Non-linear Fractional Knapsack Problem Research Articles

Learning automata-based solutions to the optimal web polling problem modelled as a nonlinear fractional knapsack problem

We consider the problem of polling web pages as a strategy for monitoring the world wide web. The problem consists of repeatedly polling a selection of web pages so that changes that occur over time are detected. In particular, we consider the case where we are constrained to poll a maximum number of web pages per unit of time, and this constraint is typically dictated by the governing communication bandwidth, and by the speed limitations associated with the processing. Since only a fraction of the web pages can be polled within a given unit of time, the issue at stake is one of determining which web pages are to be polled, and we attempt to do it in a manner that maximizes the number of changes detected. We solve the problem by first modelling it as a stochastic nonlinear fractional knapsack problem. We then present an online learning automata (LA) system, namely, the hierarchy of twofold resource allocation automata (H-TRAA), whose primitive component is a twofold resource allocation automaton (TRAA). Both the TRAA and the H-TRAA have been proven to be asymptotically optimal. Finally, we demonstrate empirically that the H-TRAA provides orders of magnitude faster convergence compared to the learning automata knapsack game (LAKG) which represents the state-of-the-art for this problem. Further, in contrast to the LAKG, the H-TRAA scales sub-linearly. Based on these results, we believe that the H-TRAA has also tremendous potential to handle demanding real-world applications, particularly those which deal with the world wide web.

Optimal sampling for estimation with constrained resources using a learning automaton-based solution for the nonlinear fractional knapsack problem

While training and estimation for Pattern Recognition (PR) have been extensively studied, the question of achieving these when the resources are both limited and constrained is relatively open. This is the focus of this paper. We consider the problem of allocating limited sampling resources in a “real-time” manner, with the explicit purpose of estimating multiple binomial proportions (the extension of these results to non-binomial proportions is, in our opinion, rather straightforward). More specifically, the user is presented with ‘n’ training sets of data points, S 1,S 2,…,S n , where the set S i has N i points drawn from two classes {ω 1,ω 2}. A random sample in set S i belongs to ω 1 with probability u i and to ω 2 with probability 1−u i , with {u i }, i=1,2,…n, being the quantities to be learnt. The problem is both interesting and non-trivial because while both n and each N i are large, the number of samples that can be drawn is bounded by a constant, c. A web-related problem which is based on this model (Snaprud et al., The Accessibility for All Conference, 2003) is intriguing because the sampling resources can only be allocated optimally if the binomial proportions are already known. Further, no non-automaton solution has ever been reported if these proportions are unknown and must be sampled.Using the general LA philosophy as a paradigm to tackle this real-life problem, our scheme improves a current solution in an online manner, through a series of informed guesses which move towards the optimal solution. We solve the problem by first modelling it as a Stochastic Non-linear Fractional Knapsack Problem. We then present a completely new on-line Learning Automata (LA) system, namely, the Hierarchy of Twofold Resource Allocation Automata (H-TRAA), whose primitive component is a Twofold Resource Allocation Automaton (TRAA), both of which are asymptotically optimal. Furthermore, we demonstrate empirically that the H-TRAA provides orders of magnitude faster convergence compared to the Learning Automata Knapsack Game (LAKG) which represents the state-of-the-art. Finally, in contrast to the LAKG, the H-TRAA scales sub-linearly. Based on these results, we believe that the H-TRAA has also tremendous potential to handle demanding real-world applications, particularly those dealing with the world wide web.

A memetic model of evolutionary PSO for computational finance applications

Motivated by the compensatory property of EA and PSO, where the latter can enhance solutions generated from the evolutionary operations by exploiting their individual memory and social knowledge of the swarm, this paper examines the implementation of PSO as a local optimizer for fine tuning in evolutionary search. The proposed approach is evaluated on applications from the field of computational finance, namely portfolio optimization and time series forecasting. Exploiting the structural similarity between these two problems and the non-linear fractional knapsack problem, an instance of the latter is generalized and implemented as the preliminary test platform for the proposed EA–PSO hybrid model. The experimental results demonstrate the positive effects of this memetic synergy and reveal general design guidelines for the implementation of PSO as a local optimizer. Algorithmic performance improvements are similarly evident when extending to the real-world optimization problems under the appropriate integration of PSO with EA.

Learning Automata-Based Solutions to the Nonlinear Fractional Knapsack Problem With Applications to Optimal Resource Allocation

This paper considers the nonlinear fractional knapsack problem and demonstrates how its solution can be effectively applied to two resource allocation problems dealing with the World Wide Web. The novel solution involves a "team" of deterministic learning automata (LA). The first real-life problem relates to resource allocation in web monitoring so as to "optimize" information discovery when the polling capacity is constrained. The disadvantages of the currently reported solutions are explained in this paper. The second problem concerns allocating limited sampling resources in a "real-time" manner with the purpose of estimating multiple binomial proportions. This is the scenario encountered when the user has to evaluate multiple web sites by accessing a limited number of web pages, and the proportions of interest are the fraction of each web site that is successfully validated by an HTML validator. Using the general LA paradigm to tackle both of the real-life problems, the proposed scheme improves a current solution in an online manner through a series of informed guesses that move toward the optimal solution. At the heart of the scheme, a team of deterministic LA performs a controlled random walk on a discretized solution space. Comprehensive experimental results demonstrate that the discretization resolution determines the precision of the scheme, and that for a given precision, the current solution (to both problems) is consistently improved until a nearly optimal solution is found--even for switching environments. Thus, the scheme, while being novel to the entire field of LA, also efficiently handles a class of resource allocation problems previously not addressed in the literature.

Experiments with dynamic programming algorithms for nonseparable problems

In this paper we report on numerical experiments with dynamic programming algorithms designed for solving problems with nonseparable objective functions of the C-programming type. The experiments were conducted with the view to evaluate the efficiency of C-programming's exclusionary rules when they are incorporated in the conventional parametric Lagrangian search method. Two classical nonseparable problems are considered: (1) the variance minimization problem and (2) the nonlinear fractional knapsack problem. The results indicate that the incorporation of C-programming's exclusionary rules in the Lagrangian procedure significantly improves the efficiency of the line search procedure.