5. Higher Order Semiring-Based Constraints

A branch-and-price approach for the nurse rostering problem with multiple units

We address the problem of scheduling n jobs on a batching machine to minimize regular scheduling criteria that are non-decreasing in the job completion times. A batching machine is a machine that can handle up to b jobs simultaneously. The jobs that are processed together form a batch, and all jobs in a batch start and complete at the same time. The processing time of a batch is equal to the largest processing time of any job in the batch. We analyse two variants: the unbounded model, where b¿n; and the bounded model, where b1; for the case with m different processing times, we give a dynamic programming algorithm that requires O(b2m22m) time. Moreover, we prove that due date based scheduling criteria give rise to NP-hard problems. Finally, we show that an arbitrary regular cost function can be minimized in polynomial time for a fixed number of batches.

We investigate the design of dynamic programming algorithms in unreliable memories, i.e., in the presence of faults that may arbitrarily corrupt memory locations during the algorithm execution. As a main result, we devise a general resilient framework that can be applied to all local dependency dynamic programming problems, where updates to entries in the auxiliary table are determined by the contents of neighboring cells. Consider, as an example, the computation of the edit distance between two strings of length n and m. We prove that, for any arbitrarily small constant " 2 (0, 1] and n m, this problem can be solved correctly with high probability in O nm + 1+" worst-case time and O(nm + n ) space, when up to memory faults can be inserted by an adversary with unbounded computational power and is the actual number of faults occurring during the computation. We also show that an optimal edit sequence can be constructed in additional time O n + 1+" . It follows that our resilient algorithms match the running time and space usage of the standard non-resilient implementations while tolerating almost linearly-many faults. 1998 ACM Subject Classification B.8 [Performance and reliability]; F.2 [Analysis of algorithms and problem complexity]; I.2.8 [Dynamic programming].

New optimality principles are put forth based on-competitive model business planning. A generalized minmax local optimum dynamic programming algorithm is presented and applied to business model computing where predictive techniques can determine local optima. Based on a systems model an enterprise is not viewed as the sum of its component elements, but the product of their interactions. The paper starts with introducing a systems approach to business modeling. A competitive business modeling technique, based on the author's planning techniques is applied. Systemic decisions are based on common organizational goals, and as such business planning and resource assignments should strive to satisfy higher organizational goals. It is critical to understand how different decisions affect and influence one another. Here, a business planning example is presented where systems thinking technique, using causal loops, are applied to complex management decisions. A preliminary optimal game modeling technique is presented in brief. Conducting gap and risk analysis can assist with this process

BackgroundThe task of spatial cluster detection involves finding spatial regions where some property deviates from the norm or the expected value. In a probabilistic setting this task can be expressed as finding a region where some event is significantly more likely than usual. Spatial cluster detection is of interest in fields such as biosurveillance, mining of astronomical data, military surveillance, and analysis of fMRI images. In almost all such applications we are interested both in the question of whether a cluster exists in the data, and if it exists, we are interested in finding the most accurate characterization of the cluster.MethodsWe present a general dynamic programming algorithm for grid-based spatial cluster detection. The algorithm can be used for both Bayesian maximum a-posteriori (MAP) estimation of the most likely spatial distribution of clusters and Bayesian model averaging over a large space of spatial cluster distributions to compute the posterior probability of an unusual spatial clustering. The algorithm is explained and evaluated in the context of a biosurveillance application, specifically the detection and identification of Influenza outbreaks based on emergency department visits. A relatively simple underlying model is constructed for the purpose of evaluating the algorithm, and the algorithm is evaluated using the model and semi-synthetic test data.ResultsWhen compared to baseline methods, tests indicate that the new algorithm can improve MAP estimates under certain conditions: the greedy algorithm we compared our method to was found to be more sensitive to smaller outbreaks, while as the size of the outbreaks increases, in terms of area affected and proportion of individuals affected, our method overtakes the greedy algorithm in spatial precision and recall. The new algorithm performs on-par with baseline methods in the task of Bayesian model averaging.ConclusionsWe conclude that the dynamic programming algorithm performs on-par with other available methods for spatial cluster detection and point to its low computational cost and extendability as advantages in favor of further research and use of the algorithm.

In the Uncapacitated Facility Location (UFL) problem, there is a fixed cost for opening a facility, and some distance matrix d that determines the cost of distributing commodities from any facility i to any consumer j. The problem is NP-hard in general and when d consists of a distance metric in a graph [7, 12]. However, for the case where the commodity transportation costs are given by path lengths in a tree, an O(n2) dynamic programming algorithm was given by [4, 7]. We improve this dynamic programming algorithm by using the geometry of piecewise linear functions and fast merging of the binary search trees used to store these functions. We achieve the complexity bound of O(n log n) for the Tree Location Problem and some related problems. Our approach gives a general method for solving tree dynamic programming problems.

