Nonlinear Resource Allocation Problem Research Articles

Distributed Algorithm Design for Nonsmooth and Nonlinear Resource Allocation Problems of Autonomous High-Order Agents and Its Application to Smart Grids

Distributed Algorithm Design for Nonsmooth and Nonlinear Resource Allocation Problems of Autonomous High-Order Agents and Its Application to Smart Grids

Integrating Statistical Simulation and Optimization for Redundancy Allocation in Smart Grid Infrastructure

It is a critical issue to allocate redundancy to critical smart grid infrastructure for disaster recovery planning. In this study, a framework to combine statistical prediction methods and optimization models for the optimal redundancy allocation problem is presented. First, statistical simulation methods to identify critical nodes of very large-scale smart grid infrastructure based on the topological features of embedding networks are developed, and then a linear integer programming model based on generalized assignment problem (GAP) for the redundancy allocation of critical nodes in smart grid infrastructure is presented. This paper aims to contribute to the field by employing a general redundancy allocation problem (GRAP) model from high-order nonlinear to linear model transformation. The model is specifically implemented in the context of smart grid infrastructure. The innovative linear integer programming model proposed in this paper capitalizes on the logarithmic multiplication property to reframe the inherently nonlinear resource allocation problem (RAP) into a linearly separable function. This reformulation markedly streamlines the problem, enhancing its suitability for efficient and effective solutions. The findings demonstrate that the combined approach of statistical simulation and optimization effectively addresses the size limitations inherent in a sole optimization approach. Notably, the optimal solutions for redundancy allocation in large grid systems highlight that the cost of redundancy is only a fraction of the economic losses incurred due to weather-related outages.

On the paper “Augmented Lagrangian algorithms for solving the continuous nonlinear resource allocation problem”

In the paper Torrealba et al. (2021) an augmented Lagrangian algorithm was proposed for resource allocation problems with the intriguing characteristic that instead of solving the box-constrained augmented Lagrangian subproblem, they propose projecting the solution of the unconstrained subproblem onto such box. A global convergence result for the quadratic case was provided, however, this is somewhat counterintuitive, as in usual augmented Lagrangian theory, this strategy can fail in solving the augmented Lagrangian subproblems. In this note we investigate further this algorithm and we show that the proposed method may indeed fail when the Hessian of the quadratic is not a multiple of the identity. In the paper, it is not clear enough that two different projections are being used: one for obtaining their convergence results and other in their implementation. However, despite the lack of theoretical convergence, their strategy works remarkably well in some classes of problems; thus, we propose a hybrid method which uses their idea as a starting point heuristics, switching to a standard augmented Lagrangian method under certain conditions. Our contribution consists in presenting an efficient way of determining when the heuristics is failing to improve the KKT residual of the problem, suggesting that the heuristic procedure should be abandoned. Numerical results are provided showing that this strategy is successful in accelerating the standard method.

Design of ANN Based Non-Linear Network Using Interconnection of Parallel Processor

Suspicious mass traffic constantly evolves, making network behaviour tracing and structure more complex. Neural networks yield promising results by considering a sufficient number of processing elements with strong interconnections between them. They offer efficient computational Hopfield neural networks models and optimization constraints used by undergoing a good amount of parallelism to yield optimal results. Artificial neural network (ANN) offers optimal solutions in classifying and clustering the various reels of data, and the results obtained purely depend on identifying a problem. In this research work, the design of optimized applications is presented in an organized manner. In addition, this research work examines theoretical approaches to achieving optimized results using ANN. It mainly focuses on designing rules. The optimizing design approach of neural networks analyzes the internal process of the neural networks. Practices in developing the network are based on the interconnections among the hidden nodes and their learning parameters. The methodology is proven best for nonlinear resource allocation problems with a suitable design and complex issues. The ANN proposed here considers more or less 46k nodes hidden inside 49 million connections employed on full-fledged parallel processors. The proposed ANN offered optimal results in real-world application problems, and the results were obtained using MATLAB.

Robust Secure Resource Allocation for RIS-Aided SWIPT Communication Systems.

Aiming at the influence of channel uncertainty, user information leakage and harvested energy improvement, this paper proposes a robust resource allocation algorithm for reconfigurable intelligent reflector (RIS) multiple-input single-output systems based on imperfect channel state information. First, considering the legal user minimum secret rate constraint, the base station maximum transmit power constraint and the RIS phase shift constraint with the bounded channel uncertainty, a joint optimization of the base station active beam, energy beam and RIS phase shift is established. A multivariate coupled nonlinear resource allocation problem for matrices is addressed. Then, using S-procedure and alternating optimization methods, the original non-convex problem is transformed into a deterministic convex optimization problem and an alternating optimization algorithm based on continuous convex approximation is proposed. The simulation results show that the proposed algorithm has better fairness harvested energy compared with the traditional robust algorithm.

