An Exact Algorithm for the Two-Constraint 0–1 Knapsack Problem

In this paper, we consider the network slicing () problem which aims to map multiple customized virtual network requests (also called services) to a common shared network infrastructure and manage network resources to meet diverse quality of service (QoS) requirements. We propose a mixed-integer nonlinear programming (MINLP) formulation for the considered NS problem that can flexibly route the traffic flow of the services on multiple paths and provide end-to-end delay and reliability guarantees for all services. To overcome the computational difficulty due to the intrinsic nonlinearity in the MINLP formulation, we transform the formulation into an equivalent mixed-integer linear programming () formulation and further show that their continuous relaxations are equivalent. In sharp contrast to the continuous relaxation of the formulation which is a nonconvex nonlinear programming problem, the continuous relaxation of the formulation is a polynomial-time solvable linear programming problem, which significantly facilitates the algorithmic design. Based on the newly proposed formulation, we develop a customized column generation () algorithm for solving the problem. The proposed algorithm is a decomposition-based algorithm and is particularly suitable for solving large-scale problems. Numerical results demonstrate the efficacy of the proposed formulations and the proposed algorithm.

In this paper, a branch-and-bound method for solving multi-dimensional quadratic 0–1 knapsack problems was studied. The method was based on the Lagrangian relaxation and the surrogate constraint technique for finding feasible solutions. The Lagrangian relaxations were solved with the maximum-flow algorithm and the Lagrangian bounds was determined with the outer approximation method. Computational results show the efficiency of the proposed method for multi-dimensional quadratic 0–1 knapsack problems.

In this paper, a new branch-and-bound algorithm based on the Lagrangian dual relaxation and continuous relaxation is proposed for discrete multi-factor portfolio selection model with roundlot restriction in financial optimization. This discrete portfolio model is of integer quadratic programming problems. The separable structure of the model is investigated by using Lagrangian relaxation and dual search. Computational results show that the algorithm is capable of solving real-world portfolio problems with data from US stock market and randomly generated test problems with up to 120 securities.

This paper presents an application of a combined genetic algorithms (GAs) and Lagrangian relaxation (LR) method for the unit commitment (UC) problem. Genetic algorithms (GAs) are a general purpose optimization technique based on principle of natural selection and natural genetics. The Lagrangian relaxation (LR) method provides a fast solution but it may suffer from numerical convergence and solution quality problems. The proposed Lagrangian relaxation and genetic algorithms (LRGA) incorporates genetic algorithms into Lagrangian relaxation method to update the Lagrangian multipliers and improve the performance of Lagrangian relaxation method in solving combinatorial optimization problems such as the UC problem. Numerical results on two cases including a system of 100 units and comparisons with results obtained using Lagrangian relaxation (LR) and genetic algorithms (GAs), show that the feature of easy implementation, better convergence, and highly near-optimal solution to the UC problem can be achieved by the LRGA.

In this paper the problem of optimally using object model information in image reconstruction is addressed. A closed form solution for the estimated object spectrum is derived using the Lagrange multiplier technique which assumes a measured image, knowledge of the optical transfer function, statistical information about the measurement noise, and a model of the object. This reconstruction algorithm is iterative in nature for two reasons: (1) because the optimal Lagrange multiplier is not generally known at the start of the problem; and (2) we can use the object estimate obtained from one step of the algorithm as the model input for the next step. In this paper we derive the estimator, describe one technique for determining the optimal Lagrange multiplier, demonstrate a stopping criterion based on the mean squared error between a noise free image and the photon-limited version of the image, and show representative results for a sparse aperture imaging application.

This paper examines the bit and power allocation problem for orthogonal frequency division multiplexing systems in which the overall transmission power is minimized by constraining the fixed data rate and bit error rate. To provide the optimal allocation with less computational complexity, we propose new bit and power allocation schemes based on the Lagrangian method. Firstly, we propose an initial search range of the bisection search method to find the optimal Lagrangian multiplier efficiently. The simulation results verify that the proposed initial search range guarantees the optimal solution with less computational complexity. Secondly, a new iterative search method for the optimal Lagrangian multiplier is proposed using Newton's search method. The simulation results demonstrate that the proposed scheme has significant computational advantages over the conventional algorithms while providing optimal performance.

Recent work has established Lagrangian relaxation (LR) based gate sizing as state-of-the-art providing the best power reduction with low run time. Gate sizing has limited potential to reduce the power when the timing constraints are tight. By adjusting the arrival times of clock signals (clock skew scheduling), the timing constraints can be relaxed facilitating more power reduction.Previous LR attempts at simultaneous gate sizing and skew scheduling solved a minimum-cost network flow problem for updating the Lagrange multipliers in each LR iteration, and for optimality assumed continuous sizes with convex delay models. We propose an alternative approach, modifying a LR discrete gate sizing formulation with table lookup non-convex delay models, which are more accurate for modern process technologies. For the Lagrange multiplier update, we use a projection heuristic that is much faster than solving the minimum cost network flow problem.On the ISPD 2012 gate sizing contest benchmark suite, our proposed approach outperforms the previous min-cost flow based approach by saving 5.3% more power and is 70x faster. Compared to sizing alone with the state-of-the-art LR gate sizer, skew scheduling with sizing saves 19.7% more power with a small runtime penalty.

