A fast approximation algorithm for solving the complete set packing problem

We study the complete set packing problem (CSPP) where the family of feasible subsets may include all possible combinations of objects. This setting arises in applications such as combinatorial auctions (for selecting optimal bids) and cooperative game theory (for finding optimal coalition structures). Although the set packing problem has been well-studied in the literature, where exact and approximation algorithms can solve very large instances with up to hundreds of objects and thousands of feasible subsets, these methods are not extendable to the CSPP since the number of feasible subsets is exponentially large. Formulating the CSPP as an MILP and solving it directly, using CPLEX for example, is impossible for problems with more than 20 objects. We propose a new mathematical formulation for the CSPP that directly leads to an efficient algorithm for finding feasible set packings (upper bounds). We also propose a new formulation for finding tighter lower bounds compared to LP relaxation and develop an efficient method for solving the corresponding large-scale MILP. We test the algorithm with the winner determination problem in spectrum auctions, the coalition structure generation problem in coalitional skill games, and a number of other simulated problems that appear in the literature.     

Upper and lower bounding procedures for the multiple knapsack assignment problem

We formulate the multiple knapsack assignment problem (MKAP) as an extension of the multiple knapsack problem (MKP), as well as of the assignment problem. Except for small instances, MKAP is hard to solve to optimality. We present a heuristic algorithm to solve this problem approximately but very quickly. We first discuss three approaches to evaluate its upper bound, and prove that these methods compute an identical upper bound. In this process, reference capacities are derived, which enables us to decompose the problem into mutually independent MKPs. These MKPs are solved euristically, and in total give an approximate solution to MKAP. Through numerical experiments, we evaluate the performance of our algorithm. Although the algorithm is weak for small instances, we find it prospective for large instances. Indeed, for instances with more than a few thousand items we usually obtain solutions with relative errors less than 0.1% within one CPU second.     

The ring-star problem: A new integer programming formulation and a branch-and-cut algorithm

A ring star in a graph is a subgraph that can be decomposed into a cycle (or ring) and a set of edges with exactly one vertex in the cycle. In the minimum ring-star problem (mrsp) the cost of a ring star is given by the sum of the costs of its edges, which vary, depending on whether the edge is part of the ring or not. The goal is to find a ring-star spanning subgraph minimizing the sum of all ring and assignment costs. In this paper we show that the mrsp can be reduced to a minimum (constrained) Steiner arborescence problem on a layered graph. This reduction is used to introduce a new integer programming formulation for the mrsp. We prove that the dual bound generated by the linear relaxation of this formulation always dominates the one provided by an early model from the literature. Based on our new formulation, we developed a branch-and-cut algorithm for the mrsp. On the primal side, we devised a grasp heuristic to generate good upper bounds for the problem. Computational tests with these algorithms were conducted on a benchmark of public domain. In these experiments both our exact and heuristics algorithms had excellent performances, noticeably in dealing with instances whose optimal solution has few vertices in the ring. In addition, we also investigate the minimum spanning caterpillar problem (mscp) which has the same input as the mrsp and admits feasible solutions that can be viewed as ring stars with paths in the place of rings. We present an easy reduction of the mscp to the mrsp, which makes it possible to solve to optimality instances of the former problem too. Experiments carried out with the mscp revealed that our branch-and-cut algorithm is capable to solve to optimality instances with up to 200 vertices in reasonable time.     

Convergence in norm of modified Krasnoselski-Mann iterations for fixed points of demicontractive mappings

This paper deals with fixed points methods related to the general class of demicontractive mappings (including the well-known classes of nonexpansive and quasi-nonexpansive mappings) in Hilbert spaces. Specifically, we point out some historical aspects concerning the concept of demicontactivity and we investigate a regularized variant of the Krasnoselski-Mann iteration that can be alternatively regarded as a simplified form of the inertial iteration (P-E. Maingé, J. Math. Anal. Appl. 344 (2008) 876-887) with non-constant relaxation factors. These two methods ensure the strong convergence of the generated sequence towards the least norm element of the set of fixed-points of demicontractive mappings. However, for convergence, our method does not require anymore the knowledge of some constant related to the involved demicontractive operator. A new and simpler proof is also proposed for its convergence even when involving non-constant relaxation factors. We point out the simplicity of this algorithm (at least from computational point of view) in comparison with other existing methods. We also present some numerical experiments concerning a convex feasibility problem, experiments that emphasize the characteristics of the considered algorithm comparing with a classical cyclic projection-type iteration.     

