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CONVEX-programming WITH SET-INCLUSIVE CONSTRAINTS AND ITS APPLICATIONS TO generalized linear AND FRACTIONAL-programming

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1982年第1期38卷 33-42页

作者： SINGH, C Associate Professor of Mathematics Saint Lawrence University Canton New York

Duality results are established in convex programming with the set-inclusive constraints studied by Soyster. The recently developed duality theory for generalized linear 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.

关键词： Duality theory convex programming set-inclusive constraints generalized linear programming fractional programming

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CONVERGENT DUALITY FOR THE TRAVELING SALESMAN PROBLEM

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OPERATIONS RESEARCH LETTERS 1991年第3期10卷 129-136页

作者： SHAPIRO, JF MIT CTR OPERAT RESCAMBRIDGEMA 02139 USA

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.

关键词： TRAVELING SALESMAN PROBLEM LAGRANGEAN RELAXATION SHORTEST ROUTE PROBLEM generalized linear programming GROUP THEORY

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NUMERICAL-SOLUTION OF ESTIMATION AND STOCHASTIC-CONTROL PROBLEMS IN A GUARANTEEING FORMULATION

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JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL 1994年第6期32卷 36-43页

作者： LIDOV, ML Russian Acad of Sciences Moscow Russian Federation

The direct and the dual problems of a priori estimation are formulated, and their connection with the problem of linear estimation is indicated. The problem of estimating the parameters of the motion of an artificial earth satellite under the action of unmodellable perturbations is considered. The direct and dual problems are reduced to problems of generalized linear programming and solved by special versions of the simplex method. The problem of constructing a synthesis of the control when a spacecraft moves in the neighborhood of an exponentially unstable conditionally periodic orbit is examined. The time of controllable motion when there are limitations on the number of times the engine is switched on and on the reserve of characteristic speed is maximized. A numerical solution is obtained for the ''Relikt-2'' project for launching a spacecraft into a halo orbit in the neighborhood of a collinear point of libration of the earth-sun system.

关键词： GUARANTEEING ESTIMATION UNMODELLABLE PERTURBATIONS generalized linear programming RELIKT-2 PROJECT

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POSYNOMIAL GEOMETRIC-programming AS A SPECIAL CASE OF SEMI-INFINITE linear-programming

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1990年第3期66卷 455-475页

作者： RAJGOPAL, J BRICKER, DL UNIV IOWA DEPT IND & MANAGEMENT ENGNIOWA CITYIA 52242

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 generalized linear 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.

关键词： Geometric programming semi-infinite linear programming duality generalized linear programming

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Violator spaces:: Structure and algorithms

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DISCRETE APPLIED MATHEMATICS 2008年第11期156卷 2124-2141页

作者： Gaertner, B. Matousek, J. Ruest, L. Skovron, P. Swiss Fed Inst Technol Inst Theoret Comp Sci CH-8092 Zurich Switzerland Charles Univ Prague Dept Appl Math Prague 11800 1 Czech Republic Charles Univ Prague Inst Theoret Comp Sci Prague 11800 1 Czech Republic

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 linear programming work in the context of violator spaces. For example, in this way we obtain the fastest known algorithm for the P-matrix generalized linear 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.

关键词： LP-type problem generalized linear programming violator space Clarkson's algorithms unique sink orientation generalized linear complementarity problem

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STABLE MONOTONE VARIATIONAL-INEQUALITIES

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MATHEMATICAL programming 1990年第2期48卷 303-338页

作者： MCLINDEN, L 1. Department of Mathematics University of Illinois at Urbana-Champaign 61801 Urbana IL USA

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.

关键词： Primary 47H05 49A29 90C33 Secondary 90C05 90C25 Monotone operators variational inequalities complementarity problems generalized linear programming stable solvability surjectivity reflexive Banach spaces locally convex Hausdorff spaces

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Split cuts for robust mixed-integer optimization

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OPERATIONS RESEARCH LETTERS 2012年第3期40卷 165-171页

作者： Haus, Utz-Uwe Pfeuffer, Frank ETH Dept Math Inst Operat Res CH-8092 Zurich Switzerland Univ Magdeburg Inst Math Optimierung D-39106 Magdeburg Germany

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.

关键词： Robust mixed integer programming Cutting planes Split cuts generalized linear programming Robust duality

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EFFICIENT GROUP CUTS FOR INTEGER PROGRAMS

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MATHEMATICAL programming 1979年第2期17卷 176-183页

作者： BELL, DE Harvard University Boston MA USA

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.

关键词： Integer programming Column Generation Group Theoretic Method Lagrangian Group Cuts generalized linear programming

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A NEW APPROACH FOR CREW PAIRING PROBLEMS BY COLUMN GENERATION WITH AN APPLICATION TO AIR TRANSPORTATION

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1988年第1期35卷 45-58页

作者： LAVOIE, S MINOUX, M ODIER, E UNIV MONTREAL CTR RECH TRANSPORTSMONTREAL H3C 3J7QUEBECCANADA UNIV PARIS 09 LAMSADEF-75016 PARISFRANCE AIR FRANCE OPERAT RES GRPPARISFRANCE

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 columns, 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. [ABSTRACT FROM AUTHOR]

关键词： air transportation Combinatorial optimization generalized linear programming scheduling

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A ONE-PHASE ALGORITHM FOR SEMI-INFINITE linear-programming

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MATHEMATICAL programming 1990年第1期46卷 85-103页

作者： HU, H 1. Department of Operations Research Stanford University 94305 Stanford CA USA

We present an algorithm for solving a large class of semi-infinite linear programming 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.

关键词： Semi-infinite linear programming generalized linear programming convex programming linear programming duality

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