Pub Date : 1999-01-01DOI: 10.1080/10556789908805760
R. Vanderbei
LOQO is a system for solving smooth constrained optimization problems. The problems can be linear or nonlinear, convex or nonconvex, constrained or unconstrained. The only real restriction is that the functions defining the problem be smooth (at the points evaluated by the algorithm). If the problem is convex, LOQO finds a globally optimal solution. Otherwise, it finds a locally optimal solution near to a given starting point. This manual describes 1. how to install LOQO on your hardware. 2. how to use AMPL together with LOQO to solve general optimization problems, 3. how to use the subroutine library to formulate and solve optimization problems, and 4. how to formulate and solve linear and quadratic programs in MPS format.
{"title":"LOQO user's manual — version 3.10","authors":"R. Vanderbei","doi":"10.1080/10556789908805760","DOIUrl":"https://doi.org/10.1080/10556789908805760","url":null,"abstract":"LOQO is a system for solving smooth constrained optimization problems. The problems can be linear or nonlinear, convex or nonconvex, constrained or unconstrained. The only real restriction is that the functions defining the problem be smooth (at the points evaluated by the algorithm). If the problem is convex, LOQO finds a globally optimal solution. Otherwise, it finds a locally optimal solution near to a given starting point. This manual describes 1. how to install LOQO on your hardware. 2. how to use AMPL together with LOQO to solve general optimization problems, 3. how to use the subroutine library to formulate and solve optimization problems, and 4. how to formulate and solve linear and quadratic programs in MPS format.","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"1999-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81566429","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-01-01DOI: 10.1080/10556789908805752
S. Bellavia, M. Macconi
In this paper we present an inexact Interior Point method for solving monotone nonlinear complementarity problems. We show that the theory presented by Kojima, Noma and Yoshise for an exact version of this method can be used to establish global convergence for the inexact form. Then we prove that local superlinear convergence can be achieved under some stronger hypotheses. The complexity of the algorithm is also studied under the assumption that the problem satisfies a scaled Lipschitz condition. It is proved that the feasible version of the algorithm is polynomial, while the infeasible one is globally convergent at a linear rate.
{"title":"An inexact interior point method for monotone NCP","authors":"S. Bellavia, M. Macconi","doi":"10.1080/10556789908805752","DOIUrl":"https://doi.org/10.1080/10556789908805752","url":null,"abstract":"In this paper we present an inexact Interior Point method for solving monotone nonlinear complementarity problems. We show that the theory presented by Kojima, Noma and Yoshise for an exact version of this method can be used to establish global convergence for the inexact form. Then we prove that local superlinear convergence can be achieved under some stronger hypotheses. The complexity of the algorithm is also studied under the assumption that the problem satisfies a scaled Lipschitz condition. It is proved that the feasible version of the algorithm is polynomial, while the infeasible one is globally convergent at a linear rate.","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"1999-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79600876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-01-01DOI: 10.1080/10556789908805741
G. Zanghirati
Large and sparse nonlinear systems arise in many areas of science and technology, very often as a core process for the model of a real world problem. Newton-like approaches to their solution imply the computation of a (possibly approximated) Jacobian: in the case of block bordered systems this results in a matrix with disjoint square blocks on the main diagonal, plus a final set of rows and columns. This sparsity class allows to develop multistage Newton-like methods (with inner and outer iterations) that are very suitable for a parallel implementation ou multiprocessors computers. Recently, Feng and Schnabel proposed an algorithm which is actually the state of the art in this field. In this paper we analyze in depth important theoretical properties of the steps generated by the Feng-Schnabel algorithm. Then we study a cheap modification that gives an improvement of the direction properties, allowing a global convergence result, as well as the extension of the convergence to a broader class of algorithms,...
{"title":"Some theoretical properties of Feng-Schnabel algorithm for block bordered nonlinear systems","authors":"G. Zanghirati","doi":"10.1080/10556789908805741","DOIUrl":"https://doi.org/10.1080/10556789908805741","url":null,"abstract":"Large and sparse nonlinear systems arise in many areas of science and technology, very often as a core process for the model of a real world problem. Newton-like approaches to their solution imply the computation of a (possibly approximated) Jacobian: in the case of block bordered systems this results in a matrix with disjoint square blocks on the main diagonal, plus a final set of rows and columns. This sparsity class allows to develop multistage Newton-like methods (with inner and outer iterations) that are very suitable for a parallel implementation ou multiprocessors computers. Recently, Feng and Schnabel proposed an algorithm which is actually the state of the art in this field. In this paper we analyze in depth important theoretical properties of the steps generated by the Feng-Schnabel algorithm. Then we study a cheap modification that gives an improvement of the direction properties, allowing a global convergence result, as well as the extension of the convergence to a broader class of algorithms,...","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"1999-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86716938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-01-01DOI: 10.1080/10556789908805747
M. Shida
We present a long-step predictor-corrector interior-point algorithm for the monotone semidefinite linear complementarity problems using the Monteiro-Zhang unified search directions. Our algorithm is based on the long-step predictor-corrector interior-point algorithm proposed by Kojima, Shida and Shindoh using the Alizadeh-Haeberly-Overton search direction, though the AHO search direction does not belong to the MZ unified search directions in general.
