Global optimization using special ordered sets

The task of finding global optima to general classes of nonconvex optimization problem is attracting increasing attention. McCormick [4] points out that many such problems can conveniently be expressed in separable form, when they can be tackled by the special methods of Falk and Soland [2] or Soland [6], or by Special Ordered Sets. Special Ordered Sets, introduced by Beale and Tomlin [1], have lived up to their early promise of being useful for a wide range of practical problems. Forrest, Hirst and Tomlin [3] show how they have benefitted from the vast improvements in branch and bound integer programming capabilities over the last few years, as a result of being incorporated in a general mathematical programming system.Nevertheless, Special Ordered Sets in their original form require that any continuous functions arising in the problem be approximated by piecewise linear functions at the start of the analysis. The motivation for the new work described in this paper is the relaxation of this requirement by allowing automatic interpolation of additional relevant points in the course of the analysis.This is similar to an interpolation scheme as used in separable programming, but its incorporation in a branch and bound method for global optimization is not entirely straightforward. Two by-products of the work are of interest. One is an improved branching strategy for general special-ordered-set problems. The other is a method for finding a global minimum of a function of a scalar variable in a finite interval, assuming that one can calculate function values and first derivatives, and also bounds on the second derivatives within any subinterval.The paper describes these methods, their implementation in the UMPIRE system, and preliminary computational experience.     

A separable integer programming problem equivalent to its continual version

The separable integer programming problem with so called nested constraints is shown to be equivalent to its continual version obtained by piecewise linear continuation of the cost functions. A new approach to solution of the latter based on its successive reduction in size is suggested. When applied to the problem with piecewise linear convex functions it leads to two algorithms for its solution applicable also to the similar integer problem. These algorithms turn out more efficient than those obtained by dynamic programming approach.     

A global optimization method for nonconvex separable programming problems

Conventional methods of solving nonconvex separable programming (NSP) problems by mixed integer programming methods requires adding numerous 0–1 variables. In this work, we present a new method of deriving the global optimum of a NSP program using less number of 0–1 variables. A separable function is initially expressed by a piecewise linear function with summation of absolute terms. Linearizing these absolute terms allows us to convert a NSP problem into a linearly mixed 0–1 program solvable for reaching a solution which is extremely close to the global optimum.     

On the inventory model with variable lead time and price–quantity discount

In this paper, we propose a mixed integer optimization approach for solving the inventory problem with variable lead time, crashing cost, and price–quantity discount. A linear programming relaxation based on piecewise linearization techniques is derived for the problem. It first converts non-linear terms into the sum of absolute terms, which are then linearized by goal programming techniques and linearization approaches. The proposed method can eliminate the complicated multiple-step solution process used in the traditional inventory models. In addition, the proposed model allows constraints to be added by the inventory decision-maker as deemed appropriate in real-world situations.     

Linear,Integer, Separable and Fuzzy Programming Problems: A Unified Approach towards Reformulation

For mathematical programming (MP) to have greater impact as a decision tool, MP software systems must offer suitable support in terms of model communication and modelling techniques. In this paper, modelling techniques that allow logical restrictions to be modelled in integer programming terms are described, and their implications discussed. In addition, it is illustrated that many classes of non-linearities which are not variable separable may be, after suitable algebraic manipulation, put in a variable separable form. The methods of reformulating the fuzzy linear programming problem as a max-min problem is also introduced. It is shown that analysis of bounds plays a key role in the following four important contexts: model reduction, reformulation of logical restrictions as 0-1 mixed integer programmes, reformulation of non-linear programmes as variable separable programmes and reformulation of fuzzy linear programmes. It is observed that, as well as incorporating an interface between the modeller and the optimizer, there is a need to make available to the modeller software facilities which support the model reformulation techniques described here.     

