physical and biological systems.

Quadratic Binary Programming and Dynamical System Approach to Determine the Predictability of Epileptic Seizures

Epilepsy is one of the most common disorders of the nervous system. The progressive entrainment between an epileptogenic focus and normal brain areas results to transitions of the brain from chaotic to less chaotic spatiotemporal states, the epileptic seizures. The entrainment between two brain sites can be quantified by the T-index from the measures of chaos (e.g., Lyapunov exponents) of the electrical activity (EEG) of the brain. By applying the optimization theory, in particular quadratic zero-one programming, we were able to select the most entrained brain sites 10 minutes before seizures and subsequently follow their entrainment over 2 hours before seizures. In five patients with 3–24 seizures, we found that over 90% of the seizures are predictable by the optimal selection of electrode sites. This procedure, which is applied to epilepsy research for the first time, shows the possibility of prediction of epileptic seizures well in advance (19.8 to 42.9 minutes) of their occurrence.

A continuous approch for globally solving linearly constrained quadratic

In a continuous approach we propose an efficient method for globally solving linearly constrained quadratic zero-one programming considered as a d.c. (difference of onvex functions) program. A combination of the d.c. optimization algorithm (DCA) which has a finite convergence, and the branch-and-bound scheme was studied. We use rectangular bisection in the branching procedure while the bounding one proceeded by applying d.c.algorithms from a current best feasible point (for the upper bound) and by minimizing a well tightened convex underestimation of the objective function on the current subdivided domain (for the lower bound). DCA generates a sequence of points in the vertex set of a new polytope containing the feasible domain of the problem being considered. Moreover if an iterate is integral then all following iterates are integral too.Our combined algorithm converges so quite often to an integer approximate solution.Finally, we present computational results of several test problems with up to 1800 variables which prove the efficiency of our method, in particular, for linear zero-one programming

QUAD01: A data-structured implementation of Hansen's quadratic zero-one programming algorithm

The QUAD01 program described here implements an implicit-enumeration algorithm for quadratic zero-one programming devised by Pierre Hansen just over twenty years ago. The present author's implementation is written in the C programming language and uses an efficient linked-list structure to store and manipulate constraint and objective data. This use, together with the increased speed of modern microcomputers and improved optimisation of generated code, has led to a marked reduction in running times compared with the original implementation (in FORTRAN) by Hansen. Further reductions of running times have been obtained by incorporating dynamic ordering of constraints into QUAD01. Problems having up to 50–100 variables and 100–200 constraints have been solved; some results are reported here.

A heuristic-based branch and bound algorithm for unconstrained quadratic zero-one programming

In this paper we describe a branch and bound algorithm for solving the unconstrained quadratic 0–1 programming problem. The salient features of it are the use of quadratic programming heuristics in the transformation of subproblems and exploiting some classes of facets of the polytope related to the quadratic problem in deriving upper bounds on the objective function. We develop facet selection procedures that form a basis of the bound computation algorithm. We present computational experience on four series of randomly generated problems and 14 real instances of a quadratic problem arising in design automation. We remark that the same ideas can also be applied to some other combinatorial optimization problems.

The location of ambivalent facilities: Use of a quadratic zero-one programming algorithm

Ambivalent facilities are defined here as facilities that have undesirable or obnoxious properties, such as the generation of noise or pollution, in addition to the desirable properties for which they were chosen. The problem studied is that of finding the site or sites on a discrete set of rectangular grid positions for a small number of these facilities (which need not all be of the same type) among other facilities that are sensitive to the undesirable properties. The criterion used is that of maximizing the minimum Euclidean distance between any ambivalent-sensitive pair. The existence of exclusion zones and the possible requirement of a minimum tolerable distance between facilities are easily modelled. An original, data-structured computer implementation of Hansen's quadratic zero-one programming algorithm has successfully been used to solve a series of test problems containing up to 100–200 sensitive locations with grid resolutions up to 1023 × 1023. The results are reported here.

A branch and bound algorithm for the maximum clique problem

A method to solve the maximum clique problem based on an unconstrained quadratic zero-one programming formulation is presented. A branch and bound algorithm for unconstrained quadratic zero-one programming is given that uses a technique to dynamically select variables for the ordering of the branching tree. Dynamic variable selection is equivalent to vertex selection in a similar branch and bound algorithm for the maximum clique problem. In this paper we compare two different rules for selecting a vertex. The first rule selects a variable corresponding to a vertex with high connectivity (a greedy approach) and the second rule selects a variable corresponding to a vertex with low connectivity (a nongreedy approach). We demonstrate that the first rule discovers a maximum clique sooner but it takes significantly longer to verify optimality. Computational results for an efficient vectorizable implementation on an IBM 3090 are provided for randomly generated graphs with up to 1000 vertices and 150,000 edges.

Construction of test problems in quadratic bivalent programming

A method of constructing test problems for constrained bivalent quadratic programming is presented. For any feasible integer point for a given domain, the method generates quadratic functions whose minimum over the given domain occurs at the selected point. Certain properties of unconstrained quadratic zero-one programs that determine the difficulty of the test problems are also discussed. In addition, a standardized random test problem generator for unconstrained quadratic zero-one programming is given.

Graph separation techniques for quadratic zero-one programming

In this paper, we consider a divide-and-conquer algorithm for quadratic zero-one programs. The algorithm is based on the use of separators of the related graph. For some classes of problems, the algorithm runs in subexponential or in polynomial time.

Computational aspects of a branch and bound algorithm for quadratic zero-one programming

In this paper we describe computational experience in solving unconstrained quadratic zero-one problems using a branch and bound algorithm. The algorithm incorporates dynamic preprocessing techniques for forcing variables and heuristics to obtain good starting points. Computational results and comparisons with previous studies on several hundred test problems with dimensions up to 200 demonstrate the efficiency of our algorithm.

A Tight Linearization and an Algorithm for Zero-One Quadratic Programming Problems

This paper is concerned with the solution of linearly constrained zero-one quadratic programming problems. Problems of this kind arise in numerous economic, location decision, and strategic planning situations, including capital budgeting, facility location, quadratic assignment, media selection, and dynamic set covering. A new linearization technique is presented for this problem which is demonstrated to yield a tighter continuous or linear programming relaxation than is available through other methods. An implicit enumeration algorithm which uses Lagrangian relaxation, Benders' cutting planes, and local explorations is designed to exploit the strength of this linearization. Computational experience is provided to demonstrate the usefulness of the proposed linearization and algorithm.

An improved enumerative algorithm for solving quadratic zero-one programming

An improved algorithm is proposed for the minimization of a quadratic function in zero-one variables under quadratic constraints which is based on the idea of additive penalties proposed by P. Hansen. At first, the quadratic function in which all the coefficients except the constant term are nonnegative is obtained by the introduction of the negative variables, and the constant term is a tighter bound of the function. Starting from the tighter bound, some properties are obtained and some efficient tests are established. To obtain a much tighter bound, a simple heuristic method is suggested instead of solving a linear programming problem. Furthermore, the flexibility dealing with some tests is discussed and it is also helpful to the algorithm.

A linear mixed-integer model for a purchasing problem involving discount structures

In a recent paper a quadratic mixed-integer programming model was presented for an industrial purchasing problem involving complicated discount structures. The solution method presented in the paper for this model involved the solution of a series of quadratic zero-one programming problems. The disadvantages of such an approach are immediately obvious to anyone familiar with the state-of-the-art of solution methods for nonlinear integer programming problems. In this paper a linear model is constructed for the same problem and its computational advantages over the nonlinear model are discussed.