A reduction approach for solving the rectangle packing area minimization problem

In the rectangle packing area minimization problem (RPAMP) we are given a set of rectangles with known dimensions. We have to determine an arrangement of all rectangles, without overlapping, inside an enveloping rectangle of minimum area. The paper presents a generic approach for solving the RPAMP that is based on two algorithms, one for the 2D Knapsack Problem (KP), and the other for the 2D Strip Packing Problem (SPP). In this way, solving an instance of the RPAMP is reduced to solving multiple SPP and KP instances. A fast constructive heuristic is used as SPP algorithm while the KP algorithm is instantiated by a tree search method and a genetic algorithm alternatively. All these SPP and KP methods have been published previously. Finally, the best variants of the resulting RPAMP heuristics are combined within one procedure. The guillotine cutting condition is always observed as an additional constraint. The approach was tested on 15 well-known RPAMP instances (above all MCNC and GSRC instances) and new best solutions were obtained for 10 instances. The computational effort remains acceptable. Moreover, 24 new benchmark instances are introduced and promising results are reported.     

On dominant poles and model reduction of second order time-delay systems

The method known as the dominant pole algorithm (DPA) has previously been successfully used in combination with model order reduction techniques to approximate standard linear time-invariant dynamical systems and second order dynamical systems. In this paper, we show how this approach can be adapted to a class of second order delay systems, which are large scale nonlinear problems whose transfer functions have an infinite number of simple poles. Deflation is a very important ingredient for this type of methods. Because of the nonlinearity, many deflation approaches for linear systems are not applicable. We therefore propose an alternative technique that essentially removes computed poles from the system?s input and output vectors. In general, this technique changes the residues, and hence, modifies the order of dominance of the poles, but we prove that, under certain conditions, the residues stay near the original residues. The new algorithm is illustrated by numerical examples.     

Global optimization algorithm for sum of generalized polynomial ratios problem

In this paper, a global optimization algorithm is proposed for solving sum of generalized polynomial ratios problem (P) which arises in various practical problems. Due to its intrinsic difficulty, less work has been devoted to globally solve the problem (P). For such problems, we present a branch and bound algorithm. In this method, by utilizing exponent transformation and new three-level linear relaxation method, a sequence of linear relaxation programming of the initial nonconvex programming problem (P) are derived which are embedded in a branch and bound algorithm. The proposed method need not introduce new variables and constraints and it is convergent to the global minimum of prime problem by means of the subsequent solutions of a series of linear programming problems. Several numerical examples in the literatures are tested to demonstrate that the proposed algorithm can systematically solve these examples to find the approximate ?-global optimum.     

Cross decomposition for mixed integer programming

Many methods for solving mixed integer programming problems are based either on primal or on dual decomposition, which yield, respectively, a Benders decomposition algorithm and an implicit enumeration algorithm with bounds computed via Lagrangean relaxation. These methods exploit either the primal or the dual structure of the problem. We propose a new approach, cross decomposition, which allows exploiting simultaneously both structures. The development of the cross decomposition method captures profound relationships between primal and dual decomposition. It is shown that the more constraints can be included in the Langrangean relaxation (provided the duality gap remains zero), the fewer the Benders cuts one may expect to need. If the linear programming relaxation has no duality gap, only one Benders cut is needed to verify optimality.     

Accelerated reflection projection algorithm and its application to the LMI problem

We discuss accelerated version of the alternating projection method which can be applied to solve the linear matrix inequality (LMI) problem. The alternating projection method is a well-known algorithm for the convex feasibility problem, and has many generalizations and extensions. Bauschke and Kruk proposed a reflection projection algorithm for computing a point in the intersection of an obtuse cone and a closed convex set. We carry on this research in two directions. First, we present an accelerated version of the reflection projection algorithm, and prove its weak convergence in a Hilbert space; second, we prove the finite termination of an algorithm which is based on the proposed algorithm and provide an explicit upper bound for the required number of iterations under certain assumptions. Numerical experiments for the LMI problem are provided to demonstrate the effectiveness and merits of the proposed algorithms.     