Global optimization of concave functions subject to quadratic constraints: An application in nonlinear bilevel programming

When the follower's optimality conditions are both necessary and sufficient, the nonlinear bilevel program can be solved as a global optimization problem. The complementary slackness condition is usually the complicating constraint in such problems. We show how this constraint can be replaced by an equivalent system of convex and separable quadratic constraints. In this paper, we propose different methods for finding the global minimum of a concave function subject to quadratic separable constraints. The first method is of the branch and bound type, and is based on rectangular partitions to obtain upper and lower bounds. Convergence of the proposed algorithm is also proved. For computational purposes, different procedures that accelerate the convergence of the proposed algorithm are analysed. The second method is based on piecewise linear approximations of the constraint functions. When the constraints are convex, the problem is reduced to global concave minimization subject to linear constraints. In the case of non-convex constraints, we use zero-one integer variables to linearize the constraints. The number of integer variables depends only on the concave parts of the constraint functions.Parts of the present paper were prepared while the second author was visiting Georgia Tech and the University of Florida.     

A reformulation framework for global optimization

In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions two versions of the so-called αBB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem will form a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.     

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.     

A patient assignment algorithm for home care services

We consider the problem of assigning patients to nurses for home care services. The aim is to balance the workload of the nurses while avoiding long travels to visit the patients. We analyse the case of the CSSS Côte-des-Neiges, Métro and Parc Extension for which a previous analysis has shown that demand fluctuations may create work overload for the nursing staff. We propose a mixed integer programming model with some non-linear constraints and a non-linear objective which we solve using a Tabu Search algorithm. A simplification of the workload measure leads to a linear mixed integer program which we optimize using CPLEX.     

Special ordered sets and an application to gas supply operations planning

Special Ordered Sets provide a powerful means of modeling nonconvex functions and discrete requirements, though there has been a tendency to think of them only in terms of multiple-choice zero-one programming. This paper emphasizes the origins and generality of the special ordered set concept, and describes an application in which type 2 sets are used in several forms to model both logical conditions and nonlinear functions.Now at IBM Almaden Research Center, San Jose, CA 95120.     

Pricing and revenue maximization over a multicommodity transportation network: the nonlinear demand case

This paper is concerned with the design of efficient exact and heuristic algorithms for addressing a bilevel network pricing problem where demand is a nonlinear function of travel cost. The exact method is based on the piecewise linear approximation of the demand function, yielding mixed integer programming formulations, while heuristic procedures are developed within a bilevel trust region framework.     

A method for solving the general parametric linear complementarity problem

This paper presents a solution method for the general (mixed integer) parametric linear complementarity problem pLCP(q(θ),M), where the matrix M has a general structure and integrality restriction can be enforced on the solution. Based on the equivalence between the linear complementarity problem and mixed integer feasibility problem, we propose a mixed integer programming formulation with an objective of finding the minimum 1-norm solution for the original linear complementarity problem. The parametric linear complementarity problem is then formulated as multiparametric mixed integer programming problem, which is solved using a multiparametric programming algorithm. The proposed method is illustrated through a number of examples.     

Locally ideal formulations for piecewise linear functions with indicator variables

In this paper, we consider mixed integer linear programming (MIP) formulations for piecewise linear functions (PLFs) that are evaluated when an indicator variable is turned on. We describe modifications to standard MIP formulations for PLFs with desirable theoretical properties and superior computational performance in this context.     

A seasonal demand inventory model with variable lead time and resource constraints

In this paper, two new methods are proposed for solving a seasonal demand problem with variable lead-time and resource constraints. Despite its significance, no study has been done on such problem to obtain the best policy. First, in order to solve the variable lead time, a linear programming relaxation using piecewise linearization techniques is derived. Then, a mixed integer program with linearization techniques is constructed for the seasonal demand problem. Finally, some illustrative examples are included to demonstrate the applicability of the proposed models.     

On the mixed integer signomial programming problems

This paper proposes an approximate method to solve the mixed integer signomial programming problem, for which the objective function and the constraints may contain product terms with exponents and decision variables, which could be continuous or integral. A linear programming relaxation is derived for the problem based on piecewise linearization techniques, which first convert a signomial term into the sum of absolute terms; these absolute terms are then linearized by linearization strategies. In addition, a novel approach is included for solving integer and undefined problems in the logarithmic piecewise technique, which leads to more usefulness of the proposed method. The proposed method could reach a solution as close as possible to the global optimum.     

