Optimal multiple-objective resource allocation using hybrid particle swarm optimization and adaptive resource bounds technique

AbstractThe multiple-objective resource allocation problem (MORAP) seeks for an allocation of resource to a number of activities such that a set of objectives are optimized simultaneously and the resource constraints are satisfied. MORAP has many applications, such as resource distribution, project budgeting, software testing, health care resource allocation, etc. This paper addresses the nonlinear MORAP with integer decision variable constraint. To guarantee that all the resource constraints are satisfied, we devise an adaptive-resource-bound technique to construct feasible solutions. The proposed method employs the particle swarm optimization (PSO) paradigm and presents a hybrid execution plan which embeds a hill-climbing heuristic into the PSO for expediting the convergence. To cope with the optimization problem with multiple objectives, we evaluate the candidate solutions based on dominance relationship and a score function. Experimental results manifest that the hybrid PSO derives solution sets which are very close to the exact Pareto sets. The proposed method also outperforms several representatives of the state-of-the-art algorithms on a simulation data set of the MORAP

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

A Class of Convex Quadratic Nonseparable Resource Allocation Problems with Generalized Bound Constraints

We study a convex quadratic nonseparable resource allocation problem that arises in the area of decentralized energy management (DEM), where unbalance in electricity networks has to be minimized. In this problem, the given resource is allocated over a set of activities that is divided into subsets, and a cost is assigned to the overall allocated amount of resources to activities within the same subset. We derive two efficient algorithms with $O(n \log  n)$ worst-case time complexity to solve this problem. For the special case where all subsets have the same size, one of these algorithms even runs in linear time given the subset size. Both algorithms are inspired by well-studied breakpoint search methods for separable convex resource allocation problems. Numerical evaluations on both real and synthetic data confirm the theoretical efficiency of both algorithms and demonstrate their suitability for integration in DEM systems

Surrogate dual search in nonlinear integer programming.

Wang, Chongyu.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (leaves 74-78).Abstract also in Chinese.Abstract --- p.1Abstract in Chinese --- p.3Acknowledgement --- p.4Contents --- p.5List of Tables --- p.7List of Figures --- p.8Chapter 1. --- Introduction --- p.9Chapter 2. --- Conventional Dynamic Programming --- p.15Chapter 2.1. --- Principle of optimality and decomposition --- p.15Chapter 2.2. --- Backward dynamic programming --- p.17Chapter 2.3. --- Forward dynamic programming --- p.20Chapter 2.4. --- Curse of dimensionality --- p.23Chapter 3. --- Surrogate Constraint Formulation --- p.26Chapter 3.1. --- Surrogate constraint formulation --- p.26Chapter 3.2. --- Singly constrained dynamic programming --- p.28Chapter 3.3. --- Surrogate dual search --- p.29Chapter 4. --- Distance Confined Path Algorithm --- p.34Chapter 4.1. --- Yen´ةs algorithm for the kth shortest path problem --- p.35Chapter 4.2. --- Application of Yen´ةs method to integer programming --- p.36Chapter 4.3. --- Distance confined path problem --- p.42Chapter 4.4. --- Application of distance confined path formulation to integer programming --- p.50Chapter 5. --- Convergent Surrogate Dual Search --- p.59Chapter 5.1. --- Algorithm for convergent surrogate dual search --- p.62Chapter 5.2. --- "Solution schemes for (Pμ{αk,αβ)) and f(x) = αk" --- p.63Chapter 5.3. --- Computational Results and Analysis --- p.68Chapter 6. --- Conclusions --- p.72Bibliography --- p.7

Sensor Location For Network Flow And Origin-Destination Estimation With Multiple Vehicle Classes

The need for multi-class origin-destination (O-D) estimation and link volume estimation requires multi-class observations from sensors. This dissertation has established a new sensor location model that includes: 1) multiple vehicle classes; 2) a variety of data types from different types of sensors; and 3) a focus on both link-based and O-D based flow estimation. The model seeks a solution that maximizes the overall information content from sensors, subject to a budget constraint. An efficient twophase metaheuristic algorithm is developed to solve the problem. The model is based on a set of linear equations that connect O-D flows, link flows and sensor observations. Concepts from Kalman filtering are used to define the information content from a set of sensors as the trace of the posterior covariance matrix of flow estimates, and to create a linear update mechanism for the precision matrix as new sensors are added or deleted from the solution set. Sensor location decisions are nonlinearly related to information content because the precision matrix must be inverted to construct the covariance matrix which is the basis for measuring information. The resulting model is a nonlinear knapsack problem. The two-phase search algorithm proposed addresses this nonlinear, nonseparable integer sensor location problem. A greedy phase generates an initial solution, feeding into a Tabu Search phase which swaps sensors along the budget constraint. The neighbor generation in Tabu search is a combination of a fixed swapout strategy with a guided random swap-in strategy. Extensive computational experiments have been performed on a standard test network. These tests verify the effectiveness of the problem formulation and solution algorithm. A case study on Rockland County, NY demonstrates that the sensor location method developed in this dissertation can successfully allocate sensors in realistic networks, and thus has significant practical value