A theoretical and empirical investigation on the Lagrangian capacities of the 0-1 multidimensional knapsack problem

We present a novel Lagrangian method to find good feasible solutions in theoretical and empirical aspects. After investigating the concept of Lagrangian capacity, which is the value of the capacity constraint that Lagrangian relaxation can find an optimal solution, we formally reintroduce Lagrangian capacity suitable to the 0-1 multidimensional knapsack problem and present its new geometric equivalent condition including a new associated property. Based on the property, we propose a new Lagrangian heuristic that finds high-quality feasible solutions of the 0-1 multidimensional knapsack problem. We verify the advantage of the proposed heuristic by experiments. We make comparisons with existing Lagrangian approaches on benchmark data and show that the proposed method performs well on large-scale data.     

Edit distance with move operations

The traditional edit-distance problem is to find the minimum number of insert-character and delete-character (and sometimes change character) operations required to transform one string into another. Here we consider the more general problem of a string represented by a singly linked list (one character per node) and being able to apply these operations to the pointer associated with a vertex as well as the character associated with the vertex. That is, in O(1) time, not only can characters be inserted or deleted, but substrings can be moved or deleted. We limit our attention to the ability to move substrings and leave substring deletions for future research. Note that O(1) time substring move operation implies O(1) substring exchange operation as well, a form of transformation that has been of interest in molecular biology. We show that this problem is NP-complete, and that a “recursive” sequence of moves can be simulated with at most a constant factor increase by a non-recursive sequence. Although a greedy algorithm is known to have poor (a polynomial factor) worst case performance, we present a polynomial time greedy algorithm for non-recursive moves which on a subclass of instances of a problem of size n achieves an approximation factor to optimal of at most O(logn). The development of this greedy algorithm shows how to reduce moves of substrings to moves of characters, and how to convert moves of characters to only inserts and deletes of characters.     

基于异步次梯度法的LR算法及其在多阶段HFSP的应用

为降低求解复杂度和缩短计算时间,针对多阶段混合流水车间总加权完成时间问题,提出了一种结合异步次梯度法的改进拉格朗日松弛算法。建立综合考虑有限等待时间和工件释放时间的整数规划数学模型,将异步次梯度法嵌入到拉格朗日松弛算法中,从而通过近似求解拉格朗日松弛问题得到一个合理的异步次梯度方向,沿此方向进行搜索,逐渐降低到最优点的距离。通过仿真实验,验证了所提算法的有效性。对比所提算法与传统的基于次梯度法的拉格朗日松弛算法,结果表明,就综合解的质量和计算效率而言,所提算法能在较短的计算时间内获得更好的近优解,尤其是对大规模问题。     

Compact finite difference method for American option pricing

A compact finite difference method is designed to obtain quick and accurate solutions to partial differential equation problems. The problem of pricing an American option can be cast as a partial differential equation. Using the compact finite difference method this problem can be recast as an ordinary differential equation initial value problem. The complicating factor for American options is the existence of an optimal exercise boundary which is jointly determined with the value of the option. In this article we develop three ways of combining compact finite difference methods for American option price on a single asset with methods for dealing with this optimal exercise boundary. Compact finite difference method one uses the implicit condition that solutions of the transformed partial differential equation be nonnegative to detect the optimal exercise value. This method is very fast and accurate even when the spatial step size h   is large (h?0.1)$(h?0.1)$. Compact difference method two must solve an algebraic nonlinear equation obtained by Pantazopoulos (1998) at every time step. This method can obtain second order accuracy for space x and requires a moderate amount of time comparable with that required by the Crank Nicolson projected successive over relaxation method. Compact finite difference method three refines the free boundary value by a method developed by Barone-Adesi and Lugano [The saga of the American put, 2003], and this method can obtain high accuracy for space x. The last two of these three methods are convergent, moreover all the three methods work for both short term and long term options. Through comparison with existing popular methods by numerical experiments, our work shows that compact finite difference methods provide an exciting new tool for American option pricing.     

Relaxed cutting plane method with convexification for solving nonlinear semi-infinite programming problems

In this paper, we present an algorithm to solve nonlinear semi-infinite programming (NSIP) problems. To deal with the nonlinear constraint, Floudas and Stein (SIAM J. Optim. 18:1187?C1208, 2007) suggest an adaptive convexification relaxation to approximate the nonlinear constraint function. The ??BB method, used widely in global optimization, is applied to construct the convexification relaxation. We then combine the idea of the cutting plane method with the convexification relaxation to propose a new algorithm to solve NSIP problems. With some given tolerances, our algorithm terminates in a finite number of iterations and obtains an approximate stationary point of the NSIP problems. In addition, some NSIP application examples are implemented by the method proposed in this paper, such as the proportional-integral-derivative controller design problem and the nonlinear finite impulse response filter design problem. Based on our numerical experience, we demonstrate that our algorithm enhances the computational speed for solving NSIP problems.     

