Revisiting <Emphasis Type="Italic">k</Emphasis>-sum optimization

In this paper, we address continuous, integer and combinatorial k-sum optimization problems. We analyze different formulations of this problem that allow to solve it through the minimization of a relatively small number of minisum optimization problems. This approach provides a general tool for solving a variety of k-sum optimization problems and at the same time, improves the complexity bounds of many ad-hoc algorithms previously reported in the literature for particular versions of this problem. Moreover, the results developed for k-sum optimization have been extended to the more general case of the convex ordered median problem, improving upon existing solution approaches.     

An analytical comparison of the LP relaxations of integer models for the k-club problem

Given an undirected graph G = (V, E), a k-club is a subset of nodes that induces a subgraph with diameter at most k. The k-club problem is to find a maximum cardinality k-club. In this study, we use a linear programming relaxation standpoint to compare integer formulations for the k-club problem. The comparisons involve formulations known from the literature and new formulations, built in different variable spaces. For the case k = 3, we propose two enhanced compact formulations. From the LP relaxation standpoint these formulations dominate all other compact formulations in the literature and are equivalent to a formulation with a non-polynomial number of constraints. Also for k = 3, we compare the relative strength of LP relaxations for all formulations examined in the study (new and known from the literature). Based on insights obtained from the comparative study, we devise a strengthened version of a recursive compact formulation in the literature for the k-club problem (k > 1) and show how to modify one of the new formulations for the case k = 3 in order to accommodate additional constraints recently proposed in the literature.     

When is rounding allowed in integer nonlinear optimization?

In this paper we consider nonlinear integer optimization problems. Nonlinear integer programming has mainly been studied for special classes, such as convex and concave objective functions and polyhedral constraints. In this paper we follow an other approach which is not based on convexity or concavity. Studying geometric properties of the level sets and the feasible region, we identify cases in which an integer minimizer of a nonlinear program can be found by rounding (up or down) the coordinates of a solution to its continuous relaxation. We call this property rounding property. If it is satisfied, it enables us (for fixed dimension) to solve an integer programming problem in the same time complexity as its continuous relaxation. We also investigate the strong rounding property which allows rounding a solution to the continuous relaxation to the next integer solution and in turn yields that the integer version can be solved in the same time complexity as its continuous relaxation for arbitrary dimensions.     

A new exact discrete linear reformulation of the quadratic assignment problem

The quadratic assignment problem (QAP) is a challenging combinatorial problem. The problem is NP-hard and in addition, it is considered practically intractable to solve large QAP instances, to proven optimality, within reasonable time limits. In this paper we present an attractive mixed integer linear programming (MILP) formulation of the QAP. We first introduce a useful non-linear formulation of the problem and then a method of how to reformulate it to a new exact, compact discrete linear model. This reformulation is efficient for QAP instances with few unique elements in the flow or distance matrices. Finally, we present optimal results, obtained with the discrete linear reformulation, for some previously unsolved instances (with the size n = 32 and 64), from the quadratic assignment problem library, QAPLIB.     

Integer programming formulation of combinatorial optimization problems

This paper considers in a somewhat general setting when a combinatorial optimization problem can be formulated as an (all-integer) integer programming (IP) problem. The main result is that any combinatorial optimization problem can be formulated as an IP problem if its feasible region S is finite but there are many rather sample problems that have no IP formulation if their S is infinite. The approach used for finite S usually gives a formulation with a relatively small number of additional variables for example, an integer polynomial of n 0?1 variables requires at most n + 1 additional variables by our approach, whereas 2ⁿ - (n + 1) additional variables at maximum are required by other existing methods. Finally, the decision problem of deciding whether an arbitrarily given combinatorial optimization problem has an IP formulation is considered and it is shown by an argument closely related to Hilbert's tenth problem (drophantine equations) that no such algorithm exists.     