Augmented Lagrangian algorithms for solving the continuous nonlinear resource allocation problem

We propose a class of algorithms for solving the continuous nonlinear resource allocation problem which is stated many times in the literature as the Knapsack problem. This problem is known for its diverse gamma of applications and we solve it by using a hybrid approach, i.e., we combine the augmented Lagrangian method with Newton’s method to solve the subproblem generated by it. In other words, at each step we minimize the augment ed Lagrangian using Newton’s method and project the solution on the box. Most of the papers in this area deal with quadratic separable problems. Our proposal is more general in the sense that the problem can be non-quadratic and non-separable. We present and discuss the convergence properties for the proposed method and we show numerical applications illustrating its competitiveness and robustness for solving different Knapsack problems.

An Exact Algorithm for Large-Scale Continuous Nonlinear Resource Allocation Problems with Minimax Regret Objectives

We study a large-scale resource allocation problem with a convex, separable, not necessarily differentiable objective function that includes uncertain parameters falling under an interval uncertainty set, considering a set of deterministic constraints. We devise an exact algorithm to solve the minimax regret formulation of this problem, which is NP-hard, and we show that the proposed Benders-type decomposition algorithm converges to an [Formula: see text]-optimal solution in finite time. We evaluate the performance of the proposed algorithm via an extensive computational study, and our results show that the proposed algorithm provides efficient solutions to large-scale problems, especially when the objective function is differentiable. Although the computation time takes longer for problems with nondifferentiable objective functions as expected, we show that good quality, near-optimal solutions can be achieved in shorter runtimes by using our exact approach. We also develop two heuristic approaches, which are partially based on our exact algorithm, and show that the merit of the proposed exact approach lies in both providing an [Formula: see text]-optimal solution and providing good quality near-optimal solutions by laying the foundation for efficient heuristic approaches.

Optimal coinsurance rates for a heterogeneous population under inequality and resource constraints

Although operations research has contributed heavily to the derivation of optimal treatment guidelines for chronic diseases, patient adherence to treatment plans is low and variable. One mechanism for improving patient adherence to guidelines is to tailor coinsurance rates for prescription medications to patient characteristics. We seek to find coinsurance rates that maximize the welfare of the heterogeneous patient population at risk for cardiovascular disease. We analyze the problem as a bilevel optimization model where the lower optimization problem has the structure of a Markov decision process that determines the optimal treatment plan for each patient class. The upper optimization problem is a nonlinear resource allocation problem with constraints on total expenditures and coinsurance inequality. We used dynamic programming with a penalty function for nonseparable constraint violations to derive the optimal coinsurance rates. We parameterized and solved this model by considering patients who are insured by Medicare and are prescribed medications for prevention of cardiovascular disease. We find that optimizing coinsurance rates can be a cost-effective intervention for improving patient adherence and health outcomes, particularly for those patients at high risk for cardiovascular disease.

Solving stochastic nonlinear resource allocation problems using continuous learning automata

This paper deals with the Stochastic Non-linear Fractional Equality Knapsack (NFEK) problem which is a fundamental resource allocation problem based on incomplete and noisy information (Granmo and Oommen, Appl Intell 33(1):3–20, 2010, IEEE Trans Comput 59(4):545–560, 2010). The NFEK problem arises in many applications such as in web polling under polling constraints, and in constrained estimation. The primary contribution of this paper is a continuous Learning Automata (LA)-based, optimal, efficient and yet simple solution to the NFEK problem. Our solution is distinct from the first-reported optimal solution to the problem due to Granmo and Oommen (Appl Intell 33(1):3–20, 2010, IEEE Trans Comput 59(4):545–560, 2010) which resorts to utilizing multiple two-action discretized LA, organized in a hierarchical manner which comes with extra implementation and computational complexity. In this work, we present an optimal solution to the problem using a continuous LA which does not involve mapping the materials onto a binary hierarchy. As opposed to the traditional family of Reward-Inaction (R-I) LA, our scheme is modified in order to accommodate non-absorbing barriers, thus guaranteeing convergence to the optimal allocation. The experimental results that we have presented for numerous simulations demonstrate the efficiency of our scheme and its superiority compared to the state-of-the-art in terms of peak performance.