TSPACK: A two-level tabu search for the three-dimensional bin packing problem

Three-dimensional orthogonal bin packing is a problem NP-hard in the strong sense where a set of boxes must be orthogonally packed into the minimum number of three-dimensional bins. We present a two-level tabu search for this problem. The first-level aims to reduce the number of bins. The second optimizes the packing of the bins. This latter procedure is based on the Interval Graph representation of the packing, proposed by Fekete and Schepers, which reduces the size of the search space. We also introduce a general method to increase the size of the associated neighborhoods, and thus the quality of the search, without increasing the overall complexity of the algorithm. Extensive computational results on benchmark problem instances show the effectiveness of the proposed approach, obtaining better results compared to the existing ones.     

Reoptimizing the 0-1 knapsack problem

In this paper we study the problem where an optimal solution of a knapsack problem on n items is known and a very small number k of new items arrive. The objective is to find an optimal solution of the knapsack problem with n+k items, given an optimal solution on the n items (reoptimization of the knapsack problem). We show that this problem, even in the case k=1, is NP-hard and that, in order to have effective heuristics, it is necessary to consider not only the items included in the previously optimal solution and the new items, but also the discarded items. Then, we design a general algorithm that makes use, for the solution of a subproblem, of an α-approximation algorithm known for the knapsack problem. We prove that this algorithm has a worst-case performance bound of , which is always greater than α, and therefore that this algorithm always outperforms the corresponding α-approximation algorithm applied from scratch on the n+k items. We show that this bound is tight when the classical Ext-Greedy algorithm and the algorithm are used to solve the subproblem. We also show that there exist classes of instances on which the running time of the reoptimization algorithm is smaller than the running time of an equivalent PTAS and FPTAS.     

On Steiner trees and minimum spanning trees in hypergraphs

The bottleneck of the state-of-the-art algorithms for geometric Steiner problems is usually the concatenation phase, where the prevailing approach treats the generated full Steiner trees as edges of a hypergraph and uses an LP-relaxation of the minimum spanning tree in hypergraph (MSTH) problem. We study this original and some new equivalent relaxations of this problem and clarify their relations to all classical relaxations of the Steiner problem. In an experimental study, an algorithm of ours which is designed for general graphs turns out to be an efficient alternative to the MSTH approach.     

An efficient linearization technique for mixed 0-1 polynomial problem

This paper addresses a new and efficient linearization technique to solve mixed 0-1 polynomial problems to achieve a global optimal solution. Given a mixed 0-1 polynomial term z=c_tx₁x₂…x_ny, where x₁,x₂,…,x_n are binary (0-1) variables and y is a continuous variable. Also, c_t can be either a positive or a negative parameter. We transform z into a set of auxiliary constraints which are linear and can be solved by exact methods such as branch and bound algorithms. For this purpose, we will introduce a method in which the number of additional constraints is decreased significantly rather than the previous methods proposed in the literature. As is known in any operations research problem decreasing the number of constraints leads to decreasing the mathematical computations, extensively. Thus, research on the reducing number of constraints in mathematical problems in complicated situations have high priority for decision makers. In this method, each n-auxiliary constraints proposed in the last method in the literature for the linearization problem will be replaced by only 3 novel constraints. In other words, previous methods were dependent on the number of 0-1 variables and therefore, one auxiliary constraint was considered per 0-1 variable, but this method is completely independent of the number of 0-1 variables and this illustrates the high performance of this method in computation considerations. The analysis of this method illustrates the efficiency of the proposed algorithm.     

Inverse integer optimization with an imperfect observation

This study develops and evaluates methods for inverse integer optimization problems with an imperfect observation where the unknown parameters are the cost coefficients. We propose a cutting plane algorithm for this problem and compare it to a heuristic which solves the inverse of the linear relaxation of the forward problem. We then propose a hybrid approach that initializes the cutting plane algorithm from the solution of the heuristic.