Cones of multipowers and combinatorial optimization problems

The cone of multipowers is dual to the cone of nonnegative polynomials. The relation of the former cone to combinatorial optimization problems is examined. Tensor extensions of polyhedra of combinatorial optimization problems are used for this purpose. The polyhedron of the MAX-2-CSP problem (optimization version of the two-variable constraint satisfaction problem) of tensor degree 4k is shown to be the intersection of the cone of 4k-multipowers and a suitable affine space. Thus, in contrast to SDP relaxations, the relaxation to a cone of multipowers becomes tight even for an extension of degree 4.     

Lower bounds for nonlinear assignment problems using many body interactions

This paper concerns lower bounding techniques for the general α-adic assignment problem. The nonlinear objective function is linearized by the introduction of additional variables and constraints, thus yielding a mixed integer linear programming formulation of the problem. The concept of many body interactions is introduced to strengthen this formulation and incorporated in a modified formulation obtained by lifting the original representation to a higher dimensional space. This process involves two steps — (i) addition of new variables and constraints and (ii) incorporation of the new variables in the objective function. If this lifting process is repeated β times on an α-adic assignment problem along with the incorporation of higher order interactions, it results in the mixed-integer formulation of an equivalent (α + β)-adic assignment problem. The incorporation of many body interactions in the higher dimensional formulation improves its degeneracy properties and is also critical to the derivation of decomposition methods for the solution of these large scale mathematical programs in the higher dimensional space. It is shown that a lower bound to the optimal solution of the corresponding linear programming relaxation can be obtained by dualizing a subset of constraints in this formulation and solving O(N^2(α+β−1)) linear assignment problems, whose coefficients depend on the dual values. Moreover, it is proved that the optimal solution to the LP relaxation is obtained if we use the optimal duals for the solution of the linear assignment problems. This concept of many body interactions could be applied in designing algorithms for the solution of formulations obtained by lifting general MILP's. We illustrate all these concepts on the quadratic assignment problems With these decomposition bounds, we have found the provably optimal solutions of two unsolved QAP's of size 32 and have also improved upon existing lower bounds for other QAP's.     

Simple solution methods for separable mixed linear and quadratic knapsack problem

Quadratic knapsack problem has a central role in integer and nonlinear optimization, which has been intensively studied due to its immediate applications in many fields and theoretical reasons. Although quadratic knapsack problem can be solved using traditional nonlinear optimization methods, specialized algorithms are much faster and more reliable than the nonlinear programming solvers. In this paper, we study a mixed linear and quadratic knapsack with a convex separable objective function subject to a single linear constraint and box constraints. We investigate the structural properties of the studied problem, and develop a simple method for solving the continuous version of the problem based on bi-section search, and then we present heuristics for solving the integer version of the problem. Numerical experiments are conducted to show the effectiveness of the proposed solution methods by comparing our methods with some state of the art linear and quadratic convex solvers.     

The balanced minimum evolution problem under uncertain data

We investigate the Robust Deviation Balanced Minimum Evolution Problem (RDBMEP), a combinatorial optimization problem that arises in computational biology when the evolutionary distances from taxa are uncertain and varying inside intervals. By exploiting some fundamental properties of the objective function, we present a mixed integer programming model to exactly solve instances of the RDBMEP and discuss the biological impact of uncertainty on the solutions to the problem. Our results give perspective on the mathematics of the RDBMEP and suggest new directions to tackle phylogeny estimation problems affected by uncertainty.     

