We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in a real vector space of dimension two. Previous work shows that optimizing a quadratic polynomia...
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We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in a real vector space of dimension two. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in a real vector space of dimension two can be done in polynomial time, whereas optimizing a quartic polynomial in the same type of region is NP-hard. We close the gap by showing that this problem can be solved in polynomial time for cubic polynomials. Furthermore, we show that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a bounded polyhedron in a real vector space of dimension two is solvable in polynomial time. We show that this holds for polynomials that can be translated into homogeneous polynomials, even when the translation vector is unknown. We demonstrate that such problems in the unbounded case can have smallest optimal solutions of exponential size in the size of the input, thus requiring a compact representation of solutions for a general polynomial time algorithm for the unbounded case.
We explore in this paper certain rich geometric properties hidden behind quadratic 0-1 programming. Especially, we derive new lower bounding methods and variable fixation techniques for quadratic 0-1 optimization prob...
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We explore in this paper certain rich geometric properties hidden behind quadratic 0-1 programming. Especially, we derive new lower bounding methods and variable fixation techniques for quadratic 0-1 optimization problems by investigating geometric features of the ellipse contour of a (perturbed) convex quadratic function. These findings further lead to some new optimality conditions for quadratic 0-1 programming. Integrating these novel solution schemes into a proposed solution algorithm of a branch-and-bound type, we obtain promising preliminary computational results.
In this paper, we prove that the Chvatal-Gomory closure of a set obtained as an intersection of a strictly convex body and a rational polyhedron is a polyhedron. Thus, we generalize a result of Schrijver [Schrijver, A...
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In this paper, we prove that the Chvatal-Gomory closure of a set obtained as an intersection of a strictly convex body and a rational polyhedron is a polyhedron. Thus, we generalize a result of Schrijver [Schrijver, A. 1980. On cutting planes. Ann. Discrete Math. 9 291-296], which shows that the Chvatal-Gomory closure of a rational polyhedron is a polyhedron.
This paper addresses two significant issues in the design of cellular manufacturing (CM) systems: (i) the availability of alternative locations for a cell, and (ii) the use of alternative routes to move part loads bet...
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This paper addresses two significant issues in the design of cellular manufacturing (CM) systems: (i) the availability of alternative locations for a cell, and (ii) the use of alternative routes to move part loads between cells when the capacity of the material transporter (MT) employed is limited. In addition, several other important factors in the design of CM systems including machine capacity limitations, batches of part demands, non-consecutive operations of parts, and maximum number of machines assigned to a cell are considered. A nonlinearprogramming model, comprised of binary and general integer variables, is formulated for the research problem. A higher-level heuristic solution algorithm based upon a concept known as 'tabu search' is presented for solving industry-size problems. Six different versions of the heuristic are developed to investigate the impact of long-term memory and the use of fixed versus variable tabu-list sizes. Explicit method-based techniques are developed to convert the original nonlinearprogramming model into an equivalent mixed (binary)-integer linear programming model in order to test the efficacy of the proposed solution technique for solving small problem instances. The solutions obtained from the heuristics have average deviation of less than 3% of the optimal solutions, and require less than a minute in comparison with optimizing methods that needed 1-10h of computation time. A carefully designed statistical experiment is used to compare the performance of the heuristics by solving three different problem structures, ranging from four to 30 parts, and three to nine locations. The experiment shows that as the problem size increases, the tabu-search-based heuristic using fixed tabu list size and long-term memory based on minimal frequency strategy is preferred over the other heuristics.
nonlinear constrained optimization problems in discrete and continuous spaces are an important class of problems studied extensively in artificial intelligence and operations research. These problems can be solved by ...
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nonlinear constrained optimization problems in discrete and continuous spaces are an important class of problems studied extensively in artificial intelligence and operations research. These problems can be solved by a Lagrange-multiplier method in continuous space and by an extended discrete Lagrange-multiplier method indiscrete space. When constraints are satisfied, these methods rely on gradient descents in the objective space to find high-quality solutions. On the other hand, when constraints are violated, these methods rely on gradient ascents in the Lagrange-multiplier space in order to increase the penalties on unsatisfied constraints and to force the constraints into satisfaction. The balance between gradient descents and gradient ascents depends on the relative weights between the objective function and the constraints, which indirectly control the convergence speed and solution quality of the Lagrangian method. To improve convergence speed without degrading solution quality, we propose an algorithm to dynamically control the relative weights between the objective and the constraints. Starting from an initial weight, the algorithm automatically adjusts the weights based on the behavior of the search progress. With this strategy, we are able to eliminate divergence, reduce oscillation, and speed up convergence. We show improved convergence behavior of our proposed algorithm on both nonlinear continuous and discrete problems. (C) 2000 Elsevier Science Inc. All rights reserved.
