Duality results are established in convex programming with the set-inclusive constraints studied by Soyster. The recently developed duality theory for generalizedlinear programs by Thuente is further generalized and ...
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Duality results are established in convex programming with the set-inclusive constraints studied by Soyster. The recently developed duality theory for generalizedlinear programs by Thuente is further generalized and also brought into the framework of Soyster's theory. Convex programming with set-inclusive constraints is further extended to fractional programming.
A constructive method is presented for optimizing exactly the Traveling Salesman Problem as a sequence of shortest route problems. The method combines group-theoretic and Lagrangean relaxation constructions.
A constructive method is presented for optimizing exactly the Traveling Salesman Problem as a sequence of shortest route problems. The method combines group-theoretic and Lagrangean relaxation constructions.
Sharir and Welzl introduced an abstract framework for optimization problems, called LP-type problems or also generalized linear programming problems, which proved useful in algorithm design. We define a new, and as we...
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Sharir and Welzl introduced an abstract framework for optimization problems, called LP-type problems or also generalized linear programming problems, which proved useful in algorithm design. We define a new, and as we believe, simpler and more natural framework: violator spaces, which constitute a proper generalization of LP-type problems. We show that Clarkson's randomized algorithms for low-dimensional linearprogramming work in the context of violator spaces. For example, in this way we obtain the fastest known algorithm for the P-matrix generalizedlinear complementarity problem with a constant number of blocks. We also give two new characterizations of LP-type problems: they are equivalent to acyclic violator spaces, as well as to concrete LP-type problems (informally, the constraints in a concrete LP-type problem are subsets of a linearly ordered ground set, and the value of a set of constraints is the minimum of its intersection). (C) 2007 Elsevier B.V. All rights reserved.
This paper develops a wholly linear formulation of the posynomial geometric programming problem. It is shown that the primal geometric programming problem is equivalent to a semi-infinite linear program, and the dual ...
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This paper develops a wholly linear formulation of the posynomial geometric programming problem. It is shown that the primal geometric programming problem is equivalent to a semi-infinite linear program, and the dual problem is equivalent to a generalizedlinear program. Furthermore, the duality results that are available for the traditionally defined primal-dual pair are readily obtained from the duality theory for semi-infinite linear programs. It is also shown that two efficient algorithms (one primal based and the other dual based) for geometric programming actually operate on the semi-infinite linear program and its dual.
Variational inequalities associated with monotone operators (possibly nonlinear and multivalued) and convex sets (possibly unbounded) are studied in reflexive Banach spaces. A variety of results are given which relate...
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Variational inequalities associated with monotone operators (possibly nonlinear and multivalued) and convex sets (possibly unbounded) are studied in reflexive Banach spaces. A variety of results are given which relate to a stability concept involving a natural parameter. These include characterizations useful as criteria for stable existence of solutions and also several characterizations of surjectivity. The monotone complementarity problem is covered as a special case, and the results are sharpened for linear monotone complementarity and for generalized linear programming.
Gomory's group relaxation for integer programs has been refined by column generation methods and dual ascent algorithms to identify a set of candidate solutions which are feasible in the relaxation but not necessa...
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Gomory's group relaxation for integer programs has been refined by column generation methods and dual ascent algorithms to identify a set of candidate solutions which are feasible in the relaxation but not necessarily so in the original integer program. Attempts at avoiding branch and bound procedures at this point have focussed on providing extra group constraints which eliminate all or most of the candidate solutions so that further ascent can take place. It will be shown that a single constraint usually of order 2 or 3, can eliminate all of the candidate solutions.
Robust optimization problems are conventionally solved by reformulation as non-robust problems. We propose a direct method to separate split cuts for robust mixed-integer programs with polyhedral uncertainty sets. The...
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Robust optimization problems are conventionally solved by reformulation as non-robust problems. We propose a direct method to separate split cuts for robust mixed-integer programs with polyhedral uncertainty sets. The method generalizes the well-known cutting plane procedure of Balas. Computational experiments show that applying cutting planes directly is favorable to the reformulation approach. It is thus viable to solve robust MIP problems in a branch-and-cut framework using a generalized linear programming oracle. (C) 2012 Elsevier B.V. All rights reserved.
We propose a new approach to crew-pairing problems arising in the context of airline companies. The problem is first formulated as a large scale set covering problem with many colums, each column representing a valid ...
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We propose a new approach to crew-pairing problems arising in the context of airline companies. The problem is first formulated as a large scale set covering problem with many colums, each column representing a valid crew-pairing. We then suggest a solution procedure for the continuous relaxation of this large scale problem, based on generalized linear programming, in which the column generation subproblem is shown to be equivalent to a shortest path problem in an associated graph. Computational results obtained on a series of real problems (involving up to 329 flight segments) are reported, confirming both computational efficiency and practical applicability of the new approach. Indeed not only were the resulting solutions observed to be integral for most test problems, but average savings of about 4 to 5% over the best available hand-built solutions were shown to be obtained.
We present an algorithm for solving a large class of semi-infinite linearprogramming problems. This algorithm has several advantages: it handles feasibility and optimality together; it has very weak restrictions on t...
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We present an algorithm for solving a large class of semi-infinite linearprogramming problems. This algorithm has several advantages: it handles feasibility and optimality together; it has very weak restrictions on the constraints; it allows cuts that are not near the most violated cut; and it solves the primal and the dual problems simultaneously. We prove the convergence of this algorithm in two steps. First, we show that the algorithm can find anε-optimal solution after finitely many iterations. Then, we use this result to show that it can find an optimal solution in the limit. We also estimate how good anε-optimal solution is compared to an optimal solution and give an upper bound on the total number of iterations needed for finding anε-optimal solution under some assumptions. This algorithm is generalized to solve a class of nonlinear semi-infinite programming problems. Applications to convex programming are discussed.
Wind integration in power grids is challenging because of the uncertain nature of wind speed. Forecasting errors may have costly consequences. Indeed, power might be purchased at highest prices to meet the load, and i...
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Wind integration in power grids is challenging because of the uncertain nature of wind speed. Forecasting errors may have costly consequences. Indeed, power might be purchased at highest prices to meet the load, and in case of surplus, power may be wasted. Energy storage may provide some recourse against the uncertainty of wind generation. Because of their sequential nature, in theory, power scheduling problems may be solved via stochastic dynamic programming. However, this scheme is limited to small networks by the so-called curse of dimensionality. This paper analyzes the management of a network composed of conventional power units and wind turbines through approximate dynamic programming, more precisely stochastic dual dynamic programming. A general power network model with ramping constraints on the conventional generators is considered. The approximate method is tested on several networks of different sizes. The numerical experiments also include comparisons with classical dynamic programming on a small network. The results show that the combination of approximation techniques enables to solve the problem in reasonable time.
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