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39
Mixed Logical/Linear Programming
 Discrete Applied Mathematics
, 1996
"... Mixed logical/linear programming (MLLP) is an extension of mixed integer/linear programming (MILP). It represents the discrete elements of a problem with logical propositions and provides a more natural modeling framework than MILP. It can also have computational advantages, partly because it elimin ..."
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Cited by 33 (10 self)
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Mixed logical/linear programming (MLLP) is an extension of mixed integer/linear programming (MILP). It represents the discrete elements of a problem with logical propositions and provides a more natural modeling framework than MILP. It can also have computational advantages, partly because it eliminates integer variables when they serve no purpose, provides alternatives to the traditional continuous relaxation, and applies logic processing algorithms. This paper surveys previous work and attempts to organize ideas associated with MLLP, some old and some new, into a coherent framework. It articulates potential advantages and disadvantages of MLLP and illustrates some of them with computational experiments. 1 Introduction Mixed logical/linear programming (MLLP) is a general approach to formulating and solving optimization problems that have both discrete and continuous elements. Mixed integer/linear programming (MILP), the traditional approach, is effective in many instances. But it unn...
A hybrid method for planning and scheduling
 In Procs. of the 10th Intern. Conference on Principles and Practice of Constraint Programming  CP 2004
, 2004
"... We combine mixed integer linear programming (MILP) and constraint programming (CP) to solve planning and scheduling problems. Tasks are allocated to facilities using MILP and scheduled using CP, and the two are linked via logicbased Benders decomposition. Tasks assigned to a facility may run in par ..."
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Cited by 26 (6 self)
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We combine mixed integer linear programming (MILP) and constraint programming (CP) to solve planning and scheduling problems. Tasks are allocated to facilities using MILP and scheduled using CP, and the two are linked via logicbased Benders decomposition. Tasks assigned to a facility may run in parallel subject to resource constraints (cumulative scheduling). We solve minimum cost problems, as well as minimum makespan problems in which all tasks have the same release date and deadline. We obtain computational speedups of several orders of magnitude relative to the state of the art in both MILP and CP. We address a fundamental class of planning and scheduling problems for manufacturing and supply chain management. Tasks must be assigned to facilities and scheduled subject to release dates and deadlines. Tasks may run in parallel on a given facility provided the total resource consumption at any time remains with limits (cumulative scheduling). In our study the objective is to minimize cost or minimize makespan. The problem naturally decomposes into an assignment portion and a scheduling portion. We exploit the relative strengths of mixed integer linear programming (MILP) and constraint programming (CP) by applying MILP to the assignment problem and CP to the scheduling problem. We then link the two with a logicbased Benders algorithm. We obtain speedups of several orders of magnitude relative to the existing state of the art in both mixed integer programming (CPLEX) and constraint programming (ILOG Scheduler). As a result we solve larger instances to optimality than could be solved previously. In minimum makespan problems, the Benders method provides a feasible solution and a lower bound on the optimal makespan even when it is terminated before finding a provably optimal solution. 1. The Basic Idea Benders decomposition solves a problem by enumerating values of certain primary variables. For each set of values enumerated, it solves the subproblem that results from fixing the primary variables to these values. Solution of the subproblem generates a Benders cut (a type of nogood) that the primary variables must satisfy in all subsequent solutions enumerated. The next set of values for the primary variables is
New Algorithms for Nonlinear Generalized Disjunctive Programming
 Computers and Chemical Engineering Journal
, 2000
"... Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative model to MINLP for representing discrete/continuous optimization problems. The basic idea of GDP consists of representing discrete decisions in the continuous space with disjunctions, and constraints in the disc ..."
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Cited by 22 (17 self)
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Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative model to MINLP for representing discrete/continuous optimization problems. The basic idea of GDP consists of representing discrete decisions in the continuous space with disjunctions, and constraints in the discrete space with logic propositions. In this paper, we describe a new convex nonlinear relaxation of the nonlinear GDP problem that relies on the use of the convex hull of each of the disjunctions involving nonlinear inequalities. The proposed nonlinear relaxation is used to reformulate the GDP problem as a tight MINLP problem, and for deriving a branch and bound method. Properties of these methods are given, and the relation of this method with the Logic Based OuterApproximation method is established. Numerical results are presented for problems in jobshop scheduling, synthesis of process networks, optimal positioning of new products and batch process design.
Logic, Optimization, and Constraint Programming
 INFORMS Journal on Computing
, 2000
"... Because of their complementary strengths, optimization and constraint programming can be profitably merged. Their integration has been the subject of increasing commercial and research activity. This paper summarizes and contrasts the characteristics of the two fields; in particular, how they use ..."
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Cited by 12 (2 self)
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Because of their complementary strengths, optimization and constraint programming can be profitably merged. Their integration has been the subject of increasing commercial and research activity. This paper summarizes and contrasts the characteristics of the two fields; in particular, how they use logical inference in di#erent ways, and how these ways can be combined. It sketches the intellectual background for recent e#orts at integration. In particular, it traces the history of logicbased methods in optimization and the development of constraint programming in artificial intelligence. It concludes with a review of recent research, with emphasis on schemes for integration, relaxation methods, and practical applications. Optimization and constraint programming are beginning to converge, despite their very di#erent origins. Optimization is primarily associated with mathematics and engineering, while constraint programming developed much more recently in the computer science an...
Generalized Convex Disjunctive Programming: Nonlinear Convex Hull Relaxation
 Computational Optimization and Applications
, 2001
"... Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative to mixedinteger programming for represent ing discrete/continuous optimization problems. The basic idea of GDP consists of representing these problems in terms of sets of disjunctions in the continuous spa ..."
