Column enumeration based decomposition techniques for a class of non-convex MINLP problems

We propose a decomposition algorithm for a special class of nonconvex mixed integer nonlinear programming problems which have an assignment constraint. If the assignment decisions are decoupled from the remaining constraints of the optimization problem, we propose to use a column enumeration approach. The master problem is a partitioning problem whose objective function coefficients are computed via subproblems. These problems can be linear, mixed integer linear, (non-)convex nonlinear, or mixed integer nonlinear. However, the important property of the subproblems is that we can compute their exact global optimum quickly. The proposed technique will be illustrated solving a cutting problem with optimum nonlinear programming subproblems.     

NONLINEAR INTEGER PROGRAMMING AND GLOBALOPTIMIZATION

1.IntroductionAlthoughthegenerallinearintegerprogrammingproblemisNP-hard,muchworkhasbeendevotedtoit(SeeNumhauserandWolsey[1988],Schrijver[1986]).Thesolutionmethodsincludethecuttingplane,theBranch-and-Bound,thedynamicprogrammingmethodsetc..However,thegeneralnonlinearintegerprogrammingproblemisdifficulttosolve.GareyandJohnson[1979]pointedoutthattheintegerprogrammingoverRewithalinearobjectivefunctionandquadraticconstraintsisundecidable.Soifanonlinearintegerprogrammingproblemishandled,itisalw…     

Bus driver duty optimization using an integer programming and evolutionary hybrid algorithm

This paper describes the details of a successful application where an integer programming and evolutionary hybrid algorithm was used to solve a bus driver duty optimization problem. The task is NP-hard, therefore theoretically optimal solutions can only be calculated for very small problem instances. Our aim is to obtain solutions of good quality within reasonable time limits. We first applied an integer programming approach to a set partitioning problem. The model was solved with a column generation algorithm in a branch and bound scheme. In order to solve larger real-life problems, we have combined the integer programming method with a greedy 1+1 steady state evolutionary algorithm. The resulting hybrid algorithm was capable of providing near-optimal solutions within reasonable timescales to larger instances of the bus driver scheduling problem. We present the results and running times of our algorithm in detail, as well as possible directions of future improvements.     

Parametric global optimisation for bilevel programming

We propose a global optimisation approach for the solution of various classes of bilevel programming problems (BLPP) based on recently developed parametric programming algorithms. We first describe how we can recast and solve the inner (follower’s) problem of the bilevel formulation as a multi-parametric programming problem, with parameters being the (unknown) variables of the outer (leader’s) problem. By inserting the obtained rational reaction sets in the upper level problem the overall problem is transformed into a set of independent quadratic, linear or mixed integer linear programming problems, which can be solved to global optimality. In particular, we solve bilevel quadratic and bilevel mixed integer linear problems, with or without right-hand-side uncertainty. A number of examples are presented to illustrate the steps and details of the proposed global optimisation strategy.     

Comparative studies on dynamic programming and integer programming approaches for concave cost production/inventory control problems

This paper is concerned with classical concave cost multi-echelon production/inventory control problems studied by W. Zangwill and others. It is well known that the problem with m production steps and n time periods can be solved by a dynamic programming algorithm in O(n ⁴ m) steps, which is considered as the fastest algorithm for solving this class of problems. In this paper, we will show that an alternative 0–1 integer programming approach can solve the same problem much faster particularly when n is large and the number of 0–1 integer variables is relatively few. This class of problems include, among others problem with set-up cost function and piecewise linear cost function with fewer linear pieces. The new approach can solve problems with mixed concave/convex cost functions, which cannot be solved by dynamic programming algorithms.     

A partitioning algorithm for the network loading problem

This paper proposes a Benders-like partitioning algorithm to solve the network loading problem. The approach is an iterative method in which the integer programming solver is not used to produce the best integer point in the polyhedral relaxation of the set of feasible capacities. Rather, it selects an integer solution that is closest to the best known integer solution. Contrary to previous approaches, the method does not exploit the original mixed integer programming formulation of the problem. The effort of computing integer solutions is entirely left to a pure integer programming solver while valid inequalities are generated by solving standard nonlinear multicommodity flow problems. The method is compared to alternative approaches proposed in the literature and appears to be efficient for computing good upper bounds.     