A continuous DC programming approach to the strategic supply chain design problem from qualified partner set

We present a new continuous approach based on the DC (difference of convex functions) programming and DC algorithms (DCA) to the problem of supply chain design at the strategic level when production of a new market opportunity has to be launched among a set of qualified partners. A well known formulation of this problem is the mixed integer linear program. In this paper, we reformulate this problem as a DC program by using an exact penalty technique. The proposed algorithm is a combination of DCA and Branch and Bound scheme. It works in a continuous domain but provides mixed integer solutions. Numerical simulations on many empirical data sets show the efficiency of our approach with respect to the standard Branch and Bound algorithm.     

Dynamic project selection and funding under risk: A decision tree based MILP approach

This paper presents a mixed integer linear programming (MILP) model for a new class of dynamic project selection and funding problems under risk given multiple scarce resources of different qualifications. The underlying stochastic decision tree concept extends classical approaches mainly in that it adds a novel node type that allows for the continuous control of discrete branching probability distributions. The control functions are piecewise linear and are convex for the costs and concave for the benefits. The MILP-model has been embedded in a prototype Decision Support System (DSS). With respect to the proposed solution the DSS provides complete probability distributions for both costs and benefits.     

Implementations of special ordered sets in MP software

Special ordered sets (SOS) have been introduced as a practical device for efficiently handling special classes of nonconvex optimization problems. They are now implemented in most commercial codes for mathematical programming (MP software). The paper gives a survey of possible applications as multiple choice restrictions, conditional multiple choice restrictions, discrete variables, discontinuous variables and piecewise linear functions, global optimization of separable programming problems, alternative right-hand sides, overlapping special ordered sets and the solution of quadratic programming problems. Alternative problem formulations are discussed. Since special ordered sets are not defined uniquely modelling facilities depend on the definition of a special orderedset in a code. The paper demonstrates the superiority of SOS to the application of binary variables if they are treated judiciously.     

Fair transfer price and inventory holding policies in two-enterprise supply chains

A key issue in supply chain optimisation involving multiple enterprises is the determination of policies that optimise the performance of the supply chain as a whole while ensuring adequate rewards for each participant.In this paper, we present a mathematical programming formulation for fair, optimised profit distribution between echelons in a general multi-enterprise supply chain. The proposed formulation is based on an approach applying the Nash bargaining solution for finding optimal multi-partner profit levels subject to given minimum echelon profit requirements.The overall problem is first formulated as a mixed integer non-linear programming (MINLP) model. A spatial and binary variable branch-and-bound algorithm is then applied to the above problem based on exact and approximate linearisations of the bilinear terms involved in the model, while at each node of the search tree, a mixed integer linear programming (MILP) problem is solved. The solution comprises inter-firm transfer prices, production and inventory levels, flows of products between echelons, and sales profiles.The applicability of the proposed approach is demonstrated by a number of illustrative examples based on industrial processes.     

对带有盒约束的二次整数规划的一种线性化方法

本文主要讨论了二次整数规划问题的线性化方法.在目标函数为二次函数的情况下,我们讨论了带有二次约束的整数规划问题的线性化方法,并将文献中对二次0-1问题的研究拓展为对带有盒约束的二次整数规划问题的研究.最终将带有盒约束的二次整数规划问题转化为线性混合本文主要讨论了二次整数规划问题的线性化方法.在目标函数为二次函数的情况下,我们讨论了带有二次约束的整数规划问题的线性化方法,并将文献中对二次0-1问题的研究拓展为对带有盒约束的二次整数规划问题的研究.最终将带有盒约束的二次整数规划问题转化为线性混合0-1整数规划问题,然后利用Ilog-cplex或Excel软件中的规划求解工具进行求解,从而解决原二次整数规划.