Convex Hull Representation Research Articles

A Hierarchy of Relaxations Leading to the Convex Hull Representation for General Discrete Optimization Problems

We consider linear mixed-integer programs where a subset of the variables are restricted to take on a finite number of general discrete values. For this class of problems, we develop a reformulation-linearization technique (RLT) to generate a hierarchy of linear programming relaxations that spans the spectrum from the continuous relaxation to the convex hull representation. This process involves a reformulation phase in which suitable products using a defined set of Lagrange interpolating polynomials (LIPs) are constructed, accompanied by the application of an identity that generalizes x(1−x) for the special case of a binary variable x. This is followed by a linearization phase that is based on variable substitutions. The constructs and arguments are distinct from those for the mixed 0-1 RLT, yet they encompass these earlier results. We illustrate the approach through some examples, emphasizing the polyhedral structure afforded by the linearized LIPs. We also consider polynomial mixed-integer programs, exploitation of structure, and conditional-logic enhancements, and provide insight into relationships with a special-structure RLT implementation.

Partial convexification cuts for 0–1 mixed-integer programs

In this research, we propose a new cut generation scheme based on constructing a partial convex hull representation for a given 0–1 mixed-integer programming problem by using the reformulation–linearization technique (RLT). We derive a separation problem that projects the extended space of the RLT formulation into the original space, in order to generate a cut that deletes a current fractional solution. Naturally, the success of such a partial convexification based cutting plane scheme depends on the process used to tradeoff the strength of the cut derived and the effort expended. Accordingly, we investigate several variable selection rules for performing this convexification, along with restricted versions of the accompanying separation problems, so as to be able to derive strong cuts within a reasonable effort. We also develop a strengthening procedure that enhances the generated cut by considering the binariness of the remaining unselected 0–1 variables. Finally, we present some promising computational results that provide insights into implementing the proposed cutting plane methodology.

Global optimization of mixed-integer bilevel programming problems

Two approaches that solve the mixed-integer nonlinear bilevel programming problem to global optimality are introduced. The first addresses problems mixed-integer nonlinear in outer variables and C2-nonlinear in inner variables. The second adresses problems with general mixed-integer nonlinear functions in outer level. Inner level functions may be mixed-integer nonlinear in outer variables, linear, polynomial, or multilinear in inner integer variables, and linear in inner continuous variables. This second approach is based on reformulating the mixed-integer inner problem as continuous via its vertex polyheral convex hull representation and solving the resulting nonlinear bilevel optimization problem by a novel deterministic global optimization framework. Computational studies illustrate proposed approaches.

Efficient MILP formulations for the optimal synthesis of chromatographic protein purification processes

The use of recombinant proteins has increased greatly in recent years, as well as the techniques used for their purification. The selection of an efficient process to purify proteins is a major bottleneck found when trying to scale up results obtained in the laboratory to a large-scale industrial process. One of the main challenges in the synthesis of downstream purification stages in biotechnological processes is the appropriate selection and sequencing of chromatographic steps. The objective of this work is to develop mixed integer linear programming models for the synthesis of protein purification processes. Models for each chromatographic technique rely on physicochemical data of a protein mixture, which contains the desired product and provide information on its potential purification. Formulations that are based on convex hull representations are proposed to calculate the minimum number of steps from a set of chromatographic techniques that must achieve a specified purity level and alternatively to maximize purity for a given number of steps. The proposed models are tested in several examples with experimental data and present time reductions of up to three orders of magnitude when compared to big-M formulations.

Logical Constraints as Cardinality Rules: Tight Representation

We derive by means of disjunctive programming the convex hull representation of logical conditions called cardinality rules given by Hong and Hooker (1999), and specify an efficient separation algorithm for the family of inequalities defining the convex hull. We then extend both the convex hull characterization and the efficient separation procedure to more general logical conditions.

Efficient MILP formulations for the optimal synthesis of chromatographic protein purification processes

The use of recombinant proteins has increased greatly in recent years, as well as the techniques used for their purification. The selection of an efficient process to purify proteins is a major bottleneck found when trying to scale up results obtained in the laboratory to a large-scale industrial process. One of the main challenges in the synthesis of downstream purification stages in biotechnological processes is the appropriate selection and sequencing of chromatographic steps. The objective of this work is to develop mixed integer linear programming models for the synthesis of protein purification processes. Models for each chromatographic technique rely on physicochemical data of a protein mixture, which contains the desired product and provide information on its potential purification. Formulations that are based on convex hull representations are proposed to calculate the minimum number of steps from a set of chromatographic techniques that must achieve a specified purity level and alternatively to maximize purity for a given number of steps. The proposed models are tested in several examples with experimental data and present time reductions of up to three orders of magnitude when compared to big- M formulations.