Extended formulations in mixed integer conic quadratic programming

In this paper we consider the use of extended formulations in LP-based algorithms for mixed integer conic quadratic programming (MICQP). Extended formulations have been used by Vielma et al. (INFORMS J Comput 20: 438–450, 2008) and Hijazi et al. (Comput Optim Appl 52: 537–558, 2012) to construct algorithms for MICQP that can provide a significant computational advantage. The first approach is based on an extended or lifted polyhedral relaxation of the Lorentz cone by Ben-Tal and Nemirovski (Math Oper Res 26(2): 193–205 2001) that is extremely economical, but whose approximation quality cannot be iteratively improved. The second is based on a lifted polyhedral relaxation of the euclidean ball that can be constructed using techniques introduced by Tawarmalani and Sahinidis (Math Programm 103(2): 225–249, 2005). This relaxation is less economical, but its approximation quality can be iteratively improved. Unfortunately, while the approach of Vielma, Ahmed and Nemhauser is applicable for general MICQP problems, the approach of Hijazi, Bonami and Ouorou can only be used for MICQP problems with convex quadratic constraints. In this paper we show how a homogenization procedure can be combined with the technique by Tawarmalani and Sahinidis to adapt the extended formulation used by Hijazi, Bonami and Ouorou to a class of conic mixed integer programming problems that include general MICQP problems. We then compare the effectiveness of this new extended formulation against traditional and extended formulation-based algorithms for MICQP. We find that this new formulation can be used to improve various LP-based algorithms. In particular, the formulation provides an easy-to-implement procedure that, in our benchmarks, significantly improved the performance of commercial MICQP solvers.     

Cardinality constrained combinatorial optimization: Complexity and polyhedra

Given a combinatorial optimization problem and a subset $N$ of nonnegative integer numbers, we obtain a cardinality constrained version of this problem by permitting only those feasible solutions whose cardinalities are elements of $N$. In this paper we briefly touch on questions that address common grounds and differences of the complexity of a combinatorial optimization problem and its cardinality constrained version. Afterwards we focus on the polyhedral aspects of the cardinality constrained combinatorial optimization problems. Maurras (1977) [5] introduced a class of inequalities, called forbidden cardinality inequalities in this paper, that can be added to a given integer programming formulation for a combinatorial optimization problem to obtain one for the cardinality restricted versions of this problem. Since the forbidden cardinality inequalities in their original form are mostly not facet defining for the associated polyhedron, we discuss some possibilities to strengthen them, based on the experiments made in Kaibel and Stephan (2007) and Maurras and Stephan (2009) [2], [3].     

Strong valid inequalities for fluence map optimization problem under dose-volume restrictions

Fluence map optimization problems are commonly solved in intensity modulated radiation therapy (IMRT) planning. We show that, when subject to dose-volume restrictions, these problems are NP-hard and that the linear programming relaxation of their natural mixed integer programming formulation can be arbitrarily weak. We then derive strong valid inequalities for fluence map optimization problems under dose-volume restrictions using disjunctive programming theory and show that strengthening mixed integer programming formulations with these valid inequalities has significant computational benefits.     

New and efficient algorithms for transfer prices and inventory holding policies in two-enterprise supply chains

We consider a multi-period problem of fair transfer prices and inventory holding policies in two enterprise supply chains. This problem was formulated as a mixed integer non-linear program by Gjerdrum et al. (Eur J Oper Res 143:582–599, 2002). Existing global optimization methods to solve this problem are computationally expensive. We propose a continuous approach based on difference of convex functions (DC) programming and DC Algorithms (DCA) for solving this combinatorial optimization problem. The problem is first reformulated as a DC program via an exact penalty technique. Afterward, DCA, an efficient local approach in non-convex programming framework, is investigated to solve the resulting problem. For globally solving this problem, we investigate a combined DCA-Branch and Bound algorithm. DCA is applied to get lower bounds while upper bounds are computed from a relaxation problem. The numerical results on several test problems show that the proposed algorithms are efficient: DCA provides a good integer solution in a short CPU time although it works on a continuous domain, and the combined DCA-Branch and Bound finds an \(\epsilon \) -optimal solution for large-scale problems in a reasonable time.     

Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio selection problems

In this paper we investigate a class of cardinality-constrained portfolio selection problems. We construct convex relaxations for this class of optimization problems via a new Lagrangian decomposition scheme. We show that the dual problem can be reduced to a second-order cone program problem which is tighter than the continuous relaxation of the standard mixed integer quadratically constrained quadratic program (MIQCQP) reformulation. We then propose a new MIQCQP reformulation which is more efficient than the standard MIQCQP reformulation in terms of the tightness of the continuous relaxations. Computational results are reported to demonstrate the tightness of the SOCP relaxation and the effectiveness of the new MIQCQP reformulation.     