Stochastic nonlinear resource allocation problem

In this paper, we deal with a resource allocation problem modeled as special case of 0-1 Quadratic Programs with joint probabilistic rectangular constraints (QPJPC) with normally distributed coefficients and independent matrix vector rows. We reformulate this problem as a completely positive problem. In addition, the optimal value of the latter problem converges to the optimal value of the original problem under certain conditions. Numerical experiments on randomly generated data are given.

Algorithms for the continuous nonlinear resource allocation problem—New implementations and numerical studies

Patriksson (2008) provided a then up-to-date survey on the continuous, separable, differentiable and convex resource allocation problem with a single resource constraint. Since the publication of that paper the interest in the problem has grown: several new applications have arisen where the problem at hand constitutes a subproblem, and several new algorithms have been developed for its efficient solution. This paper therefore serves three purposes. First, it provides an up-to-date extension of the survey of the literature of the field, complementing the survey in Patriksson (2008) with more then 20 books and articles. Second, it contributes improvements of some of these algorithms, in particular with an improvement of the pegging (that is, variable fixing) process in the relaxation algorithm, and an improved means to evaluate subsolutions. Third, it numerically evaluates several relaxation (primal) and breakpoint (dual) algorithms, incorporating a variety of pegging strategies, as well as a quasi-Newton method. Our conclusion is that our modification of the relaxation algorithm performs the best. At least for problem sizes up to 30 million variables the practical time complexity for the breakpoint and relaxation algorithms is linear.

Solving the continuous nonlinear resource allocation problem with an interior point method

Resource allocation problems are usually solved with specialized methods exploiting their general sparsity and problem-specific algebraic structure. We show that the sparsity structure alone yields a closed-form Newton search direction for the generic primal-dual interior point method. Computational tests show that the interior point method consistently outperforms the best specialized methods when no additional algebraic structure is available.

A Newton-Type Algorithm for Solving Problems of Search Theory

In the survey of the continuous nonlinear resource allocation problem, Patriksson pointed out that Newton-type algorithms have not been proposed for solving the problem of search theory in the theoretical perspective. In this paper, we propose a Newton-type algorithm to solve the problem. We prove that the proposed algorithm has global and superlinear convergence. Some numerical results indicate that the proposed algorithm is promising.

Solving Stochastic Nonlinear Resource Allocation Problems Using a Hierarchy of Twofold Resource Allocation Automata

In a multitude of real-world situations, resources must be allocated based on incomplete and noisy information. However, in many cases, incomplete and noisy information render traditional resource allocation techniques ineffective. The decentralized Learning Automata Knapsack Game (LAKG) was recently proposed for solving one such class of problems, namely the class of Stochastic Nonlinear Fractional Knapsack Problems. Empirically, the LAKG was shown to yield a superior performance when compared to methods which are based on traditional parameter estimation schemes. This paper presents a completely new online Learning Automata (LA) system, namely the Hierarchy of Twofold Resource Allocation Automata (H-TRAA). In terms of contributions, we first of all, note that the primitive component of the H-TRAA is a Twofold Resource Allocation Automaton (TRAA) which possesses novelty in the field of LA. Second, the paper contains a formal analysis of the TRAA, including a rigorous proof for its convergence. Third, the paper proves the convergence of the H-TRAA itself. Finally, we demonstrate empirically that the H-TRAA provides orders of magnitude faster convergence compared to the LAKG for simulated data pertaining to two-material unit-value functions. Indeed, in contrast to the LAKG, the H-TRAA scales sublinearly. Consequently, we believe that the H-TRAA opens avenues for handling demanding real-world applications such as the allocation of sampling resources in large-scale Web accessibility assessment problems. We are currently working on applying the H-TRAA solution to the web-polling and sample-size detection problems applicable to the world wide web.

A survey on the continuous nonlinear resource allocation problem

Our problem of interest consists of minimizing a separable, convex and differentiable function over a convex set, defined by bounds on the variables and an explicit constraint described by a separable convex function. Applications are abundant, and vary from equilibrium problems in the engineering and economic sciences, through resource allocation and balancing problems in manufacturing, statistics, military operations research and production and financial economics, to subproblems in algorithms for a variety of more complex optimization models. This paper surveys the history and applications of the problem, as well as algorithmic approaches to its solution. The most common techniques are based on finding the optimal value of the Lagrange multiplier for the explicit constraint, most often through the use of a type of line search procedure. We analyze the most relevant references, especially regarding their originality and numerical findings, summarizing with remarks on possible extensions and future research.

