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154
Deterministic JobShop Scheduling: Past, Present and Future
 European Journal of Operational Research
, 1998
"... : Due to the stubborn nature of the deterministic jobshop scheduling problem many solutions proposed are of hybrid construction cutting across the traditional disciplines. The problem has been investigated from a variety of perspectives resulting in several analytical techniques combining generic ..."
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Cited by 105 (2 self)
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: Due to the stubborn nature of the deterministic jobshop scheduling problem many solutions proposed are of hybrid construction cutting across the traditional disciplines. The problem has been investigated from a variety of perspectives resulting in several analytical techniques combining generic as well as problem specific strategies. We seek to assess a subclass of this problem in which the objective is minimising makespan, by providing an overview of the history, the techniques used and the researchers involved. The sense and direction of their work is evaluated by assessing the reported results of their techniques on the available benchmark problems. From these results the current situation and pointers for future work are provided. KEYWORDS: Scheduling Theory; JobShop; Review; Computational Study; 1. INTRODUCTION Current market trends such as consumer demand for variety, shorter product life cycles and competitive pressure to reduce costs have resulted in the need for zero i...
Complete search in continuous global optimization and constraint satisfaction
 ACTA NUMERICA 13
, 2004
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CHVATAL CLOSURES FOR MIXED INTEGER PROGRAMMING PROBLEMS
, 1990
"... Chvátal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. T ..."
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Cited by 87 (0 self)
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Chvátal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. The basic ingredient in these cuttingplane proofs is that for a polyhedron P and integral vector w, if max(wx]x ~ P, wx integer} = t, then wx ~ t is valid for all integral vectors in P. We consider the variant of this step where the requirement that wx be integer may be replaced by the requirement that #x be integer for some other integral vector #. The cuttingplane proofs thus obtained may be seen either as an abstraction of Gomory's mixed integer cuttingplane technique or as a proof version of a simple class of the disjunctive cutting planes studied by Balas and Jeroslow. Our main result is that for a given polyhedron P, the set of vectors that satisfy every cutting plane for P with respect to a specified subset of integer variables is again a polyhedron. This allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron, analogous to the process of repeatedly taking Chvátal closures in the integer programming case. These results are illustrated with a number of examples from combinatorial optimization. Our work can be seen as a continuation of that of Nemhauser and Wolsey on mixed integer cutting planes.
EVBDDbased algorithms for integer linear programming, spectral transformation, and function decomposition
 IEEE TRANSACTIONS ON COMPUTERAIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS
, 1994
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Optimizing over the split closure
 CARNEGIE MELLON UNIVERSITY
, 2006
"... The polyhedron defined by all the split cuts obtainable directly (i.e. without iterated cut generation) from the LPrelaxation P of a mixed integer program (MIP) is termed the (elementary, or rank 1) split closure of P. This paper deals with the problem of optimizing over the split closure. This is ..."
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Cited by 37 (2 self)
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The polyhedron defined by all the split cuts obtainable directly (i.e. without iterated cut generation) from the LPrelaxation P of a mixed integer program (MIP) is termed the (elementary, or rank 1) split closure of P. This paper deals with the problem of optimizing over the split closure. This is accomplished by repeatedly solving the following separation problem: given a fractional point, say x, find a rank1 split cut violated by x or show that none exists. We show that this separation problem can be formulated as a parametric mixed integer linear program (PMILP) with a single parameter in the objective function and the right hand side. We develop an algorithmic framework to deal with the resulting PMILP by creating and maintaining a dynamically updated grid of parameter values, and use the corresponding mixed integer programs to generate rank 1 split cuts. Our approach was implemented in the COINOR framework using CPLEX 9.0 as a general purpose MIP solver. We report our computational results on wellknown benchmark instances from MIPLIB 3.0 and Capacitated Warehouse Location Problems from OrLib. Our computational results show that rank1 split cuts close more than 98 % of the duality
A Scheme for Unifying Optimization and Constraint Satisfaction Methods
, 2000
"... Optimization and constraint satisfaction methods are complementary to a large extent, and there has been much recent interest in combining them. Yet no generally accepted principle or scheme for their merger has evolved. We propose a scheme based on two fundamental dualities, the duality of search a ..."
