A polyhedral study of the semi-continuous knapsack problem

We study the convex hull of the feasible set of the semi-continuous knapsack problem, in which the variables belong to the union of two intervals. Such structure arises in a wide variety of important problems, e.g. blending and portfolio selection. We show how strong inequalities valid for the semi-continuous knapsack polyhedron can be derived and used in a branch-and-cut scheme for problems with semi-continuous variables. To demonstrate the effectiveness of these inequalities, which we call collectively semi-continuous cuts, we present computational results on real instances of the unit commitment problem, as well as on a number of randomly generated instances of linear programming with semi-continuous variables.     

A local relaxation method for the cardinality constrained portfolio optimization problem

The NP-hard nature of cardinality constrained mean-variance portfolio optimization problems has led to a number of different algorithms with varying degrees of success in reaching optimality given limited computational resources and under the presence of strict time constraints present in practice. The proposed local relaxation algorithm explores the inherent structure of the objective function. It solves a sequence of small, local, quadratic-programs by first projecting asset returns onto a reduced metric space, followed by clustering in this space to identify sub-groups of assets that best accentuate a suitable measure of similarity amongst different assets. The algorithm can either be cold started using a suitable heuristic method such as the centroids of initial clusters or be warm started based on the last output. Results, using a basket of up to 3,000 stocks and with different cardinality constraints, indicates that the proposed algorithm can lead to significant performance gain over popular branch-and-cut methods. One key application of this algorithm is in dealing with large scale cardinality constrained portfolio optimization under tight time constraint, such as for the purpose of index tracking or index arbitrage at high frequency.     

A branch and cut approach to the cardinality constrained circuit problem

The Cardinality Constrained Circuit Problem (CCCP) is the problem of finding a minimum cost circuit in a graph where the circuit is constrained to have at most k edges. The CCCP is NP-Hard. We present classes of facet-inducing inequalities for the convex hull of feasible circuits, and a branch-and-cut solution approach using these inequalities. Received: April 1998 / Accepted: October 2000?Published online October 26, 2001     

Clique-based facets for the precedence constrained knapsack problem

We consider a knapsack problem with precedence constraints imposed on pairs of items, known as the precedence constrained knapsack problem (PCKP). This problem has applications in manufacturing and mining, and also appears as a subproblem in decomposition techniques for network design and related problems. We present a new approach for determining facets of the PCKP polyhedron based on clique inequalities. A comparison with existing techniques, that lift knapsack cover inequalities for the PCKP, is also presented. It is shown that the clique-based approach generates facets that cannot be found through the existing cover-based approaches, and that the addition of clique-based inequalities for the PCKP can be computationally beneficial, for both PCKP instances arising in real applications, and applications in which PCKP appears as an embedded structure.     

A reactive local search-based algorithm for the disjunctively constrained knapsack problem

In this paper, we propose a reactive local search-based algorithm for the disjunctively constrained knapsack problem (DCKP). DCKP is a variant of the standard knapsack problem, an NP-hard combinatorial optimization problem, with special disjunctive constraints. A disjunctive constraint is a couple of items for which only one item is packed. The proposed algorithm is based upon a reactive local search, where an explicit check for the repetition of configurations is added to the search process. Initially, two complementary greedy procedures are applied in order to construct a starting solution. Second, a degrading procedure is introduced in order (i) to escape to local optima and (ii) to introduce a diversification in the search space. Finally, a memory list is added in order to forbid the repetition of configurations. The performance of two versions of the algorithm has been evaluated on several problem instances and compared to the results obtained by running the Cplex solver. Encouraging results have been obtained.     

Lifting valid inequalities for the precedence constrained knapsack problem

This paper considers the precedence constrained knapsack problem. More specifically, we are interested in classes of valid inequalities which are facet-defining for the precedence constrained knapsack polytope. We study the complexity of obtaining these facets using the standard sequential lifting procedure. Applying this procedure requires solving a combinatorial problem. For valid inequalities arising from minimal induced covers, we identify a class of lifting coefficients for which this problem can be solved in polynomial time, by using a supermodular function, and for which the values of the lifting coefficients have a combinatorial interpretation. For the remaining lifting coefficients it is shown that this optimization problem is strongly NP-hard. The same lifting procedure can be applied to (1,k)-configurations, although in this case, the same combinatorial interpretation no longer applies. We also consider K-covers, to which the same procedure need not apply in general. We show that facets of the polytope can still be generated using a similar lifting technique. For tree knapsack problems, we observe that all lifting coefficients can be obtained in polynomial time. Computational experiments indicate that these facets significantly strengthen the LP-relaxation. Received July 10, 1997 / Revised version received January 9, 1999? Published online May 12, 1999     

Extended formulations for the cardinality constrained subtree of a tree problem

Given a tree with n nodes, we consider the problem of finding the most profitable subtree of that tree with at most K nodes which is known as the Cardinality Subtree of a Tree Problem. We present a new exact linear extended formulation with O(nK) two-indexed variables and O(nK) constraints.     

Online cardinality constrained scheduling

In online load balancing problems, jobs arrive over a list. Upon arrival of a job, the algorithm is required to assign it immediately and irrevocably to a machine. We consider such a makespan minimization problem with an additional cardinality constraint, i.e., at most k jobs may be assigned to each machine, where k is a parameter of the problem. We present both upper and lower bounds on the competitive ratio of online algorithms for this problem with identical machines.     