A particle swarm optimization approach to the nonlinear resource allocation problem

The resource allocation problem seeks to find an optimal allocation of a limited amount of resource to a number of activities for optimizing the objective under the resource constraint. Most existing methods use mathematical programming techniques, but they may fail to derive exact solutions for large-sized problems with reasonable time. An alternative is to use meta-heuristic algorithms for obtaining approximate solutions. This paper presents a particle swarm optimization (PSO) algorithm for conquering the nonlinear resource allocation problem. To ensure the resource constraint is satisfied, we propose adaptive resource bounds for guiding the search. The experimental results manifest that the proposed method is more effective and efficient than a genetic algorithm. The convergence behavior of the proposed method is analyzed by observing the variations of particle entropy. Finally, a worst-case analysis is conducted to provide a reliable performance guarantee.

Ant colony optimization for the nonlinear resource allocation problem

The nonlinear resource allocation problem addresses the important issue which seeks to find an optimal allocation of a limited amount of resource to a number of tasks for optimizing a nonlinear objective over the given resource constraint. Relevant literature has been focused on the use of mathematical programming approaches, few researches based on meta-heuristic algorithms have been conducted. In this paper we present an ant colony optimization algorithm for conquering the nonlinear resource allocation problem. To ensure the resource constraint is satisfied, we incorporate adaptive resource bounds to guide the search. The experimental results manifest that the proposed method is more effective and efficient than a genetic algorithm. Also, our method converges at a fast rate and a reliable performance guarantee is provided through a worst-case analysis.

A specially structured nonlinear integer resource allocation problem

AbstractWe present an algorithm for solving a specially structured nonlinear integer resource allocation problem. This problem was motivated by a capacity planning study done at a large Health Maintenance Organization in Texas. Specifically, we focus on a class of nonlinear resource allocation problems that involve the minimization of a convex function over one general convex constraint, a set of block diagonal convex constraints, and bounds on the integer variables. The continuous variable problem is also considered. The continuous problem is solved by taking advantage of the structure of the Karush‐Kuhn‐Tucker (KKT) conditions. This method for solving the continuous problem is then incorporated in a branch and bound algorithm to solve the integer problem. Various reoptimization results, multiplier bounding results, and heuristics are used to improve the efficiency of the algorithms. We show how the algorithms can be extended to obtain a globally optimal solution to the nonconvex version of the problem. We further show that the methods can be applied to problems in production planning and financial optimization. Extensive computational testing of the algorithms is reported for a variety of applications on continuous problems with up to 1,000,000 variables and integer problems with up to 1000 variables. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 770–792, 2003.

The nonlinear knapsack problem – algorithms and applications

We present a survey of algorithms and applications for the nonlinear knapsack problem (or, the nonlinear resource allocation problem). In its most general form, the nonlinear knapsack problem will be defined as a nonlinear optimization problem with just one constraint, bounds on the variables, and, in some cases, a set of specially structured constraints such as generalized upper bounds (GUBs). This problem is encountered either directly, or as a subproblem, in a variety of applications, including production planning, financial modeling, stratified sampling, and capacity planning in manufacturing, health care, and computer networks. By taking advantage of the special structure of the problem, efficient solution methods can be developed. Problem classes addressed here include continuous and integer problems, convex and nonconvex problems, separable and nonseparable problems, and problems with additional specially structured constraints.

A pegging algorithm for the nonlinear resource allocation problem

In this paper we present a new algorithm for solving the nonlinear resource allocation problem. The nonlinear resource allocation problem is defined as the minimization of a convex function over a single convex constraint and bounded integer variables. We first present a pegging algorithm for solving the continuous variable problem, and then incorporate the pegging method in a branch and bound algorithm for solving the integer variable problem. We compare the computational performance of the pegging branch and bound algorithm with three other methods: a multiplier search branch and bound algorithm, dynamic programming, and a 0,1 linearization method. The computational results demonstrate that the pegging branch and bound algorithm advances the state of the art in methods for solving the nonlinear resource allocation problem. Scope and purpose The nonlinear resource allocation problem (i.e., a knapsack problem where both the objective function and single constraint may be nonlinear) is encountered in a variety of applications, including financial modeling, sampling, production and inventory management, and queueing network models of manufacturing and computer systems. Despite its importance to these and other applications, the nonlinear resource allocation problem has received limited attention in the literature. Therefore, we develop methods for solving continuous and integer variable versions of the problem and report extensive computational testing of the algorithms.