Enhanced Model Formulations for Optimal Facility Layout

This paper presents an improved mixed-integer programming (MIP) model and effective solution strategies for the facility layout problem and is motivated by the work of Meller et al. (1999). This class of problems seeks to determine a least-cost layout of departments having various size and area requirements within a rectangular building, and it is challenging even for small instances. The difficulty arises from the disjunctive constraints that prevent departmental overlaps and the nonlinear area constraints for each department, which existing models have failed to approximate with adequate accuracy. We develop several modeling and algorithmic enhancements that are demonstrated to produce more accurate solutions while also decreasing the solution effort required. We begin by deriving a novel polyhedral outer approximation scheme that can provide as accurate a representation of the area requirements as desired. We also design alternative methods for reducing problem symmetry, evaluate the performance of several classes of valid inequalities, explore the construction of partial convex hull representations for the disjunctive constraints, and investigate judicious branching variable selection priority schemes. The results indicate a substantial increase in the accuracy of the layout produced, while at the same time providing a dramatic reduction in computational effort. In particular, three previously unsolved test problems from the literature for which Meller et al.'s algorithm terminated prematurely after 24 cpu hours of computation (on a SUN Ultra 2 workstation with 390 MB RAM) with respective optimality gaps of 10.14%, 26.45%, and 40%, have been solved to exact optimality with reasonable effort using our proposed approach.

A modification of Benders' decomposition algorithm for discrete subproblems: An approach for stochastic programs with integer recourse

In this paper, we modify Benders' decomposition method by using concepts from the Reformulation-Linearization Technique (RLT) and lift-and-project cuts in order to develop an approch for solving discrete optimization problems that yield integral subproblems, such as those that arise in the case of two-stage stochastic programs with integer recourse. We first demonstrate that if a particular convex hull representation of the problem's constrained region is available when binariness is enforced on only the second-stage (or recourse) variables, then the regular Benders' algorithm is applicable. The proposed procedure is based on sequentially generating a suitable partial description of this convex hull representation as needed in the process of deriving valid Benders' cuts. The key idea is to solve the subproblems using an RLT or lift-and-project cutting plane scheme, but to generate and store the cuts as functions of the first-stage variables. Hence, we are able to re-use these cutting planes from one subproblem solution to the next simply by updating the values of the first-stage decisions. The proposed Benders' cuts also recognize these RLT or lift-and-project cuts as functions of the first-stage variables, and are hence shown to be globally valid, thereby leading to an overall finitely convergent solution procedure. Some illustrative examples are provided to elucidate the proposed approach. The focus of this paper is on developing such a finitely convergent Benders' approach for problems having 0-1 mixed-integer subproblems as in the aforementioned context of two-stage stochastic programs with integer recourse. A second part of this paper will deal with related computational experiments.

Convex hull representations of models for computing collisions between multiple bodies

In this paper, we consider a collision detection problem that frequently arises in the field of robotics. Given a set of bodies with their initial positions and trajectories, we wish to identify the first collision that occurs between any two bodies, or to determine that none exists. For the case of bodies having linear trajectories, we construct a convex hull representation of the integer programming model of S.Z. Selim and H.A. Almohamad [European Journal of Operational Research 119 (1) (1999) 121–129], and compare the relative effectiveness in solving this problem via the resultant linear program. We also extend this analysis to model a situation in which bodies move along piecewise linear trajectories, possibly rotating at the end of each linear segment. For this case, we again compare an integer programming approach with its linear programming convex hull representation, and exhibit the effectiveness of solving a sequence of mathematical programs for each time segment over a global programming scheme which considers all segments at once. We provide computational results to illustrate the effect of various numbers of bodies present in the collision scenarios, as well as the times at which the first collision occurs.

On mixed-integer zero-one representations for separable lower-semicontinuous piecewise-linear functions

In a recent paper, Padberg (Oper. Res. Lett. 27(1) (2000) 1) has provided some insights into constructing locally ideal formulations for continuous piecewise-linear approximations to separable nonlinear programs. He shows that in contrast with such representations, the standard text-book modeling strategy is weak with respect to its linear programming relaxation. We propose a new pedagogically simpler modification of the latter formulation that constructs its convex hull representation, thereby rendering it locally ideal. Moreover, this modeling strategy imparts a totally unimodular structure to the formulation, it readily extends to representing separable lower-semicontinuous piecewise-linear functions, and it also facilitates a reduced locally ideal representation based on a piecewise-linear convex decomposition of the function. In the special case of continuous piecewise-linear functions, we exhibit a nonsingular linear transformation that equivalently converts the proposed model into Padberg's formulation.