A polyhedral study of the maximum edge subgraph problem

The study of cohesive subgroups is an important aspect of social network analysis. Cohesive subgroups are studied using different relaxations of the notion of clique in a graph. For instance, given a graph and an integer k, the maximum edge subgraph problem consists in finding a k-vertex subset such that the number of edges within the subset is maximum. This work proposes a polyhedral approach for this NP-hard problem. We study the polytope associated to an integer programming formulation of the problem, present several families of facet-inducing valid inequalities, and discuss the separation problem associated to these families.     

A polyhedral study of the generalized vertex packing problem

The traditional vertex packing problem defined on an undirected graph identifies the largest weighted independent set of nodes, that is, a set of nodes whose induced subgraph contains no edges. In this paper, we examine a generalized vertex packing problem (GVP-k) in which k ``violations' to the independent set restriction are permitted, whereby k edges may exist within the subgraph induced by the chosen set of nodes. A particular context in which such problems arise is in the national airspace planning model of Sherali, Smith, and Trani (2000), where a set of flight-plans need to be composed for various flights subject to conflict, workload, and equity considerations. The GVP-k structure arises in modeling the air-traffic control sector workload restrictions, which stipulate that for each sector and during each discretized time-slot, the number of aircraft conflicts that would need to be resolved should not exceed k, for some k≥1. We derive several classes of facetial valid inequalities for GVP-k for certain specially structured subgraphs, identifying polynomial-sized convex hull representations for some of these cases. Related constraint generation routines are also developed, and some computational results are provided to demonstrate the efficacy of utilizing the proposed valid inequalities in solving GVP-k for different values of k.     

The constrained compartmentalized knapsack problem: mathematical models and solution methods

The constrained compartmentalized knapsack problem can be seen as an extension of the constrained knapsack problem. However, the items are grouped into different classes so that the overall knapsack has to be divided into compartments, and each compartment is loaded with items from the same class. Moreover, building a compartment incurs a fixed cost and a fixed loss of the capacity in the original knapsack, and the compartments are lower and upper bounded. The objective is to maximize the total value of the items loaded in the overall knapsack minus the cost of the compartments. This problem has been formulated as an integer non-linear program, and in this paper, we reformulate the non-linear model as an integer linear master problem with a large number of variables. Some heuristics based on the solution of the restricted master problem are investigated. A new and more compact integer linear model is also presented, which can be solved by a branch-and-bound commercial solver that found most of the optimal solutions for the constrained compartmentalized knapsack problem. On the other hand, heuristics provide good solutions with low computational effort.     

A polyhedral study on chance constrained program with random right-hand side

Mathematical Programming - The essential structure of the mixed-integer programming formulation for chance-constrained program (CCP) is the intersection of multiple mixing sets with a 0–1...     

A computational study of exact knapsack separation for the generalized assignment problem

The Generalized Assignment Problem is a well-known NP-hard combinatorial optimization problem which consists of minimizing the assignment costs of a set of jobs to a set of machines satisfying capacity constraints. Most of the existing algorithms are of a Branch-and-Price type, with lower bounds computed through Dantzig–Wolfe reformulation and column generation.     

Heuristic algorithms for the cardinality constrained efficient frontier

This paper examines the application of genetic algorithm, tabu search and simulated annealing metaheuristic approaches to finding the cardinality constrained efficient frontier that arises in financial portfolio optimisation. We consider the mean-variance model of Markowitz as extended to include the discrete restrictions of buy-in thresholds and cardinality constraints. Computational results are reported for publicly available data sets drawn from seven major market indices involving up to 1318 assets. Our results are compared with previous results given in the literature illustrating the effectiveness of the proposed metaheuristics in terms of solution quality and computation time.     

A multi-objective genetic algorithm for cardinality constrained fuzzy portfolio selection

This paper presents a new procedure that extends genetic algorithms from their traditional domain of optimization to fuzzy ranking strategy for selecting efficient portfolios of restricted cardinality. The uncertainty of the returns on a given portfolio is modeled using fuzzy quantities and a downside risk function is used to describe the investor's aversion to risk. The fitness functions are based both on the value and the ambiguity of the trapezoidal fuzzy number which represents the uncertainty on the return. The soft-computing approach allows us to consider uncertainty and vagueness in databases and also to incorporate subjective characteristics into the portfolio selection problem. We use a data set from the Spanish stock market to illustrate the performance of our approach to the portfolio selection problem.     

A polyhedral study of the single-item lot-sizing problem with continuous start-up costs

In this work we consider the single-item single-machine lot-sizing problem with continuous start-up costs. A continuous start-up cost is generated in a period whenever there is a nonzero production in the period and the production capacity in the previous period is not saturated. This concept of start-up does not correspond to the standard (discrete) start-up considered in previous models, thus motivating a polyhedral study of this problem. We introduce a natural integer programming formulation for this problem, we study some general properties and facet-inducing inequalities of the associated polytope, and we state relationships with known lotsizing polytopes.     

A hybrid algorithm for the unbounded knapsack problem

This paper presents a new approach for exactly solving the Unbounded Knapsack Problem (UKP) and proposes a new bound that was proved to dominate the previous bounds on a special class of UKP instances. Integrating bounds within the framework of sparse dynamic programming led to the creation of an efficient and robust hybrid algorithm, called EDUK2. This algorithm takes advantage of the majority of the known properties of UKP, particularly the diverse dominance relations and the important periodicity property. Extensive computational results show that, in all but a very few cases, EDUK2 significantly outperforms both MTU2 and EDUK, the currently available UKP solvers, as well the well-known general purpose mathematical programming optimizer CPLEX of ILOG. These experimental results demonstrate that the class of hard UKP instances needs to be redefined, and the authors offer their insights into the creation of such instances.