Exact and approximation algorithms for a soft rectangle packing problem

We consider the problem of finding an arrangement of rectangles with given areas that minimizes the total length of all inner and outer border lines. We present a polynomial time approximation algorithm and derive an upper bound estimation on its approximation ratio. Furthermore, we give a formulation of the problem as mixed-integer nonlinear program and show that it can be approximatively reformulated as linear mixed-integer program. On a test set of problem instances, we compare our approximation algorithm with another one from the literature. Using a standard numerical mixed-integer linear solver, we show that adding the solutions from the approximation algorithm as advanced starter helps to reduce the overall solution time for proven global optimality, or gives better primal and dual bounds if a certain time-limit is reached before.     

The variable neighborhood search for the two machine flow shop problem with a passive prefetch

We consider the two machine flow shop scheduling problem with passive loading of the buffer on the second machine. To compute lower bounds for the global optimum, we present four integer linear programming formulations of the problem. Three local search methods with variable neighborhoods are developed for obtaining upper bounds. Some new large neighborhood is designed. Our methods use this neighborhood along with some other well-known neighborhoods. For computational experiments, we present a new class of test instances with known global optima. Computational results indicate a high efficiency of the proposed approach for the new class of instances as well as for other classes of instances of the problem.     

Nonanticipative duality,relaxations, and formulations for chance-constrained stochastic programs

We propose two new Lagrangian dual problems for chance-constrained stochastic programs based on relaxing nonanticipativity constraints. We compare the strength of the proposed dual bounds and demonstrate that they are superior to the bound obtained from the continuous relaxation of a standard mixed-integer programming (MIP) formulation. For a given dual solution, the associated Lagrangian relaxation bounds can be calculated by solving a set of single scenario subproblems and then solving a single knapsack problem. We also derive two new primal MIP formulations and demonstrate that for chance-constrained linear programs, the continuous relaxations of these formulations yield bounds equal to the proposed dual bounds. We propose a new heuristic method and two new exact algorithms based on these duals and formulations. The first exact algorithm applies to chance-constrained binary programs, and uses either of the proposed dual bounds in concert with cuts that eliminate solutions found by the subproblems. The second exact method is a branch-and-cut algorithm for solving either of the primal formulations. Our computational results indicate that the proposed dual bounds and heuristic solutions can be obtained efficiently, and the gaps between the best dual bounds and the heuristic solutions are small.     

Decomposition strategy for the stochastic pooling problem

The stochastic pooling problem is a type of stochastic mixed-integer bilinear program arising in the integrated design and operation of various important industrial networks, such as gasoline blending, natural gas production and transportation, water treatment, etc. This paper presents a rigorous decomposition method for the stochastic pooling problem, which guarantees finding an ${\epsilon}$ -optimal solution with a finite number of iterations. By convexification of the bilinear terms, the stochastic pooling problem is relaxed into a lower bounding problem that is a potentially large-scale mixed-integer linear program (MILP). Solution of this lower bounding problem is then decomposed into a sequence of relaxed master problems, which are MILPs with much smaller sizes, and primal bounding problems, which are linear programs. The solutions of the relaxed master problems yield a sequence of nondecreasing lower bounds on the optimal objective value, and they also generate a sequence of integer realizations defining the primal problems which yield a sequence of nonincreasing upper bounds on the optimal objective value. The decomposition algorithm terminates finitely when the lower and upper bounds coincide (or are close enough), or infeasibility of the problem is indicated. Case studies involving two example problems and two industrial problems demonstrate the dramatic computational advantage of the proposed decomposition method over both a state-of-the-art branch-and-reduce global optimization method and explicit enumeration of integer realizations, particularly for large-scale problems.     

Optimal value range in interval linear programming

We deal with the linear programming problem in which input data can vary in some given real compact intervals. The aim is to compute the exact range of the optimal value function. We present a general approach to the situation the feasible set is described by an arbitrary linear interval system. Moreover, certain dependencies between the constraint matrix coefficients can be involved. As long as we are able to characterize the primal and dual solution set (the set of all possible primal and dual feasible solutions, respectively), the bounds of the objective function result from two nonlinear programming problems. We demonstrate our approach on various cases of the interval linear programming problem (with and without dependencies).     

