A branch-and-bound algorithm for multi-dimensional quadratic 0–1 knapsack problems

This paper is dedicated to a study of different extensions of the classical knapsack problem to the case when different elements of the problem formulation are subject to a degree of uncertainty described by random variables. This brings the knapsack problem into the realm of stochastic programming. Two different model formulations are proposed, based on the introduction of probability constraints. The first one is a static quadratic knapsack with a probability constraint on the capacity of the knapsack. The second one is a two-stage quadratic knapsack model, with recourse, where we introduce a probability constraint on the capacity of the knapsack in the second stage. As far as we know, this is the first time such a constraint has been used in a two-stage model. The solution techniques are based on the semidefinite relaxations. This allows for solving large instances, for which exact methods cannot be used. Numerical experiments on a set of randomly generated instances are discussed below.

We consider a knapsack problem in which each item has two types of weight and the container has two types of capacity. We discuss the surrogate, Lagrangian, and continuous relaxations, and give an effective method to determine the optimal Lagrangian and surrogate multipliers associated with the continuous relaxation of the problem. These results are used to obtain an exact branch-and-bound algorithm, which also includes heuristic procedures and a reduction technique. The performance of bounds and algorithms is evaluated through extensive computational experiments.

Many optimization problems encountered in practice are multi-objective by nature, i.e., different objectives are conflicting and equally important. Many times, it is not desirable to drop some of them or to optimize them in a composite single objective or hierarchical manner. Furthermore, cost parameters change over time which makes optimization problems harder. For instance, in the transport sector, travel costs are a function of travel time which changes depending on the time of the day a vehicle is travelling (e.g., due to road congestion). Road congestion results in tremendous delays which lead to a decrease in the service quality and the responsiveness of logistic service providers. In Chapter 2, we develop a generic approach to deal with Multi-Objective Scheduling Problems (MOSPs) with State-Dependent Cost Parameters. The aim is to determine the set of Pareto solutions that capture the trade offs between the different conflicting objectives. Due to the complexity of MOSPs, an efficient approximation based on dynamic programming is developed. The approximation has a provable worse case performance guarantee. Even though the generated approximate Pareto front consist of fewer solutions, it still represents a good coverage of the true Pareto front. Furthermore, considerable gains in computation times are achieved. In Chapter 3, the developed methodology is validated on the multi-objective timedependent knapsack problem. In the classical knapsack problem, the input consists of a knapsack with a finite capacity and a set of items, each with a certain weight and a cost. A feasible solution to the knapsack problem is a selection of items such that their total weight does not exceed the knapsack capacity. The goal is to maximize the single objective function consisting of the total pro t of the selected items. We extend the classical knapsack problem in two ways. First, we consider time-dependent profits (e.g., in a retail environment profit depends on whether it is Christmas or not).

Variants to a knapsack decomposition heuristic for solving R&D pipeline management problems

The application of valid inequalities to provide relaxations which can produce tight bounds, is now common practice in Combinatorial Optimisation. This paper attempts to complement current theoretical investigations in this regard. We experimentally search for "valid" equalities which have the potential of strengthening the problem's formulation. Recently, Martello and Toth included cardinality constraints to derive tight upper bounds for the 0-1 Knapsack Problem. Encouraged by their results, we partition the search space by using equality cardinality constraints. Instead of solving the original problem, an equivalent problem, which consists of one or more 0-1 Knapsack Problem with an exact cardinality bound, is solved. By explicitly including a bound on the cardinality, one is able to reduce the size of each subproblem and compute tight upper bounds. Good feasible solutions found along the way are employed to reduce the computational effort by reducing the number of trees searched and the size of the subsequent search trees. We give a brief description of two Lagrangian-based Branch-and-Bound algorithms proposed in Kruger for solving the exact cardinality bounded subproblems and report on results of numerical experiments with a sequential implementation. Implications for and strategies towards parallel implementation are also given.