{"title":"On long-step predictor-corrector interior-point algorithm for semidefinite programming with Monteiro-Zhang unified search directions","authors":"M. Shida","doi":"10.1080/10556789908805747","DOIUrl":"https://doi.org/10.1080/10556789908805747","url":null,"abstract":"We present a long-step predictor-corrector interior-point algorithm for the monotone semidefinite linear complementarity problems using the Monteiro-Zhang unified search directions. Our algorithm is based on the long-step predictor-corrector interior-point algorithm proposed by Kojima, Shida and Shindoh using the Alizadeh-Haeberly-Overton search direction, though the AHO search direction does not belong to the MZ unified search directions in general.","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"1999-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91117050","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-01-01DOI: 10.1080/10556789908805740
A. Rubinov, M. Andramonov
We propose a general scheme of reduction of a Lipschitz programming problem to a problem of minimizing increasing convex-along-rays function. It is based on the positively homogeneous extension of degree p of the objective function and projective transformation of onto the unit simplex. The application of cutting angle method to Lipschitz programming is considered.
{"title":"Lipschitz programming via increasing convex-along-rays functions *","authors":"A. Rubinov, M. Andramonov","doi":"10.1080/10556789908805740","DOIUrl":"https://doi.org/10.1080/10556789908805740","url":null,"abstract":"We propose a general scheme of reduction of a Lipschitz programming problem to a problem of minimizing increasing convex-along-rays function. It is based on the positively homogeneous extension of degree p of the objective function and projective transformation of onto the unit simplex. The application of cutting angle method to Lipschitz programming is considered.","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"1999-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83983986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-01-01DOI: 10.1080/10556789908805730
Luigi Grippof, M. Sciandrone
In this paper we define new classes of globally convergent block-coordinate techniques for the unconstrained minimization of a continuously differentiable function. More specifically, we first describe conceptual models of decomposition algorithms based on the interconnection of elementary operations performed on the block components of the variable vector. Then we characterize the elementary operations defined through a suitable line search or the global minimization in a component subspace. Using these models, we establish new results on the convergence of the nonlinear Gauss–Seidel method and we prove that this method with a two-block decomposition is globally convergent towards stationary points, even in the absence of convexity or uniqueness assumptions. In the general case of nonconvex objective function and arbitrary decomposition we define new globally convergent line-search-based schemes that may also include partial global inimizations with respect to some component. Computational aspects are di...
{"title":"Globally convergent block-coordinate techniques for unconstrained optimization","authors":"Luigi Grippof, M. Sciandrone","doi":"10.1080/10556789908805730","DOIUrl":"https://doi.org/10.1080/10556789908805730","url":null,"abstract":"In this paper we define new classes of globally convergent block-coordinate techniques for the unconstrained minimization of a continuously differentiable function. More specifically, we first describe conceptual models of decomposition algorithms based on the interconnection of elementary operations performed on the block components of the variable vector. Then we characterize the elementary operations defined through a suitable line search or the global minimization in a component subspace. Using these models, we establish new results on the convergence of the nonlinear Gauss–Seidel method and we prove that this method with a two-block decomposition is globally convergent towards stationary points, even in the absence of convexity or uniqueness assumptions. In the general case of nonconvex objective function and arbitrary decomposition we define new globally convergent line-search-based schemes that may also include partial global inimizations with respect to some component. Computational aspects are di...","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"1999-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85056923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-01-01DOI: 10.1080/10556789908805751
J. Bonnans, C. Pola, Raja Rébaï
The path following algorithms of predictor corrector type have proved to be very effective for solving linear optimization problems. However, the assumption that the Newton direction (corresponding to a centering or affine step) is computed exactly is unrealistic. Indeed, for large scale problems, one may need to use iterative algorithms for computing the Newton step. In this paper, we study algorithms in which the computed direction is the solution of the usual linear system with an error in the right-hand-side. We give precise and explicit estimates of the error under which the computational complexity is the same as for the standard case. We also give explicit estimates that guarantee an asymptotic linear convergence at an arbitrary rate. Finally, we present some encouraging numerical results. Because our results are in the framework of monotone linear complementarity problems, our results apply to convex quadratic optimization as well.