Linear programming for weighted deviation problems using compact basis techniques

Weighted deviation problems are linear programs in which weights (or penalties) are attached to deviations from upper and lower bounds on particular linear expressions. In turn the deviations may be bracketed by secondary bounds. These problems include statistical problems of minimizing weighted sums of absolute deviations, standard and extended “goal programming” problems, problems with upper bounds on absolute values of linear affine functions, problems with arbitrarily bounded variables, and combinations of these.Previous specialized linear programming methods for related problems have been restricted to specialized cases that involve only a single basis configuration, or else, by means of “extended GUB” techniques, accommodate a diverse variety of basis structures at the cost of substantially increased computation. We show that, of the several basis configurations that can arise for this problem, precisely three are essential. Special rules are identified to allow transitions between these three structures, to yield valid compact versions of both the primal and the dual simplex methods. Finally, we show how these results lead to improved efficiency as well as reduced problem size.     

An improved Benders decomposition algorithm for the logistics facility location problem with capacity expansions

We investigate a logistics facility location problem to determine whether the existing facilities remain open or not, what the expansion size of the open facilities should be and which potential facilities should be selected. The problem is formulated as a mixed integer linear programming model (MILP) with the objective to minimize the sum of the savings from closing the existing facilities, the expansion costs, the fixed setup costs, the facility operating costs and the transportation costs. The structure of the model motivates us to solve the problem using Benders decomposition algorithm. Three groups of valid inequalities are derived to improve the lower bounds obtained by the Benders master problem. By separating the primal Benders subproblem, different types of disaggregated cuts of the primal Benders cut are constructed in each iteration. A high density Pareto cut generation method is proposed to accelerate the convergence by lifting Pareto-optimal cuts. Computational experiments show that the combination of all the valid inequalities can improve the lower bounds significantly. By alternately applying the high density Pareto cut generation method based on the best disaggregated cuts, the improved Benders decomposition algorithm is advantageous in decreasing the total number of iterations and CPU time when compared to the standard Benders algorithm and optimization solver CPLEX, especially for large-scale instances.     

Column generation decomposition with the degenerate constraints in the subproblem

In this paper, we propose a new Dantzig–Wolfe decomposition for degenerate linear programs with the non degenerate constraints in the master problem and the degenerate ones in the subproblem. We propose three algorithms. The first one, where some set of variables of the original problem are added to the master problem, corresponds to the Improved Primal Simplex algorithm (IPS) presented recently by Elhallaoui et al. [7]. In the second one, some extreme points of the subproblem are added as columns in the master problem. The third algorithm is a mixed implementation that adds some original variables and some extreme points of a subproblem to the master problem. Experimental results on some degenerate instances show that the proposed algorithms yield computational times that are reduced by an average factor ranging from 3.32 to 13.16 compared to the primal simplex of CPLEX.     

Static arbitrage bounds on basket option prices

We consider the problem of computing upper and lower bounds on the price of an European basket call option, given prices on other similar options. Although this problem is hard to solve exactly in the general case, we show that in some instances the upper and lower bounds can be computed via simple closed-form expressions, or linear programs. We also introduce an efficient linear programming relaxation of the general problem based on an integral transform interpretation of the call price function. We show that this relaxation is tight in some of the special cases examined before.     

Duality in disjunctive programming via vector optimization

In this paper, we develop a new duality theory for families of linear programs with an emphasis on disjunctive linear optimization by proposing a vector optimization problem as dual problem. We establish that the well-known relations between primal and dual problems hold in this context. We show that our method generalizes the duality results of Borwein on families of linear programs, of Balas on disjunctive programs, and of Patkar and Stancu-Minasian on disjunctive linear fractional programs. Moreover, we can derive some duality results for integer and for fractional programs where the denominator is not assumed (as usual) to be greater than zero for each feasible point.     

