The constrained compartmentalized knapsack problem: mathematical models and solution methods

The constrained compartmentalized knapsack problem can be seen as an extension of the constrained knapsack problem. However, the items are grouped into different classes so that the overall knapsack has to be divided into compartments, and each compartment is loaded with items from the same class. Moreover, building a compartment incurs a fixed cost and a fixed loss of the capacity in the original knapsack, and the compartments are lower and upper bounded. The objective is to maximize the total value of the items loaded in the overall knapsack minus the cost of the compartments. This problem has been formulated as an integer non-linear program, and in this paper, we reformulate the non-linear model as an integer linear master problem with a large number of variables. Some heuristics based on the solution of the restricted master problem are investigated. A new and more compact integer linear model is also presented, which can be solved by a branch-and-bound commercial solver that found most of the optimal solutions for the constrained compartmentalized knapsack problem. On the other hand, heuristics provide good solutions with low computational effort. (C) 2011 Elsevier BM. All rights reserved.

A general Trotter-Kato formula for a class of evolution operators

In this article we prove new results concerning the existence and various properties of an evolution system U(A+B)(t, s)0 <= s <= t <= T generated by the sum -(A(t) + B(t)) of two linear, time-dependent, and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing L(B) for the algebra of all linear bounded operators on B, we can express U(A+B)(t, s)0 <= s <= t <= T as the strong limit in C(8) of a product of the holomorphic contraction semigroups generated by -A (t) and - B(t), respectively, thereby proving a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t) + B(t)) to evolve with time provided there exists a fixed set D subset of boolean AND(t is an element of)[0,T] D(A(t) + B(t)) everywhere dense in B. We obtain a special case of our formula when B(t) = 0, which, in effect, allows us to reconstruct U(A)(t, s)0 <=(s)<=(t)<=(T) very simply in terms of the semigroup generated by -A(t). We then illustrate our results by considering various examples of nonautonomous parabolic initial-boundary value problems, including one related to the theory of timedependent singular perturbations of self-adjoint operators. We finally mention what we think remains an open problem for the corresponding equations of Schrodinger type in quantum mechanics.

Approximating a class of combinatorial problems with rational objective function

In the late seventies, Megiddo proposed a way to use an algorithm for the problem of minimizing a linear function a(0) + a(1)x(1) + ... + a(n)x(n) subject to certain constraints to solve the problem of minimizing a rational function of the form (a(0) + a(1)x(1) + ... + a(n)x(n))/(b(0) + b(1)x(1) + ... + b(n)x(n)) subject to the same set of constraints, assuming that the denominator is always positive. Using a rather strong assumption, Hashizume et al. extended Megiddo`s result to include approximation algorithms. Their assumption essentially asks for the existence of good approximation algorithms for optimization problems with possibly negative coefficients in the (linear) objective function, which is rather unusual for most combinatorial problems. In this paper, we present an alternative extension of Megiddo`s result for approximations that avoids this issue and applies to a large class of optimization problems. Specifically, we show that, if there is an alpha-approximation for the problem of minimizing a nonnegative linear function subject to constraints satisfying a certain increasing property then there is an alpha-approximation (1 1/alpha-approximation) for the problem of minimizing (maximizing) a nonnegative rational function subject to the same constraints. Our framework applies to covering problems and network design problems, among others.

Aplicação de técnicas de aprendizado de máquina no reconhecimento de classes estruturais de proteínas

Nowadays, classifying proteins in structural classes, which concerns the inference of patterns in their 3D conformation, is one of the most important open problems in Molecular Biology. The main reason for this is that the function of a protein is intrinsically related to its spatial conformation. However, such conformations are very difficult to be obtained experimentally in laboratory. Thus, this problem has drawn the attention of many researchers in Bioinformatics. Considering the great difference between the number of protein sequences already known and the number of three-dimensional structures determined experimentally, the demand of automated techniques for structural classification of proteins is very high. In this context, computational tools, especially Machine Learning (ML) techniques, have become essential to deal with this problem. In this work, ML techniques are used in the recognition of protein structural classes: Decision Trees, k-Nearest Neighbor, Naive Bayes, Support Vector Machine and Neural Networks. These methods have been chosen because they represent different paradigms of learning and have been widely used in the Bioinfornmatics literature. Aiming to obtain an improvment in the performance of these techniques (individual classifiers), homogeneous (Bagging and Boosting) and heterogeneous (Voting, Stacking and StackingC) multiclassification systems are used. Moreover, since the protein database used in this work presents the problem of imbalanced classes, artificial techniques for class balance (Undersampling Random, Tomek Links, CNN, NCL and OSS) are used to minimize such a problem. In order to evaluate the ML methods, a cross-validation procedure is applied, where the accuracy of the classifiers is measured using the mean of classification error rate, on independent test sets. These means are compared, two by two, by the hypothesis test aiming to evaluate if there is, statistically, a significant difference between them. With respect to the results obtained with the individual classifiers, Support Vector Machine presented the best accuracy. In terms of the multi-classification systems (homogeneous and heterogeneous), they showed, in general, a superior or similar performance when compared to the one achieved by the individual classifiers used - especially Boosting with Decision Tree and the StackingC with Linear Regression as meta classifier. The Voting method, despite of its simplicity, has shown to be adequate for solving the problem presented in this work. The techniques for class balance, on the other hand, have not produced a significant improvement in the global classification error. Nevertheless, the use of such techniques did improve the classification error for the minority class. In this context, the NCL technique has shown to be more appropriated