Binary knapsack problem has received considerable attention in combinational optimization. Various meta-heuristic algorithms are dedicated to solve this problem in the literature. Recently, a binary flower pollination algorithm (BFPA) was proposed, which has been successfully applied to solve 0-1 knapsack problem. In this paper, two new time-varying transfer functions are proposed to improve the exploration and exploitation capability of the BFPA with the best solution and short computing time. Based on small, medium, and high-dimensional scales of the knapsack problem, the computational results reveal that the proposed time-varying transfer functions not only to find the best possible solutions but also to have less computational time. Compared to the standard transfer functions, the efficiency of the proposed time-varying transfer functions is superior, especially in the high-dimensional scales.

A novel canonical duality theory (CDT) is presented for solving general bilevel mixed integer nonlinear optimization governed by linear and quadratic knapsack problems. It shows that the challenging knapsack problems can be solved analytically in term of their canonical dual solutions. The existence and uniqueness of these analytical solutions are proved. NP-hardness of the knapsack problems is discussed. A powerful CDT algorithm combined with an alternative iteration and a volume reduction method is proposed for solving the NP-hard bilevel knapsack problems. Application is illustrated by benchmark problems in optimal topology design. The performance and novelty of the proposed method are compared with the popular commercial codes.

An improved Lagrangian relaxation (LR) solution to the thermal unit commitment problem (UCP) is proposed in this paper. The algorithm is characterized by: (1) a new Matlab function to determine the optimal path of the dual problem, (2) new initialization procedure of Lagrangian multipliers, based on both unit and time interval classification, (3) a flexible adjustment of Lagrangian multipliers, and (4) a dynamic search for uncertain stage scheduling, using a Lagrangian relaxation - dynamic programming method (LR-DP). After the LR best feasible solution is reached, and when identical or similar units exist, a unit decommitment is used to adjust the solution. The proposed algorithm is tested and compared to conventional Lagrangian relaxation (LR), genetic algorithm (GA), evolutionary programming (EP), Lagrangian relaxation and genetic algorithm (LRGA), and genetic algorithm based on unit characteristic classification (GAUC) on systems with the number of generating units in the range of 10 to 100. The total system production cost of the proposed algorithm is less than the others especially for the larger number of generating units. Computational time was found to increase almost linearly with system size, which is favorable for large-scale implementation.

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.

We present a novel technique for obtaining global solutions to discrete min-max problems that arise naturally in the receding horizon control of unmanned craft in which the controls can be adjusted only in notches, e.g., stop, half forward, full forward, left or right $$60^{\circ }$$60?, and in the finite precision global solution of certain classes of semi-infinite min-max problems. The technique consists of a method for transcribing min-max problems over discrete sets into a matrix game and matrix game-specific adaptations of the Method of Outer Approximations and the Method of Adaptive Approximations, which are normally used for solving optimal control and semi-infinite min-max problems. The efficiency of the Method of Outer Approximations depends on having a good initial approximation to a solution. To this end, we make use of adaptive approximation techniques to decompose a large matrix game into a sequence of lower dimensional games, the solution of each giving rise to a very good initial approximation to a solution for the next game. We show that a basic approach for solving a min-max matrix game, in which one maximizes over the elements of columns and minimizes over the elements of the rows of a matrix, requires a number of function evaluations which is equal to the product of the number of rows and the number of columns of the matrix, while with our new approach our experimental results require only a number of function evaluations which is the product of the number rows and a number ranging from 2 to 20, shortening computing times from years to fractions of a minute.