A new Lagrangean approach to the pooling problem

We present a new Lagrangean approach for the pooling problem. The relaxation targets all nonlinear constraints, and results in a Lagrangean subproblem with a nonlinear objective function and linear constraints, that is reformulated as a linear mixed integer program. Besides being used to generate lower bounds, the subproblem solutions are exploited within Lagrangean heuristics to find feasible solutions. Valid cuts, derived from bilinear terms, are added to the subproblem to strengthen the Lagrangean bound and improve the quality of feasible solutions. The procedure is tested on a benchmark set of fifteen problems from the literature. The proposed bounds are found to outperform or equal earlier bounds from the literature on 14 out of 15 tested problems. Similarly, the Lagrangean heuristics outperform the VNS and MALT heuristics on 4 instances. Furthermore, the Lagrangean lower bound is equal to the global optimum for nine problems, and on average is 2.1% from the optimum. The Lagrangean heuristics, on the other hand, find the global solution for ten problems and on average are 0.043% from the optimum.     

Semidefinite relaxations for non-convex quadratic mixed-integer programming

We present semidefinite relaxations for unconstrained non-convex quadratic mixed-integer optimization problems. These relaxations yield tight bounds and are computationally easy to solve for medium-sized instances, even if some of the variables are integer and unbounded. In this case, the problem contains an infinite number of linear constraints; these constraints are separated dynamically. We use this approach as a bounding routine in an SDP-based branch-and-bound framework. In case of a convex objective function, the new SDP bound improves the bound given by the continuous relaxation of the problem. Numerical experiments show that our algorithm performs well on various types of non-convex instances.     

Approximate Farkas lemmas and stopping rules for iterative infeasible-point algorithms for linear programming

In exact arithmetic, the simplex method applied to a particular linear programming problem instance with real data either shows that it is infeasible, shows that its dual is infeasible, or generates optimal solutions to both problems. Most interior-point methods, on the other hand, do not provide such clear-cut information. If the primal and dual problems have bounded nonempty sets of optimal solutions, they usually generate a sequence of primal or primaldual iterates that approach feasibility and optimality. But if the primal or dual instance is infeasible, most methods give less precise diagnostics. There are methods with finite convergence to an exact solution even with real data. Unfortunately, bounds on the required number of iterations for such methods applied to instances with real data are very hard to calculate and often quite large. Our concern is with obtaining information from inexact solutions after a moderate number of iterations. We provide general tools (extensions of the Farkas lemma) for concluding that a problem or its dual is likely (in a certain well-defined sense) to be infeasible, and apply them to develop stopping rules for a homogeneous self-dual algorithm and for a generic infeasible-interior-point method for linear programming. These rules allow precise conclusions to be drawn about the linear programming problem and its dual: either near-optimal solutions are produced, or we obtain certificates that all optimal solutions, or all feasible solutions to the primal or dual, must have large norm. Our rules thus allow more definitive interpretation of the output of such an algorithm than previous termination criteria. We give bounds on the number of iterations required before these rules apply. Our tools may also be useful for other iterative methods for linear programming. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

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.     

Strictly feasible solutions and strict complementarity in multiple objective linear optimization

Recently, Luc defined a dual program for a multiple objective linear program. The dual problem is also a multiple objective linear problem and the weak duality and strong duality theorems for these primal and dual problems have been established. Here, we use these results to prove some relationships between multiple objective linear primal and dual problems. We extend the available results on single objective linear primal and dual problems to multiple objective linear primal and dual problems. Complementary slackness conditions for efficient solutions, and conditions for the existence of weakly efficient solution sets and existence of strictly primal and dual feasible points are established. We show that primal-dual (weakly) efficient solutions satisfying strictly complementary conditions exist. Furthermore, we consider Isermann’s and Kolumban’s dual problems and establish conditions for the existence of strictly primal and dual feasible points. We show the existence of primal-dual feasible points satisfying strictly complementary conditions for Isermann’s dual problem. Also, we give an alternative proof to establish necessary conditions for weakly efficient solutions of multiple objective programs, assuming the Kuhn–Tucker (KT) constraint qualification. We also provide a new condition to ensure the KT constraint qualification.     

An algorithm and upper bounds for the weighted maximal planar graph problem

In this paper, we investigate the weighted maximal planar graph (WMPG) problem. Given a complete, edge-weighted, simple graph, the WMPG problem involves finding a subgraph with the highest sum of edge weights that is maximal planar, namely, it can be embedded in the plane without any of its edges intersecting, and no additional edge can be added to the subgraph without violating its planarity. We present a new integer linear programming (ILP) model for this problem. We then develop a cutting-plane algorithm to solve the WMPG problem based on the proposed ILP model. This algorithm enables the problem to be solved more efficiently than previously reported algorithms. New upper bounds are also provided, which are useful in evaluating the quality of heuristic solutions or in generating initial solutions for meta-heuristics. Computational results are reported for a set of 417 test instances of size varying from 6 to 100 nodes including 105 instances from the literature and 312 randomly generated instances. The computational results indicate that instances with up to 24 nodes can be solved optimally in reasonable computational time and the new upper bounds for larger instances significantly improve existing upper bounds.     