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Cited by 12 (3 self)
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Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative to mixedinteger programming for represent ing discrete/continuous optimization problems. The basic idea of GDP consists of representing these problems in terms of sets of disjunctions in the continuous space, and logic propositions in terms of Boolean variables. In this paper we consider GDP problems involving convex nonlinear inequalities in the disjunctions. Based on the work by Stubbs and Mehrotra [19] and Ceria and Soares [5], we propose a con vex nonlinear relaxation of the nonlinear convex GDP problem that relies on the convex hull of each of the disjunctions that is obtained by variable disaggregation and reformulation of the inequalities. The proposed nonlinear relaxation is used to formulate the GDP problem as a MixedInteger Nonlinear Programming (MINLP) problem that is shown to be tighter than the conventional "bigM" formulation. A disjunctive branch and bound method is also presented, and numerical results are given for a set of test problems.
An Iterative Aggregation/Disaggregation Approach for the Solution of a Mixed Integer Nonlinear Oilfield Infrastructure Planning Model
, 1999
"... A multiperiod MINLP model for offshore oilfield infrastructure planning is presented where nonlinear reservoir behavior is incorporated directly into the formulation. Discrete decisions include the selection of production platforms, well platforms and wells to be installed/drilled, as well as the dr ..."
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Cited by 11 (8 self)
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A multiperiod MINLP model for offshore oilfield infrastructure planning is presented where nonlinear reservoir behavior is incorporated directly into the formulation. Discrete decisions include the selection of production platforms, well platforms and wells to be installed/drilled, as well as the drilling schedule for the wells over the planning horizon. Continuous decisions include the capacities of the platforms, as well as the production profile for each well in each time period. For the solution of this model, an iterative aggregation/disaggregation algorithm is proposed in which logicbased methods, a bilevel decomposition technique, the use of convex envelopes and aggregation of time periods are integrated. Furthermore, a novel dynamic programming subproblem is proposed to improve the aggregation scheme at each iteration in order to obtain an aggregate problem that resembles the disaggregate problem more closely. A number of examples are presented to illustrate the performance of the proposed method. Keywords Oilfield planning, MINLP, aggregation, decomposition
Algorithms and software for convex mixed integer nonlinear programs, IMA Volumes
"... Abstract. This paper provides a survey of recent progress and software for solving convex mixed integer nonlinear programs (MINLP)s, where the objective and constraints are defined by convex functions and integrality restrictions are imposed on a subset of the decision variables. Convex MINLPs have ..."
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Cited by 10 (2 self)
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Abstract. This paper provides a survey of recent progress and software for solving convex mixed integer nonlinear programs (MINLP)s, where the objective and constraints are defined by convex functions and integrality restrictions are imposed on a subset of the decision variables. Convex MINLPs have received sustained attention in recent years. By exploiting analogies to wellknown techniques for solving mixed integer linear programs and incorporating these techniques into software, significant improvements have been made in the ability to solve these problems. Key words. Mixed Integer Nonlinear Programming; Branch and Bound; AMS(MOS) subject classifications.
Optimization models for operative planning in drinking water networks
 IN PREPARATION
, 2004
"... The topic of this paper is minimum cost operative planning of pressurized water supply networks over a finite horizon and under reliable demand forecast. Since this is a very hard problem, it is desirable to employ sophisticated mathematical algorithms, which in turn calls for carefully designed m ..."
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Cited by 9 (3 self)
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The topic of this paper is minimum cost operative planning of pressurized water supply networks over a finite horizon and under reliable demand forecast. Since this is a very hard problem, it is desirable to employ sophisticated mathematical algorithms, which in turn calls for carefully designed models with suitable properties. The paper develops a nonlinear mixed integer model and a nonlinear programming model with favorable properties for gradientbased optimization methods, based on smooth component models for the network elements. In combination with further nonlinear programming techniques (to be reported elsewhere), practically satisfactory nearoptimum solutions even for large networks can be generated in acceptable time using standard optimization software on a PC workstation. Such an optimization system is in operation at Berliner Wasserbetriebe.
Disjuntive multiperiod optimization methods for design and planning of chemical process systems
 Computers and Chemical Engineering
, 1999
"... In this paper, we present a general disjunctive multiperiod nonlinear optimization model, which incorporates design, as well as operation and expansion planning, and takes into account the corresponding costs incurred in each time period. This model is formulated with the use of logic and disjunctiv ..."
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Cited by 8 (3 self)
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In this paper, we present a general disjunctive multiperiod nonlinear optimization model, which incorporates design, as well as operation and expansion planning, and takes into account the corresponding costs incurred in each time period. This model is formulated with the use of logic and disjunctive programming, and includes Boolean variables for design, operation planning and expansion planning. We propose two algorithms for the solution of these problems. The first is a logicbased Outer Approximation (OA) algorithm for multiperiod problems. The second is a bilevel decomposition algorithm, that exploits the problem structure by decomposing it into an upper level design problem and a lower level operation and expansion planning problem, each of which is solved with the logicbased OA algorithm. Applications are considered in the areas of design and planning of process networks, as well as retrofit design for multiproduct batch plants. The results show that the disjunctive logicbased OA algorithm performs best for small problems, while the disjunctive bilevel decomposition algorithm is superior for larger problems. In both cases, a significant decrease in MILP solution time and total solution time is achieved compared to DICOPT++. Results also show that problems with a significant number of time periods can be solved in