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Cited by 34 (6 self)
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Optimization and constraint satisfaction methods are complementary to a large extent, and there has been much recent interest in combining them. Yet no generally accepted principle or scheme for their merger has evolved. We propose a scheme based on two fundamental dualities, the duality of search and inference and the duality of strengthening and relaxation. Optimization as well as constraint satisfaction methods can be seen as exploiting these dualities in their respective ways. Our proposal is that rather than employ either type of method exclusively, one can focus on how these dualities can be exploited in a given problem class. The resulting algorithm is likely to contain elements from both optimization and constraint satisfaction, and perhaps new methods that belong to neither.
Cuts for mixed 01 conic programming
, 2005
"... In this we paper we study techniques for generating valid convex constraints for mixed 01 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 01 linear programs, such as the Gomory cuts, the liftandproject cuts, and cuts from other hierarchies of ti ..."
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Cited by 31 (0 self)
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In this we paper we study techniques for generating valid convex constraints for mixed 01 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 01 linear programs, such as the Gomory cuts, the liftandproject cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 01 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 01 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.
A probabilistic approach to optimal robust path planning with obstacles
 in Proceedings of the American Control Conference
, 2006
"... Abstract — Autonomous vehicles need to plan trajectories to a specified goal that avoid obstacles. Previous approaches that used a constrained optimization approach to solve for finite sequences of optimal control inputs have been highly effective. For robust execution, it is essential to take into ..."
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Cited by 31 (11 self)
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Abstract — Autonomous vehicles need to plan trajectories to a specified goal that avoid obstacles. Previous approaches that used a constrained optimization approach to solve for finite sequences of optimal control inputs have been highly effective. For robust execution, it is essential to take into account the inherent uncertainty in the problem, which arises due to uncertain localization, modeling errors, and disturbances. Prior work has handled the case of deterministically bounded uncertainty. We present here an alternative approach that uses a probabilistic representation of uncertainty, and plans the future probabilistic distribution of the vehicle state so that the probability of collision with obstacles is below a specified threshold. This approach has two main advantages; first, uncertainty is often modeled more naturally using a probabilistic representation (for example in the case of uncertain localization); second, by specifying the probability of successful execution, the desired level of conservatism in the plan can be specified in a meaningful manner. The key idea behind the approach is that the probabilistic obstacle avoidance problem can be expressed as a Disjunctive Linear Program using linear chance constraints. The resulting Disjunctive Linear Program has the same complexity as that corresponding to the deterministic path planning problem with no representation of uncertainty. Hence the resulting problem can be solved using existing, efficient techniques, such that planning with uncertainty requires minimal additional computation. Finally, we present an empirical validation of the new method with a number of aircraft obstacle avoidance scenarios. I.
A probabilistic particle control approach to optimal, robust predictive control
 In Proceedings of the AIAA Guidance, Navigation and Control Conference
, 2006
"... Autonomous vehicles need to be able to plan trajectories to a specified goal that avoid obstacles, and are robust to the inherent uncertainty in the problem. This uncertainty arises due to uncertain state estimation, disturbances and modeling errors. Previous solutions to the robust path planning pr ..."
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Cited by 30 (8 self)
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Autonomous vehicles need to be able to plan trajectories to a specified goal that avoid obstacles, and are robust to the inherent uncertainty in the problem. This uncertainty arises due to uncertain state estimation, disturbances and modeling errors. Previous solutions to the robust path planning problem solved this problem using a finite horizon optimal stochastic control approach. This approach finds the optimal path subject to chance constraints, which ensure that the probability of collision with obstacles is below a given threshold. This approach is limited to problems where all uncertain distributions are Gaussian, and typically result in highly conservative plans. In many cases, however, the Gaussian assumption is invalid; for example in the case of localization, the belief state about a vehicle’s position can consist of highly nonGaussian, even multimodal, distributions. In this paper we present a novel method for finite horizon stochastic control of dynamic systems subject to chance constraints. The method approximates the distribution of the system state using a finite number of particles. By expressing these particles in terms of the control variables, we are able to approximate the original stochastic control problem as a deterministic one; furthermore the approximation becomes exact as the number of particles tends to infinity. For a general class of chance constrained problems with linear system dynamics, we show that the approximate problem can be solved using efficient MixedInteger Linear Programming techniques. We apply the new method to aircraft control in turbulence, and show simulation results that demonstrate the efficacy of the approach. I.