Unboundedness in Integer and Discrete Programming L.P. Relaxations

Commercial branch and bound codes for solving the general mixed integer linear programming problem commence by solving the linear programming relaxation of the submitted problem, terminating if the relaxation is unbounded. It is assumed that the submitted problem is either unbounded or has no feasible solutions. It is shown that the assumption is correct for all integer programming problems which can be submitted to the currently available codes (though counter examples which cannot be so submitted are given), but that the assumption is generally incorrect for discrete linear programming problems (using for example the special ordered set construct). Sufficient conditions on formulations to ensure its correctness are given. One possible formulation approach, applicable to special ordered set situations, is discussed.     

Polynomial Algorithms for a Class of Discrete Minmax Linear Programming Problems

We consider the problem of obtaining integer solutions to a minmax linear programming problem. Although this general problem is NP-complete, it is shown that a restricted version of this problem can be solved in polynomial time. For this restricted class of problems two polynomial time algorithms are suggested, one of which is strongly polynomial whenever its continuous analogue and an associated linear programming problem can be solved by a strongly polynomial algorithm. Our algorithms can also be used to obtain integer solutions for the minmax transportation problem with an inequality budget constraint. The equality constrained version of this problem is shown to be NP-complete. We also provide some new insights into the solution procedures for the continuous minmax linear programming problem.     

Multi-objective integer programming: A general approach for generating all non-dominated solutions

In this paper we develop a general approach to generate all non-dominated solutions of the multi-objective integer programming (MOIP) Problem. Our approach, which is based on the identification of objective efficiency ranges, is an improvement over classical ε-constraint method. Objective efficiency ranges are identified by solving simpler MOIP problems with fewer objectives. We first provide the classical ε-constraint method on the bi-objective integer programming problem for the sake of completeness and comment on its efficiency. Then present our method on tri-objective integer programming problem and then extend it to the general MOIP problem with k objectives. A numerical example considering tri-objective assignment problem is also provided.     

An exact approach for solving integer problems under probabilistic constraints with random technology matrix

This paper addresses integer programming problems under probabilistic constraints involving discrete distributions. Such problems can be reformulated as large scale integer problems with knapsack constraints. For their solution we propose a specialized Branch and Bound approach where the feasible solutions of the knapsack constraint are used as partitioning rules of the feasible domain. The numerical experience carried out on a set covering problem with random covering matrix shows the validity of the solution approach and the efficiency of the implemented algorithm.     

A heuristic decomposition approach to optimal control in a water supply model

The optimal pump control problem in a water supply system can be formulated as a mixed integer programming problem. In general, this problem is very difficult to solve by conventional integer programming algorithms, because the number of decision variables is as large as the total number of combinations of pump stations and control periods. However, it possesses a certain block triangular structure, which offers an attractive computational scheme. Taking advantage of this structure, this paper proposes a heuristic decomposition algorithm for finding a good feasible solution to this type of mixed integer programming problems. Numerical results for an actual pump control problem are also reported.     

The reformulation of two mixed integer programming problems

Two practical problems are described, each of which can be formulated in more than one way as a mixed integer programming problem. The computational experience with two formulations of each problem is given. It is pointed out how in each case a reformulation results in the associated linear programming problem being more constrained. As a result the reformulated mixed integer problem is easier to solve. The problems are a multi-period blending problem and a mining investment problem.     

Constraint Programming and Hybrid Formulations for Three Life Designs

Conway's game of Life provides an interesting testbed for exploring issues in formulation, symmetry, and optimization with constraint programming and hybrid constraint programming/integer programming methods. We consider three Life pattern-creation problems: finding maximum density still-Lifes, finding smallest immediate predecessor patterns, and finding period-2 oscillators. For the first two problems, integrating integer programming and constraint programming approaches provides a much better solution procedure than either individually. For the final problem, the constraint programming formulation provides the better approach.     