Indefinite Knapsack Separable Quadratic Programming: Methods and Applications

Quadratic programming (QP) has received significant consideration due to an extensive list of applications. Although polynomial time algorithms for the convex case have been developed, the solution of large scale QPs is challenging due to the computer memory and speed limitations. Moreover, if the QP is nonconvex or includes integer variables, the problem is NP-hard. Therefore, no known algorithm can solve such QPs efficiently. Alternatively, row-aggregation and diagonalization techniques have been developed to solve QP by a sub-problem, knapsack separable QP (KSQP), which has a separable objective function and is constrained by a single knapsack linear constraint and box constraints. KSQP can therefore be considered as a fundamental building-block to solve the general QP and is an important class of problems for research. For the convex KSQP, linear time algorithms are available. However, if some quadratic terms or even only one term is negative in KSQP, the problem is known to be NP-hard, i.e. it is notoriously difficult to solve. The main objective of this dissertation is to develop efficient algorithms to solve general KSQP. Thus, the contributions of this dissertation are five-fold. First, this dissertation includes comprehensive literature review for convex and nonconvex KSQP by considering their computational efficiencies and theoretical complexities. Second, a new algorithm with quadratic time worst-case complexity is developed to globally solve the nonconvex KSQP, having open box constraints. Third, the latter global solver is utilized to develop a new bounding algorithm for general KSQP. Fourth, another new algorithm is developed to find a bound for general KSQP in linear time complexity. Fifth, a list of comprehensive applications for convex KSQP is introduced, and direct applications of indefinite KSQP are described and tested with our newly developed methods. Experiments are conducted to compare the performance of the developed algorithms with that of local, global, and commercial solvers such as IBM CPLEX using randomly generated problems in the context of certain applications. The experimental results show that our proposed methods are superior in speed as well as in the quality of solutions

Bydraes tot die oplossing van die veralgemeende knapsakprobleem

Text in AfikaansIn this thesis contributions to the solution of the generalised knapsack problem are given and discussed. Attention is given to problems with functions that are calculable but not necessarily in a closed form. Algorithms and test problems can be used for problems with closed-form functions as well. The focus is on the development of good heuristics and not on exact algorithms. Heuristics must be investigated and good test problems must be designed. A measure of convexity for convex functions is developed and adapted for concave functions. A test problem generator makes use of this measure of convexity to create challenging test problems for the concave, convex and mixed knapsack problems. Four easy-to-interpret characteristics of an S-function are used to create test problems for the S-shaped as well as the generalised knapsack problem. The in uence of the size of the problem and the funding ratio on the speed and the accuracy of the algorithms are investigated. When applicable, the in uence of the interval length ratio and the ratio of concave functions to the total number of functions is also investigated. The Karush-Kuhn-Tucker conditions play an important role in the development of the algorithms. Suf- cient conditions for optimality for the convex knapsack problem with xed interval lengths is given and proved. For the general convex knapsack problem, the key theorem, which contains the stronger necessary conditions, is given and proved. This proof is so powerful that it can be used to proof the adapted key theorems for the mixed, S-shaped and the generalised knapsack problems as well. The exact search-lambda algorithm is developed for the concave knapsack problem with functions that are not in a closed form. This algorithm is used in the algorithms to solve the mixed and S-shaped knapsack problems. The exact one-step algorithm is developed for the convex knapsack problem with xed interval length. This algorithm is O(n). The general convex knapsack problem is solved by using the pivot algorithm which is O(n2). Optimality cannot be proven but in all cases the optimal solution was found and for all practical reasons this problem will be considered as being concluded. A good heuristic is developed for the mixed knapsack problem. Further research can be done on this heuristic as well as on the S-shaped and generalised knapsack problems.Mathematical SciencesD. Phil. (Operasionele Navorsing