Quantization of (2+1)-spinning particles and bifermionic constraint problem

This work is a natural continuation of our recent study in quantizing relativistic particles. There it was demonstrated that, by applying a consistent quantization scheme to the classical model of a spinless relativistic particle as well as to the Berezin-Marinov model of a 3 + 1 Dirac particle, it is possible to obtain a consistent relativistic quantum mechanics of such particles. In the present paper, we apply a similar approach to the problem of quantizing the massive 2 + 1 Dirac particle. However, we stress that such a problem differs in a nontrivial way from the one in 3 + 1 dimensions. The point is that in 2 + 1 dimensions each spin polarization describes different fermion species. Technically this fact manifests itself through the presence of a bifermionic constant and of a bifermionic first-class constraint. In particular, this constraint does not admit a conjugate gauge condition at the classical level. The quantization problem in 2 + 1 dimensions is also interesting from the physical viewpoint (e.g., anyons). In order to quantize the model, we first derive a classical formulation in an effective phase space, restricted by constraints and gauges. Then the condition of preservation of the classical symmetries allows us to realize the operator algebra in an unambiguous way and construct an appropriate Hilbert space. The physical sector of the constructed quantum mechanics contains spin-1/2 particles and antiparticles without an infinite number of negative-energy levels, and exactly reproduces the one-particle sector of the 2 + 1 quantum theory of a spinor field.

The application of preprocessing and cutting plane techniques for a class of production planning problems

This paper investigates properties of integer programming models for a class of production planning problems. The models are developed within a decision support system to advise a sales team of the products on which to focus their efforts in gaining new orders in the short term. The products generally require processing on several manufacturing cells and involve precedence relationships. The cells are already (partially) committed with products for stock and to satisfy existing orders and therefore only the residual capacities of each cell in each time period of the planning horizon are considered. The determination of production recommendations to the sales team that make use of residual capacities is a nontrivial optimization problem. Solving such models is computationally demanding and techniques for speeding up solution times are highly desirable. An integer programming model is developed and various preprocessing techniques are investigated and evaluated. In addition, a number of cutting plane approaches have been applied. The performance of these approaches which are both general and application specific is examined.

Bound states of the Dirac equation for a class of effective quadratic plus inversely quadratic potentials

The Dirac equation is exactly solved for a pseudoscalar linear plus Coulomb-like potential in a two-dimensional world. This sort of potential gives rise to an effective quadratic plus inversely quadratic potential in a Sturm-Liouville problem, regardless the sign of the parameter of the linear potential, in sharp contrast with the Schrodinger case. The generalized Dirac oscillator already analyzed in a previous work is obtained as a particular case. (C) 2004 Elsevier B.V. All rights reserved.

Lagrangian heuristic for a class of the generalized assignment problems

A Lagrangian based heuristic is proposed for many-to-many assignment problems taking into account capacity limits for task and agents. A modified Lagrangian bound studied earlier by the authors is presented and a greedy heuristic is then applied to get a feasible Lagrangian-based solution. The latter is also used to speed up the subgradient scheme to solve the modified Lagrangian dual problem. A numerical study is presented to demonstrate the efficiency of the proposed approach. (C) 2010 Elsevier Ltd. All rights reserved.

A class of multiobjective control problems

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Comparison of Lagrangian Bounds for One Class of Generalized Assignment Problems

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Variational studies and replica symmetry breaking in the generalization problem of the binary perceptron

We analyze the average performance of a general class of learning algorithms for the nondeterministic polynomial time complete problem of rule extraction by a binary perceptron. The examples are generated by a rule implemented by a teacher network of similar architecture. A variational approach is used in trying to identify the potential energy that leads to the largest generalization in the thermodynamic limit. We restrict our search to algorithms that always satisfy the binary constraints. A replica symmetric ansatz leads to a learning algorithm which presents a phase transition in violation of an information theoretical bound. Stability analysis shows that this is due to a failure of the replica symmetric ansatz and the first step of replica symmetry breaking (RSB) is studied. The variational method does not determine a unique potential but it allows construction of a class with a unique minimum within each first order valley. Members of this class improve on the performance of Gibbs algorithm but fail to reach the Bayesian limit in the low generalization phase. They even fail to reach the performance of the best binary, an optimal clipping of the barycenter of version space. We find a trade-off between a good low performance and early onset of perfect generalization. Although the RSB may be locally stable we discuss the possibility that it fails to be the correct saddle point globally. ©2000 The American Physical Society.

Asset accumulation by the middle class and the poor in Latin America: political economy and governance dimensions

Includes bibliography

Existence of solutions for a class of degenerate quasilinear elliptic equation in R-N with vanishing potentials

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Mass-Imbalanced Three-Body Systems in 2D: Bound States and the Analytical Approach to the Adiabatic Potential

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Strategies and Precise Vocabulary Knowledge: Exploring the Relationships Among Mathematics Vocabulary, Problem Solving, and Confidence

In this action research study I focused on my eighth grade pre-algebra students’ abilities to attack problems with enthusiasm and self confidence whether they completely understand the concepts or not. I wanted to teach them specific strategies and introduce and use precise vocabulary as a part of the problem solving process in hopes that I would see students’ confidence improve as they worked with mathematics. I used non-routine problems and concept-related open-ended problems to teach and model problem solving strategies. I introduced and practiced communication with specific and precise vocabulary with the goal of increasing student confidence and lowering student anxiety when they were faced with mathematics problem solving. I discovered that although students were working more willingly on problem solving and more inclined to attempt word problems using the strategies introduced in class, they were still reluctant to use specific vocabulary as they communicated to solve problems. As a result of this research, my style of teaching problem solving will evolve so that I focus more specifically on strategies and use precise vocabulary. I will spend more time introducing strategies and necessary vocabulary at the beginning of the year and continue to focus on strategies and process in order to lower my students’ anxiety and thus increase their self confidence when it comes to doing mathematics, especially problem solving.