Reduced first-level representations via the reformulation-linearization technique: results, counterexamples, and computations

In this paper, we consider the reformulation-linearization technique (RLT) of Sherali and Adams (SIAM J. Discrete Math. 3 (3) (1990) 411–430, Discrete Appl. Math. 52 (1994) 83–106) and explore the generation of reduced first-level representations for 0–1 mixed-integer programs that tend to retain the strength of the full first-level linear programming relaxation. The motivation for this study is provided by the computational success of the first-level RLT representation (in full or partial form) experienced by several researchers working on various classes of problems. We show that there exists a first-level representation having only about half the RLT constraints that yields the same lower bound value via its relaxation. Accordingly, we attempt to a priori predict the form of this representation and identify many special cases for which this prediction is accurate. However, using various counter examples, we show that this prediction as well as several variants of it are not accurate in general, even for the case of a single binary variable. In addition, since the full first-level relaxation produces the convex hull representation for the case of a single binary variable, we investigate whether this is the case with respect to the reduced first-level relaxation as well, showing similarly that it holds true only for some special cases. Some empirical results on the relative merit and prediction capability of the reduced, versus the full, first-level representation are also provided.

Discrete equal-capacityp-median problem

This paper examines the discrete equal-capacity p-median problem that seeks to locate p new facilities (medians) on a network, each having a given uniform capacity, in order to minimize the sum of distribution costs while satisfying the demand on the network. Such problems arise, for example, in local access and transport area telecommunication network design problems where any number of a set of p facility units can be constructed at the specified candidate sites (hence, the net capacity is an integer multiple of a given unit capacity). We develop various valid inequalities, a separation routine for generating cutting planes that are specific members of such inequalities, as well as an enhanced reformulation that constructs a partial convex hull representation that subsumes an entire class of valid inequalities via its linear programming relaxation. We also propose suitable heuristic schemes for this problem, based on sequentially rounding the continuous relaxation solutions obtained for the various equivalent formulations of the problem. Extensive computational results are provided to demonstrate the effectiveness of the proposed valid inequalities, enhanced formulations, and heuristic schemes. The results indicate that the proposed schemes for tightening the underlying relaxations play a significant role in enhancing the performance of both exact and heuristic solution methods for this class of problems. © 2000 John & Sons, Inc. Naval Research Logistics 47: 166–183, 2000.

Tight mixed-integer optimization models for the solution of linear and nonlinear systems of disjunctive equations

This paper considers the solution of systems of algebraic equations that are expressed by a global rectangular system of equations, and a set of conditional equations that are expressed as disjunctions. These disjunctions are given by equations and inequalities, where the latter define the domain of validity of the equations. The solution of such a system is defined by variable values satisfying the rectangular equations, and exactly one set of equations for each of the disjunctions This paper addresses first the solution of systems of linear disjunctive equations. Using a convex hull representation of each of the disjunctions, it is shown that these equations can be converted into an MILP problem. A sufficient condition is presented under which this model is shown to be solvable as an LP problem. An extension to nonlinear disjunctive equations is presented by incorporating the proposed MILP formulation within a Newton iterative scheme. The application of the proposed algorithms is illustrated with several examples, including piecewise linear mass balances in process networks, and pipe networks with different flow regimes and check valves.

Exploiting Special Structures in Constructing a Hierarchy of Relaxations for 0-1 Mixed Integer Problems

A new hierarchy of relaxations is presented that provides a unifying framework for constructing a spectrum of continuous relaxations spanning from the linear programming relaxation to the convex hull representation for linear mixed integer 0-1 problems. This hierarchy is an extension of the Reformulation-Linearization Technique (RLT) of Sherali and Adams (1990, 1994a), and is particularly designed to exploit special structures. Specifically, inherent special structures are exploited by identifying particular classes of multiplicative factors that are applied to the original problem to reformulate it as an equivalent polynomial programming problem, and subsequently, this resulting problem is linearized to produce a tighter relaxation in a higher dimensional space. This general framework permits us to generate an explicit hierarchical sequence of tighter relaxations leading up to the convex hull representation. (A similar hierarchy can be constructed for polynomial mixed integer 0-1 problems.) Additional ideas for further strengthening RLT-based constraints by using conditional logical implications, as well as relationships with sequential lifting, are also explored. Several examples are presented to demonstrate how underlying special structures, including generalized and variable upper bounding, covering, partitioning, and packing constraints, as well as sparsity, can be exploited within this framework. For some types of structures, low level relaxations are exhibited to recover the convex hull of integer feasible solutions.

