We propose a new convex optimization formulation for the Fisher market problem with linear utilities. like the Eisenberg-Gale formulation, the set of feasible points is a polyhedral convex set while the cost function ...
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We propose a new convex optimization formulation for the Fisher market problem with linear utilities. like the Eisenberg-Gale formulation, the set of feasible points is a polyhedral convex set while the cost function is nonlinear;however, the optimum is always attained at a vertex of this polytope. The convex cost function depends only on the initial endowments of the buyers. This formulation yields an easy, simplex-like algorithm which is provably strongly polynomial for many special cases. The algorithm can also be interpreted as a complementary pivot algorithm resembling the classical Lemke-Howson algorithm for computing Nash equilibrium of two-person bimatrix games.
Separated continuous linear programs (SCLP) are a class of continuous linear programs which, among other things, can serve as a useful model for dynamic network problems where storage is permitted at the nodes. Recent...
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Separated continuous linear programs (SCLP) are a class of continuous linear programs which, among other things, can serve as a useful model for dynamic network problems where storage is permitted at the nodes. Recent work on SCLP has produced a detailed duality theory, conditions under which an optimal solution exists with a finite number of breakpoints, a purification algorithm, as well as a convergent algorithm for solving SCLP under certain assumptions on the problem data. This paper combines much of this work to develop a possible approach for solving a wider range of SCLP problems, namely those with fairly general costs. The techniques required to implement the algorithm are no more than standard (finite-dimensional) linear programming and line searching, and the resulting algorithm is simplex-like in nature. We conclude the paper with the numerical results obtained by using a simple implementation of the algorithm to solve a small problem.
This paper discusses a class of continuous linear programs posed in a function space called separated continuous linear programs (SCLP). A dual linear program and a corresponding discrete approximation are introduced ...
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This paper discusses a class of continuous linear programs posed in a function space called separated continuous linear programs (SCLP). A dual linear program and a corresponding discrete approximation are introduced followed by a discussion of their properties. The discrete approximation gives rise to an improvement step which is constructed from any given feasible (nonoptimal) solution for SCLP. A strong duality result follows from this. There are a variety of possible implementations of an algorithm for solving SCLP problems using this improvement step. Finally some computational results are given from one possible implementation.
A simplex-like algorithm is developed for the relaxation labeling process. The algorithm is simple and has a fast convergence property which is summarized as a one-more-step theorem. The algorithm is based on fully ex...
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A simplex-like algorithm is developed for the relaxation labeling process. The algorithm is simple and has a fast convergence property which is summarized as a one-more-step theorem. The algorithm is based on fully exploiting the linearity of the variational inequality and the linear convexity of the consistent-labeling search space in a manner similar to the operation of the simplexalgorithm in linear programming.","doi":"10.1109/34.41370","publicationTitle":"IEEE Transactions on Pattern Analysis and Machine Intelligence","startPage":"1316","endPage":"1321","rightsLink":"http://***/AppDispatchServlet?publisherName=ieee&publication=0162-8828&title=A+simplex-like+algorithm+for+the+relaxation+labeling+process&isbn=&publicationDate=Dec.+1989&author=X.+Zhuang&ContentID=10.1109/34.41370&orderBeanReset=true&startPage=1316&endPage=1321&volumeNum=11&issueNum=12","displayPublicationTitle":"IEEE Transactions on Pattern Analysis and Machine Intelligence","pdfPath":"/iel1/34/1583/***","keywords":[{"type":"IEEE Keywords","kwd":["Labeling","Linearity","Convergence","Linear programming"]},{"type":"INSPEC: Controlled Indexing","kwd":["relaxation theory","convergence of numerical methods","linear programming","pattern recognition"]},{"type":"INSPEC: Non-Controlled Indexing","kwd":["linear programming","simplex-like algorithm","relaxation labeling process","fast convergence property","one-more-step theorem","variational inequality","consistent-labeling search space"]}],"allowComments":false,"pubLink":"/xpl/***?punumber=34","issueLink":"/xpl/***?isnumber=1583","standardTitle":"A simplex-like algorithm for the relaxation labeling process
This paper briefly reviews the literature on necessary optimality conditions for optimal control problems with state-variable inequality constraints. Then, it attempts to unify the treatment of linear optimal control ...
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This paper briefly reviews the literature on necessary optimality conditions for optimal control problems with state-variable inequality constraints. Then, it attempts to unify the treatment of linear optimal control problems with state-variable inequality constraints in the framework of continuous linear programming. The duality theory in this framework makes it possible to relate the adjoint variables arising in different formulations of a problem; these relationships are illustrated by the use of a simple example. This framework also allows more general problems and admits a simplex-like algorithm to solve these problems.
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