Integer programming models and linearizations for the traveling car renter problem

The traveling car renter problem (CaRS) is an extension of the classical traveling salesman problem (TSP) where different cars are available for use during the salesman’s tour. In this study we present three integer programming formulations for CaRS, of which two have quadratic objective functions and the other has quadratic constraints. The first model with a quadratic objective function is grounded on the TSP interpreted as a special case of the quadratic assignment problem in which the assignment variables refer to visitation orders. The second model with a quadratic objective function is based on the Gavish and Grave’s formulation for the TSP. The model with quadratic constraints is based on the Dantzig–Fulkerson–Johnson’s formulation for the TSP. The formulations are linearized and implemented in two solvers. An experiment with 50 instances is reported.     

Quadratic cone cutting surfaces for quadratic programs with on–off constraints

We study the convex hull of a set arising as a relaxation of difficult convex mixed integer quadratic programs (MIQP). We characterize the extreme points of the convex hull of the set and the extreme points of its continuous relaxation. We derive four quadratic cutting surfaces that improve the strength of the continuous relaxation. Each of the cutting surfaces is second-order-cone representable. Via a shooting experiment, we provide empirical evidence as to the importance of each inequality type in improving the relaxation. Computational results that employ the new cutting surfaces to strengthen the relaxation for MIQPs arising from portfolio optimization applications are promising.     

An annotated bibliography of combinatorial optimization problems with fixed cardinality constraints

In this paper, we consider combinatorial optimization problems with additional cardinality constraints. In k-cardinality combinatorial optimization problems, a cardinality constraint requires feasible solutions to contain exactly k elements of a finite set E. Problems of this type have applications in many areas, e.g. in the mining and oil industry, telecommunications, circuit layout, and location planning. We formally define the problem, mention some examples and summarize general results. We provide an annotated bibliography of combinatorial optimization problems of which versions with cardinality constraint have been considered in the literature.     

Continuous quadratic programming formulations of optimization problems on graphs

Four NP-hard optimization problems on graphs are studied: The vertex separator problem, the edge separator problem, the maximum clique problem, and the maximum independent set problem. We show that the vertex separator problem is equivalent to a continuous bilinear quadratic program. This continuous formulation is compared to known continuous quadratic programming formulations for the edge separator problem, the maximum clique problem, and the maximum independent set problem. All of these formulations, when expressed as maximization problems, are shown to follow from the convexity properties of the objective function along the edges of the feasible set. An algorithm is given which exploits the continuous formulation of the vertex separator problem to quickly compute approximate separators. Computational results are given.     

On convex quadratic programs with linear complementarity constraints

The paper shows that the global resolution of a general convex quadratic program with complementarity constraints (QPCC), possibly infeasible or unbounded, can be accomplished in finite time. The method constructs a minmax mixed integer formulation by introducing finitely many binary variables, one for each complementarity constraint. Based on the primal-dual relationship of a pair of convex quadratic programs and on a logical Benders scheme, an extreme ray/point generation procedure is developed, which relies on valid satisfiability constraints for the integer program. To improve this scheme, we propose a two-stage approach wherein the first stage solves the mixed integer quadratic program with pre-set upper bounds on the complementarity variables, and the second stage solves the program outside this bounded region by the Benders scheme. We report computational results with our method. We also investigate the addition of a penalty term y ^T Dw to the objective function, where y and w are the complementary variables and D is a nonnegative diagonal matrix. The matrix D can be chosen effectively by solving a semidefinite program, ensuring that the objective function remains convex. The addition of the penalty term can often reduce the overall runtime by at least 50 %. We report preliminary computational testing on a QP relaxation method which can be used to obtain better lower bounds from infeasible points; this method could be incorporated into a branching scheme. By combining the penalty method and the QP relaxation method, more than 90 % of the gap can be closed for some QPCC problems.