Logic-based MINLP algorithms for the optimal synthesis of process networks

In this paper, the MINLP problem for the optimal synthesis of process networks is modeled as a discrete optimization problem involving logic disjunctions with nonlinear equations and pure logic relations. The logic disjunctions allow the conditional modeling of equations (e.g. if a unit is selected, apply mass/heat balances; otherwise, set the flow variables to zero). It is first shown that this framework for representing discrete optimization problems greatly simplifies the step of modeling. The outer approximation algorithm is then used as a basis to derive a new logic-based OA solution method which naturally gives rise to NLP sub-problems that avoid zero flows and a disjunctive LP master problem. The initial NLP sub-problems, that provide linearizations for all the terms in the disjunctions, are selected through a set-covering problem for which we consider both the cases of disjunctive and conjunctive normal form logic. The master problem, on the other hand, is converted to mixed-integer form using a convex-hull representation. Furthermore, based on some interesting relations of outer approximation with generalized Benders decomposition, it is also shown that it is possible to derive a logic-based method for the latter algorithm. The proposed algorithm has been tested on several structural optimization problems, including a flowsheet example showing distinct advantages in robustness and computational efficiency when compared to standard MINLP models and algorithms.

Solution of algebraic systems of disjunctive equations

This paper considers the solution of systems of equations that are expressed by the two sets: a global rectangular system of equations involving more variables than equations, and a set of conditional equations that are expressed as disjunctions. The set of disjunctions are given by equations and inequalities, where the latter define the domain of validity of the equations. In this way the solution of such a system is defined by variables x satisfying the rectangular equations, and exactly one set of equations for each of the disjunctions. This paper focuses mainly in the solution of systems of linear disjunctive equations. Using a convex hull representation of the disjunctions, the disjunctive system of equations is converted into an MILP problem. A sufficient condition is presented under which the model is shown to be solvable as an LP problem. The extension of the proposed method to nonlinear disjunctive equations is also discussed. The application of the proposed algorithms are illustrated with several examples.

OPTIMAL PARALLEL ALGORITHMS FOR TRIANGULATED SIMPLE POLYGONS

We provide optimal parallel solutions to several shortest path and visibility problems set in triangulated simple polygons. Let P be a triangulated simple polygon with n vertices, preprocessed to support shortest path queries. We can find the shortest path tree from any point inside P in O(log n) time using O(n/log n) processors. In the game bounds, we can preprocess P for shooting queries (a query can be answered in O(log n) time by a uniprocessor). Given a set S of m points inside P, we can find an implicit representation of the relative convex hull of S in O(log(nm)) time with O(m) processors. If the relative convex hull has k edges, we can explicitly produce these edges in O(log(nm)) time with O(k/log(nm)) processors. All of these algorithms are deterministic and use the CREW PRAM model.

A Hierarchy of Relaxations between the Continuous and Convex Hull Representations for Zero-One Programming Problems

In this paper a reformulation technique is presented that takes a given linear zero-one programming problem, converts it into a zero-one polynomial programming problem, and then relinearizes it into an extended linear program. It is shown that the strength of the resulting reformulation depends on the degree of the terms used to produce the polynomial program at the intermediate step of this method. In fact, as this degree varies from one up to the number of variables in the problem, a hierarchy of sharper representations is obtained with the final relaxation representing the convex hull of feasible solutions. The reformulation technique readily extends to produce a similar hierarchy of linear relaxations for zero-one polynomial programming problems. A characterization of the convex hull in the original variable space is also available through a projection process. The structure of this convex hull characterization (or its other relaxations) can be exploited to generate strong or facetial valid inequalities through appropriate surrogates in a computational framework. The surrogation process can also be used to study various classes of facets for different combinatorial optimization problems. Some examples are given to illustrate this point.

Polynomial bases for quadratic and cubic polynomials which yield control points with small convex hulls

Of the many useful properties of Bézier curves and surfaces, one used very often in computer graphic render operations is the convex hull property. That is, a Bézier curve or surface lies within the convex hull of its control points. However, this convex hull is a very poor approximation to the space occupied by the curve. We present here, a polynomial basis for the quadratic and cubic curves whose convex hulls contain the curve, but with a size of about 77% and 42% of the respective Bézier convex hulls. We then show how to convert between Bézier and this small convex hull representation, and how to subdivide these curves. Such curve representations may allow large savings in rendering techniques which depend on the convex hull of the control points.

A Minimum-Cost Feature-Selection Algorithm for Binary-Valued Features

An algorithm to select the minimum-cost collection of binary-valued features for use with a linear pattern classifier is presented. The feature-selection algorithm is motivated by the convex-hull representation of pattern-space separability. Combinatorial analysis and linear programming are used to find the minimum-cost collection of binary-valued features associated with a given set of preclassified patterns. A description of the interaction between these algorithm components is provided. The algorithm guarantees that its optimal feature set will correctly classify every pattern in the classifier's training sample. Coinputational considerations associated with algorithm use are discussed. An application of the algorithm to a three-feature classifier is presented in detail.