In the medium to long term, China must conduct deep emission reduction actions to transform the country's economy and meet the conditions of the Paris Agreement. As an advanced emission reduction technology, carbo...
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In the medium to long term, China must conduct deep emission reduction actions to transform the country's economy and meet the conditions of the Paris Agreement. As an advanced emission reduction technology, carbon capture and storage (CCS) is undoubtedly an important means of achieving this goal. China must consider how to retrofit existing thermal power plants to install CCS technology. Thus, the expected future roadmap for power plants with CO2 capture is of significant interest. To achieve this aim, we propose a new CCS project investment model, which helps design the CCS development roadmap for achieving the emission targets for 2050. Reductions in the cost of technologies as a result of learning-by-doing is considered to enrich the model to describe the reality more sensibly. Through some mathematical skills, we transform the original continuous problem into a nonlinear integer programming problem. By solving the model, we are trying to answer questions about when to adopt CCS technology and the cost. The results reveal that early large-scale CCS demonstrations are not necessary. The peak investment period of CCS is around 2035. Operating costs account for 80% of the overall cost of CCS, and thus, policymakers must fully motivate the development and investment of operating technologies, especially capture technologies. The results also show that when the scale of CCS technology is promoted, flexible installation levels can be considered to improve efficiency and reduce costs.
Tool reliability plays an important role in the performance and justification of flexible manufacturing systems (FMSs). Failure of a single tool can cause downtimes over the entire system. This would cause due dates t...
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Tool reliability plays an important role in the performance and justification of flexible manufacturing systems (FMSs). Failure of a single tool can cause downtimes over the entire system. This would cause due dates to be missed and can result in inferior products. Therefore, in order to justify the large capital investment associated with FMSs, the system must perform in a reliable manner to give an acceptable or required rate of return on the investment. In order to arrive at this objective, FMS reliability must be studied at the planning and design stages, tool failures pose a major obstacle to achieving this objective. In this paper, a mathematical model has been developed to determine the spare tooling requirement for the tooling system in an FMS, so that a desired system reliability is achieved and the cost is minimised. The influence of tool sharing on cost, reliability, spares requirement, and tool magazine capacity of the FMS are analysed. The tools and tool transporter are subject to general failure distributions.
Partner selection problem seeks to find a best combination of enterprises by optimising a nonlinear objective over the given constraints. In this paper the partner selection problem is modelled as a nonlinearinteger ...
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Partner selection problem seeks to find a best combination of enterprises by optimising a nonlinear objective over the given constraints. In this paper the partner selection problem is modelled as a nonlinear integer programming problem and an Ant Colony Optimization (ACO) algorithm embedded project scheduling is presented for solving the problem with the lead time, subproject cost and risk factor constraints in Virtual Enterprises (VE). Genetic Algorithm (GA) and enumeration algorithm are introduced for comparison to check the effectiveness of the ACO algorithm. A case study is implemented to verify the feasibility of the proposed approach and the computational results are satisfactory.
We present heuristics for solving a difficult nonlinear integer programming (NIP) model arising from a multi-item single machine dynamic lot-sizing problem. The heuristic obtains a local optimum for the continuous rel...
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We present heuristics for solving a difficult nonlinear integer programming (NIP) model arising from a multi-item single machine dynamic lot-sizing problem. The heuristic obtains a local optimum for the continuous relaxation of the NIP model and rounds the resulting fractional solution to a feasible integer solution by solving a series of shortest path problems. We also implement two benchmarks: a version of the well-known Feasibility Pump heuristic and the Surrogate Method developed for stochastic discrete optimization problems. Computational experiments reveal that our shortest path based rounding procedure finds better production plans than the previously developed myopic heuristic and the benchmarks. (C) 2018 Elsevier Ltd. All rights reserved.
We generalize the classical group testing problem to incorporate costs associated with pooling and inspection, both of which are significant factors in actual applications. We formulate the expected cost model as a no...
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We generalize the classical group testing problem to incorporate costs associated with pooling and inspection, both of which are significant factors in actual applications. We formulate the expected cost model as a nonlinear integer programming problem, prove several propositions and a theorem concerning when pooling is more efficient than individual testing, and determine the optimal group size such that the expected cost is minimized. (C) 2009 Elsevier Ltd. All rights reserved.
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