An algorithm for the multiple objective integer linear programming problem

A technique is presented for solving the multiple objective integer linear programming problem. The technique can be used to identify some or all efficient solutions. While the technique is applicable with any integer programming algorithm, it is well suited for implementation using integer postoptimality techniques. Such an implementation, based on Balas' Additive algorithm, is described for problems with zero-one variables.     

Solving Fixed-Charge Network Flow Problems with a Hybrid Optimization and Constraint Programming Approach

We apply to fixed charge network flow (FCNF) problems a general hybrid solution method that combines constraint programming and linear programming. FCNF problems test the hybrid approach on problems that are already rather well suited for a classical 0–1 model. They are solved by means of a global constraint that generates specialized constraint propagation algorithms and a projected relaxation that can be rapidly solved as a minimum cost network flow problem. The hybrid approach ran about twice as fast as a commercial mixed integer programming code on fixed charge transportation problems with its advantage increasing with problem size. For general fixed charge transshipment problems, however, it has no effect because the implemented propagation methods are weak.     

An approach to the modeling and analysis of multiobjective generalized networks

Generalized networks can often provide substantial advantages in both the modeling and solution of integer programming problems. In this paper we present a straightforward approach which combines generalized networks with goal programming so as to achieve a modeling and solution methodology for multiobjective generalized networks. Such an approach also encompasses the solution to weighted integer foal programming as well as lexicographic integer goal programming problems. In ongoing research, the resulting hybrid algorithms have indicated superior performance, for a number of problems, over that obtained by more conventional approaches. A particularly attractive feature of the methodology is its relative simplicity and robustness.     

A decomposition approach for optimal control problems with integer innerpoint state constraints

A large class of optimal control problems for hybrid dynamic systems can be formulated as mixed‐integer optimal control problems (MIOCPs). A decomposition approach is suggested to solve a special subclass of MIOCPs with mixed integer inner point state constraints. It is the intrinsic combinatorial complexity of the discrete variables in addition to the high nonlinearity of the continuous optimal control problem that forms the challenges in the theoretical and numerical solution of MIOCPs. During the solution procedure the problem is decomposed at the inner time points into a multiphase problem with mixed integer boundary constraints and phase transitions at unknown switching points. Due to a discretization of the state space at the switching points the problem can be decoupled into a family of continuous optimal control problems (OCPs) and a problem similar to the asymmetric group traveling salesman problem (AGTSP). The OCPs are transcribed by direct collocation to large‐scale nonlinear programming problems, which are solved efficiently by an advanced SQP method. The results are used as weights for the edges of the graph of the corresponding TSP‐like problem, which is solved by a Branch‐and‐Cut‐and‐Price (BCP) algorithm. The proposed approach is applied to a hybrid optimal control benchmark problem for a motorized traveling salesman. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

DEA based dimensionality reduction for classification problems satisfying strict non-satiety assumption

This study shows how data envelopment analysis (DEA) can be used to reduce vertical dimensionality of certain data mining databases. The study illustrates basic concepts using a real-world graduate admissions decision task. It is well known that cost sensitive mixed integer programming (MIP) problems are NP-complete. This study shows that heuristic solutions for cost sensitive classification problems can be obtained by solving a simple goal programming problem by that reduces the vertical dimension of the original learning dataset. Using simulated datasets and a misclassification cost performance metric, the performance of proposed goal programming heuristic is compared with the extended DEA-discriminant analysis MIP approach. The holdout sample results of our experiments shows that the proposed heuristic approach outperforms the extended DEA-discriminant analysis MIP approach.     

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.     

The timetable constrained distance minimization problem

We define the timetable constrained distance minimization problem (TCDMP) which is a sports scheduling problem applicable for tournaments where the total travel distance must be minimized. The problem consists of finding an optimal home-away assignment when the opponents of each team in each time slot are given. We present an integer programming, a constraint programming formulation and describe two alternative solution methods: a hybrid integer programming/constraint programming approach and a branch and price algorithm. We test all four solution methods on benchmark problems and compare the performance. Furthermore, we present a new heuristic solution method called the circular traveling salesman approach (CTSA) for solving the traveling tournament problem. The solution method is able to obtain high quality solutions almost instantaneously, and by applying the TCDMP, we show how the solutions can be further improved.