Handbook of Constraint Programming

Accurate RNA secondary structure information is the cornerstone of gene function research and RNA tertiary structure prediction. However, most traditional RNA secondary structure prediction algorithms are based on the dynamic programming (DP) algorithm, according to the minimum free energy theory, with both hard and soft constraints. The accuracy is particularly dependent on the accuracy of soft constraints (from experimental data like chemical and enzyme detection). With the elongation of the RNA sequence, the time complexity of DP-based algorithms will increase geometrically, as a result, they are not good at coping with relatively long sequences. Furthermore, due to the complexity of the pseudoknots structure, the secondary structure prediction method, based on traditional algorithms, has great defects which cannot predict the secondary structure with pseudoknots well. Therefore, few algorithms have been available for pseudoknots prediction in the past. The ATTfold algorithm proposed in this article is a deep learning algorithm based on an attention mechanism. It analyzes the global information of the RNA sequence via the characteristics of the attention mechanism, focuses on the correlation between paired bases, and solves the problem of long sequence prediction. Moreover, this algorithm also extracts the effective multi-dimensional features from a great number of RNA sequences and structure information, by combining the exclusive hard constraints of RNA secondary structure. Hence, it accurately determines the pairing position of each base, and obtains the real and effective RNA secondary structure, including pseudoknots. Finally, after training the ATTfold algorithm model through tens of thousands of RNA sequences and their real secondary structures, this algorithm was compared with four classic RNA secondary structure prediction algorithms. The results show that our algorithm significantly outperforms others and more accurately showed the secondary structure of RNA. As the data in RNA sequence databases increase, our deep learning-based algorithm will have superior performance. In the future, this kind of algorithm will be more indispensable.

This paper explores the ability of physics-informed neural networks (PINNs) to solve forward and inverse problems of contact mechanics for small deformation elasticity. We deploy PINNs in a mixed-variable formulation enhanced by output transformation to enforce Dirichlet and Neumann boundary conditions as hard constraints. Inequality constraints of contact problems, namely Karush–Kuhn–Tucker (KKT) type conditions, are enforced as soft constraints by incorporating them into the loss function during network training. To formulate the loss function contribution of KKT constraints, existing approaches applied to elastoplasticity problems are investigated and we explore a nonlinear complementarity problem (NCP) function, namely Fischer–Burmeister, which possesses advantageous characteristics in terms of optimization. Based on the Hertzian contact problem, we show that PINNs can serve as pure partial differential equation (PDE) solver, as data-enhanced forward model, as inverse solver for parameter identification, and as fast-to-evaluate surrogate model. Furthermore, we demonstrate the importance of choosing proper hyperparameters, e.g. loss weights, and a combination of Adam and L-BFGS-B optimizers aiming for better results in terms of accuracy and training time.

This paper presents a new methodology for maintenance scheduling that takes into account the uncertain constraints of the unit maintenance problem. If the spinning reserve is enough, this model can obtain an optimal solution based on reliability or cost concerned. Since the power development is more difficult than ever before, scheduling the unit maintenance under tight spinning reserve is a difficult task. In such a situation treating various constraints, such as the spinning reserve, duration of maintenance, the maintenance crew, etc., as soft constraints, a fuzzy 0-1 integer programming model is adopted to find the minimum violation solution. In order to verify that this model can be implemented in the real system, a case study of the Taiwan Power System is carried out using the algorithm.

We extend the Constraint Logic Programming (CLP) formalism in order to handle semiring-based constraints. This allows us to perform in the same language both constraint solving and optimization. In fact, constraints based on semirings are able to model both classical constraint solving and more sophisticated features like uncertainty, probability, fuzziness, and optimization. We then provide this class of languages with three equivalent semantics: model-theoretic, fix-point, and proof-theoretic, in the style of classical CLP programs.

A dynamic programming algorithm is presented for simultaneously computing the maximum-likelihood (ML) estimator of the number of multiple sources and their directions of arrival (DOA) in sensor array processing. Based on the projection-matrix decomposition technique, the multivariate log-likelihood maximization problem is transformed into a multistage one-dimensional maximization problem. The estimation procedure is obtained with relatively light computational load. In addition, the estimator gives satisfactory performance in the case of both coherent and incoherent sources. The simulation results also show that, when the element SNR is low and/or the number of snapshots is small, the DP-ML algorithm is much superior to MUSIC. >

Evolutionary algorithm approaches to solving multiobjective optimization problems have received much attention in recent years. This is one of three books published in the last year or so on the subject, and Multiobjective Evolutionary Algorithms and Applications is distinguished by its claim to being written for a wide readership of engineers, researchers, senior undergraduate students and graduate students. The book is organized into 14 chapters, with the first and second chapters providing a brief introduction into multi-objective optimization and a historical summary of the basic similarities and differences between several significant multiobjective evolutionary algorithm (MOEA) approaches. Chapter 3 provides a more in-depth look at the important subject of distribution preservation. Chapters 4, 5 and 6 introduce the author's own MOEA features and approaches. In Chapter 4, the author's suggestions for implementing hard and soft goal prioritizations and problem constraints and an adaptive fitness sharing method are presented. Chapters 5 and 6 do the same for adaptive dynamic population sizes and distributed cooperative coevolution respectively. Chapter 7 introduces potential advantages from adaptive adjustment of the size of the search space.

Soft constraints are very flexible and expressive. However, they also are very complex to handle. For this reason, it may reasonable in several cases to pass to an abstract version of a given soft problem, and then to bring some useful information from the abstract problem to the concrete one. This will hopefully make the search for a solution, or for an optimal solution, of the concrete problem, faster. In this paper we review the main concepts and properties of our abstraction framework for soft constraints, and we show how it can be used to import constraint propagation algorithms from the abstract scenario to the concrete one. This may be useful when we don't have any (or any efficient) propagation algorithm in the concrete setting.

The constant modulus adaptive blind equalization algorithms presented in this paper are shown to correspond to an error performance surface which is much improved upon that of existing algorithms such as the well-known constant modulus or Godard algorithm. Many undesirable local solutions ULSs are avoided by careful derivation. We use a deterministic optimization criterion with a soft constraint to obtain an update equation which contains a normalized gradient vector and a particular continuous non-linearity. This approach is extended to multiple constraints to yield faster converging algorithms. An autoregressive AR channel model is studied to demonstrate analytically the absence of a class of ULSs. Finally, the findings are verified experimentally for various AR and moving-average MA channels. © 1998 John Wiley & Sons, Ltd.