This paper discusses the multiple jobs scheduling problem with simultaneous resources. The problem involves one or more jobs with each job consist of a set of operations. Each operation is performed by more than one resource simultaneously. Number of units of each resource used for performing an operation is one or more units. The problem deals with determining a schedule of operations minimizing total weighted tardiness. In this paper, solution techniques based on Lagrangian relaxation are proposed. In general, the Lagrangian relaxation technique consists of three parts run iteratively, i.e., (1) solving individual job problems, (2) obtaining a feasible solution, and (3) solving a Lagrangian dual problem. For solving the individual job problems, two approaches are applied, i.e., enumeration and dynamic program¬ming. In this paper, the Lagrangian relaxation technique using the enumeration and dynamic programming approaches are called RL1 and RL2, respectively. The solution techniques proposed are examined using a set of hypothetical instances. Numerical experiments are carried out to compare the performance of RL1, RL2, and two others solution techniques (optimal and genetic algorithm techniques). Numerical experiments show that RL2 is more efficient than RL1. In terms of the solution quality, it is shown that RL2 gives same results compared to the optimal technique and genetic algorithm. However, both RL2 and genetic algorithm can handle larger problems efficiently.

This paper first proposes a discrete differential evolution algorithm for discounted {0-1} knapsack problems (D{0-1}KP) based on feasible solutions represented by the 0-1 vector. Subsequently, based on two encoding mechanisms of transforming a real vector into an integer vector, two new algorithms for solving D{0-1}KP are given through using integer vectors defined on {0, 1, 2, 3}n to represent feasible solutions of the problem. Finally, the paper conducts a comparative study on the performance between our proposed three discrete differential evolution algorithms and those developed by common genetic algorithms for solving several types of large scale D{0-1}KP problems. The paper confirms the feasibility and effectiveness of designing discrete differential evolution algorithms for D{0-1}KP by encoding conversion approaches.

The combinatorial properties of the set of solutions to the bounded knapsack problem are considered. As in the general case, this problem is an NP-complete combinatorial optimization problem and its exact solution requires the use of search algorithms with the decomposition of a set of feasible solutions. In this regard, the question of determining and evaluating the properties of the set of acceptable solutions to the problem is relevant. In this paper, formulas are obtained which allow to calculate the average value of the functional of a problem on the set of its feasible solutions and the power of this set through the number of solutions of subtasks of smaller dimension. The basic technique for obtaining results is the method of generating functions. We also consider the knapsack problem with arbitrary values of variables, in which the coefficients of the constraint vector and the objective function coincide. For this, the “continuity” of the set of solutions is assumed. Estimates of the values of the functional in this problem are found. The results may be of interest for the design of computational algorithms for finding and estimating the number of solutions and the value of the functional for optimal solutions. The expressions found can also be used in auxiliary procedures to evaluate the optimality of the solution in decomposition or heuristic algorithms for solving the knapsack problem.

A lot of uncertainties and complexities exist in real life problem. Unfortunately, the world approaches such intricate realistic life problems using traditional methods which has failed to offer robust solutions. In recent times, researchers look beyond classical techniques. There is a model shift from the use of classical techniques to the use of standardized intelligent biological systems or evolutionary biology. Genetic Algorithm (GA) has been recognized as a prospective technique capable of handling uncertainties and providing optimized solutions in diverse area, especially in homes, offices, stores and industrial operations. This research is focused on the appraisal of GA and its application in real life problem. The scenario considered is the application of GA in 0-1 knapsack problem. From the solution of the GA model, it was observed that there is no combination that would give the exact weight or capacity the 35kg bag can carry but the possible range from the solution model is 34kg and 36kg. Since the weight of the bag is 35kg, the feasible or near optimal solution weight of items the bag can carry would be 34kg at benefit of 16. Additional load beyond 34kg could lead to warping of the bag.

This paper presents an application of combined Lagrangian relaxation (LR) method and particle swarm optimization (PSO) technique for the Unit Commitment (UC) problem. This hybrid technique uses LR as the main algorithm. The LR method provides a fast solution, but it may suffer from numerical convergence and solution quality problem. The proposed algorithm uses PSO to update the Lagrangian multipliers and improve the performance of LR method. In this paper, ramp rate limit is also considered besides taking minimum up time and minimum down time of thermal units. The proposed method is implemented for solving two examples of UC test problems. The study results have shown that the approach developed is feasible and efficient.