{"title":"Perturbed path following predictor-corrector interior point algorithms","authors":"J. Bonnans, C. Pola, Raja Rébaï","doi":"10.1080/10556789908805751","DOIUrl":"https://doi.org/10.1080/10556789908805751","url":null,"abstract":"The path following algorithms of predictor corrector type have proved to be very effective for solving linear optimization problems. However, the assumption that the Newton direction (corresponding to a centering or affine step) is computed exactly is unrealistic. Indeed, for large scale problems, one may need to use iterative algorithms for computing the Newton step. In this paper, we study algorithms in which the computed direction is the solution of the usual linear system with an error in the right-hand-side. We give precise and explicit estimates of the error under which the computational complexity is the same as for the standard case. We also give explicit estimates that guarantee an asymptotic linear convergence at an arbitrary rate. Finally, we present some encouraging numerical results. Because our results are in the framework of monotone linear complementarity problems, our results apply to convex quadratic optimization as well.","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"1999-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75077635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-01-01DOI: 10.1080/10556789908805731
G. Pönisch, U. Schnabel, H. Schwetlick
A point (x *,λ*) is called a turning point of multiplicity p ≥1 of the nonlinear system if and if the Ljapunov–Schmidt reduced function has the normal form . A minimally extended system F(x, λ)=0 F(x, λ)=0 is proposed for defining turning points of multiplicity p, where is a scalar function which is related to the pth order partial derivatives of g with respect to ξ. When Fdepends on m≤p-1 additional parameters the system F(x, λ α)=0 can be inflated by m + 1 scalar equations f 1(x, λ α)=0,…f m+1(x, λ α)=0 The functions depend on certain partial derivatives of gwith respect to ξ where f m+1 corresponds to f The regular solution (x *, λ *,α *) of the extended system of n+m+1 equations delivers the desired turning point (x *, λ*). For numerically solving these systems, two-stage New tonype methods are proposed, where only one LU decomposition of an (n+1) ×(++1) matrix and some back substitutions have to be preformed per iteration step if Gaussian elimination is used for solving the linear systems. Moreover, ...
{"title":"Computing multiple turning points by using simple extended systems and computational differentiation","authors":"G. Pönisch, U. Schnabel, H. Schwetlick","doi":"10.1080/10556789908805731","DOIUrl":"https://doi.org/10.1080/10556789908805731","url":null,"abstract":"A point (x *,λ*) is called a turning point of multiplicity p ≥1 of the nonlinear system if and if the Ljapunov–Schmidt reduced function has the normal form . A minimally extended system F(x, λ)=0 F(x, λ)=0 is proposed for defining turning points of multiplicity p, where is a scalar function which is related to the pth order partial derivatives of g with respect to ξ. When Fdepends on m≤p-1 additional parameters the system F(x, λ α)=0 can be inflated by m + 1 scalar equations f 1(x, λ α)=0,…f m+1(x, λ α)=0 The functions depend on certain partial derivatives of gwith respect to ξ where f m+1 corresponds to f The regular solution (x *, λ *,α *) of the extended system of n+m+1 equations delivers the desired turning point (x *, λ*). For numerically solving these systems, two-stage New tonype methods are proposed, where only one LU decomposition of an (n+1) ×(++1) matrix and some back substitutions have to be preformed per iteration step if Gaussian elimination is used for solving the linear systems. Moreover, ...","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"1999-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80842798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-01-01DOI: 10.1080/10556789908805759
R. Vanderbei
This paper describes a software package, called LOQO, which implements a primal-dual interior-point method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex and general nonlinear programming, since a detailed paper describing these extensions was published recently elsewhere. In particular, we emphasize the importance of establishing and maintaining symmetric quasidefiniteness of the reduced KKT system. We show that the industry standard MPS format can be nicely formulated in such a way to provide quasidefiniteness. Computational results are included for a variety of linear and quadratic programming problems.
{"title":"LOQO:an interior point code for quadratic programming","authors":"R. Vanderbei","doi":"10.1080/10556789908805759","DOIUrl":"https://doi.org/10.1080/10556789908805759","url":null,"abstract":"This paper describes a software package, called LOQO, which implements a primal-dual interior-point method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex and general nonlinear programming, since a detailed paper describing these extensions was published recently elsewhere. In particular, we emphasize the importance of establishing and maintaining symmetric quasidefiniteness of the reduced KKT system. We show that the industry standard MPS format can be nicely formulated in such a way to provide quasidefiniteness. Computational results are included for a variety of linear and quadratic programming problems.","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"1999-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84946699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1999-01-01DOI: 10.1080/10556789908805761
S. Benson, Yinyu Yeb, Xiong Zhang
We use the semidefinite relaxation to approximate combinatorial and quadratic optimization problems subject to linear, quadratic, as well as boolean constraints. We present a dual potential reduction algorithm and show how to exploit the sparse structure of various problems. Coupled with randomized and heuristic methods, we report computational results for approximating graph-partition and quadratic problems with dimensions 800 to 10,000. This finding, to the best of our knowledge, is the first computational evidence of the effectiveness of these approximation algorithms for solving large-scale problems.
{"title":"Mixed linear and semidefinite programming for combinatorial and quadratic optimization","authors":"S. Benson, Yinyu Yeb, Xiong Zhang","doi":"10.1080/10556789908805761","DOIUrl":"https://doi.org/10.1080/10556789908805761","url":null,"abstract":"We use the semidefinite relaxation to approximate combinatorial and quadratic optimization problems subject to linear, quadratic, as well as boolean constraints. We present a dual potential reduction algorithm and show how to exploit the sparse structure of various problems. Coupled with randomized and heuristic methods, we report computational results for approximating graph-partition and quadratic problems with dimensions 800 to 10,000. This finding, to the best of our knowledge, is the first computational evidence of the effectiveness of these approximation algorithms for solving large-scale problems.","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"1999-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75844956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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