Upper and lower bounding procedures for the minimum caterpillar spanning problem

A spanning caterpillar in a graph is a tree composed by a path such that all vertices not in the path are leaves. In the Minimum Spanning Caterpillar Problem (MSCP) each edge has two costs: a path cost when it belongs to the path and a connection cost when it is incident to a leaf. The goal is to find a spanning caterpillar minimizing the sum of all path and connection costs. In this paper we formulate the as a minimum Steiner arborescence problem. This reduction is the basis for the development of an efficient branch-and-cut algorithm for the MSCP. We als developed a GRASP heuristic to generate primal bounds. Experiments carried out on instances adapted from TSPLIB 2.1 revealed that the exact algorithm is capable to solve to optimality instances with up to 300 vertices in reasonable time. They also showed that our heuristic yields very high quality solutions.     

Improved state space relaxation for constrained two-dimensional guillotine cutting problems

Christofides and Hadjiconstantinou (1995) introduced a dynamic programming state space relaxation for obtaining upper bounds for the Constrained Two-dimensional Guillotine Cutting Problem. The quality of those bounds depend on the chosen item weights, they are adjusted using a subgradient-like algorithm. This paper proposes Algorithm X, a new weight adjusting algorithm based on integer programming that provably obtains the optimal weights. In order to obtain even better upper bounds, that algorithm is generalized into Algorithm X2 for obtaining optimal two-dimensional item weights. We also present a full hybrid method, called Algorithm X2D, that computes those strong upper bounds but also provides feasible solutions obtained by: (1) exploring the suboptimal solutions hidden in the dynamic programming matrices; (2) performing a number of iterations of a GRASP based primal heuristic; and (3) executing X2H, an adaptation of Algorithm X2 to transform it into a primal heuristic. Extensive experiments with instances from the literature and on newly proposed instances, for both variants with and without item rotation, show that X2D can consistently deliver high-quality solutions and sharp upper bounds. In many cases the provided solutions are certified to be optimal.     

A node-based layered graph approach for the Steiner tree problem with revenues,budget and hop-constraints

The Steiner tree problem with revenues, budget and hop-constraints (STPRBH) is a variant of the classical Steiner tree problem. The goal is to find a tree maximizing the collected revenue, which is associated with nodes, subject to a given budget for the edge cost of the tree and a hop-limit for the distance between the given root node and any other node in that tree. In this work, we introduce a novel generic way to model hop-constrained tree problems as integer linear programs and apply it to the STPRBH. Our approach is based on the concept of layered graphs that gained widespread attention in the recent years, due to their computational advantage when compared to previous formulations for modeling hop-constraints. Contrary to previous MIP formulations based on layered graphs (that are arc-based models), our model is node-based. Thus it contains much less variables and allows to tackle large-scale instances and/or instances with large hop-limits, for which the size of arc-based layered graph models may become prohibitive. The aim of our model is to provide a good compromise between quality of root relaxation bounds and the size of the underlying MIP formulation. We implemented a branch-and-cut algorithm for the STPRBH based on our new model. Most of the instances available for the DIMACS challenge, including 78 (out of 86) previously unsolved ones, can be solved to proven optimality within a time limit of 1000 s, most of them being solved within a few seconds only. These instances contain up to 500 nodes and 12,500 edges, with hop-limit up to 25.     

A branch-and-bound algorithm for instrumental variable quantile regression

This paper studies a statistical problem called instrumental variable quantile regression (IVQR). We model IVQR as a convex quadratic program with complementarity constraints and—although this type of program is generally NP-hard—we develop a branch-and-bound algorithm to solve it globally. We also derive bounds on key variables in the problem, which are valid asymptotically for increasing sample size. We compare our method with two well known global solvers, one of which requires the computed bounds. On random instances, our algorithm performs well in terms of both speed and robustness.