In this paper we present an evolutionary algorithm for the multidimensional 0–1 knapsack problem. Our algorithm incorporates a heuristic operator which computes problem-specific knowledge. The design of this operator is based on the general technique used to design greedy-like heuristics for this problem, that is, the surrogate multipliers approach of Pirkul (see [7]). The main difference with work previously done is that our heuristic operator is computed following a genetic strategy -suggested by the greedy solution of the one dimensional knapsack problem- instead of the commonly used simplex method. Experimental results show that our evolutionary algorithm is capable of obtaining high quality solutions for large size problems requiring less amount of computational effort than other evolutionary strategies supported by heuristics founded on linear programming calculation of surrogate multipliers.

Shuffled frog leaping algorithm and its application to 0/1 knapsack problem

The advancements of mobile devices, public networks and the Internet of creature huge amounts of complex data, both construct & unstructured are being captured in trust to allow organizations to produce better business decisions as data is now pivotal for an organizations success. These enormous amounts of data are referred to as Big Data, which enables a competitive advantage over rivals when processed and analyzed appropriately. However Big Data Analytics has a few concerns including Management of Data, Privacy & Security, getting optimal path for transport data, and Data Representation. However, the structure of network does not completely match transportation demand, i.e., there still exist a few bottlenecks in the network. This paper presents a new approach to get the optimal path of valuable data movement through a given network based on the knapsack problem. This paper will give value for each piece of data, it depends on the importance of this data (each piece of data defined by two arguments size and value), and the approach tries to find the optimal path from source to destination, a mathematical models are developed to adjust data flows between their shortest paths based on the 0 - 1 knapsack problem. We also take out computational experience using the commercial software Gurobi and a greedy algorithm (GA), respectively. The outcome indicates that the suggest models are active and workable. This paper introduced two different algorithms to study the shortest path problems: the first algorithm studies the shortest path problems when stochastic activates and activities does not depend on weights. The second algorithm studies the shortest path problems depends on weights.

Growing evidence indicates that only a sparse subset from a pool of sensory neurons is active for the encoding of visual stimuli at any instant in time. Traditionally, to replicate such biological sparsity, generative models have been using the ℓ1 norm as a penalty due to its convexity, which makes it amenable to fast and simple algorithmic solvers. In this work, we use biological vision as a test-bed and show that the soft thresholding operation associated to the use of the ℓ1 norm is highly suboptimal compared to other functions suited to approximating ℓp with 0 ≤ p < 1 (including recently proposed continuous exact relaxations), in terms of performance. We show that ℓ1 sparsity employs a pool with more neurons, i.e. has a higher degree of overcompleteness, in order to maintain the same reconstruction error as the other methods considered. More specifically, at the same sparsity level, the thresholding algorithm using the ℓ1 norm as a penalty requires a dictionary of ten times more units compared to the proposed approach, where a non-convex continuous relaxation of the ℓ0 pseudo-norm is used, to reconstruct the external stimulus equally well. At a fixed sparsity level, both ℓ0- and ℓ1-based regularization develop units with receptive field (RF) shapes similar to biological neurons in V1 (and a subset of neurons in V2), but ℓ0-based regularization shows approximately five times better reconstruction of the stimulus. Our results in conjunction with recent metabolic findings indicate that for V1 to operate efficiently it should follow a coding regime which uses a regularization that is closer to the ℓ0 pseudo-norm rather than the ℓ1 one, and suggests a similar mode of operation for the sensory cortex in general.

The purpose of this paper is to provide strong reformulations for binary quadratic problems. We propose a first methodological analysis on a family of reformulations combining Dantzig–Wolfe and Quadratic Convex optimization principles. We show that a few reformulations of our family yield continuous relaxations that are strong in terms of dual bounds and computationally efficient to optimize. As a representative case study, we apply them to a cardinality constrained quadratic knapsack problem, providing extensive experimental insights. We report and analyze in depth a particular reformulation providing continuous relaxations whose solutions turn out to be integer optima in all our tests.

It has been reported for multi-objective knapsack problems that the recombination of similar parents often improves the performance of evolutionary multi-objective optimization (EMO) algorithms. Recently performance improvement was also reported by exchanging only a small number of genes between two parents (i.e., crossover with a very small gene exchange probability) without choosing similar parents. In this paper, we examine these performance improvement schemes through computational experiments where NSGA-II is applied to 500-item knapsack problems with 2-10 objectives. We measure the parent-parent distance and the parent-offspring distance in computational experiments. Clear performance improvement is observed when the parent-offspring distance is small. To further examine this observation, we implement a distance-based crossover operator where the parent-offspring distance is specified as a user-defined parameter. Performance of NSGA-II is examined for various parameter values. Experimental results show that an appropriate parameter value (parent-offspring distance) is surprisingly small. It is also shown that a very small parameter value is beneficial for diversity maintenance.

Among the most commonly occurring mixed-integer problems in operations research are linear programs with fixed charge objective functions. In this paper special structures of the equivalent form of such problems obtained from optimal solutions to their continuous relaxations are characterized and exploited in a series of penalty procedures for branch-and-bound type algorithms. The selection, construction, and solution of such penalty problems are discussed, and computational experience with the procedures is presented.

Augmented Lagrangian algorithms for solving the continuous nonlinear resource allocation problem