Pub Date : 1999-01-01DOI: 10.1080/10556789908805739
Aniekan Ebiefung
We show that the vertical linear complementarity problem can be solved, under certain conditions, by solving a perturbed problem when the associated matrix is a vertical block P o-matrix. The main conditions do not depend on nondegeneracy of the problem or on the matrix class to which the P o matrix may also belong. In the special case of the K o-matrices, the least element solution of the perturbed problem converges to the least element solution of the original problem.
{"title":"The vertical linear complementarity problem associated with P o-matrices","authors":"Aniekan Ebiefung","doi":"10.1080/10556789908805739","DOIUrl":"https://doi.org/10.1080/10556789908805739","url":null,"abstract":"We show that the vertical linear complementarity problem can be solved, under certain conditions, by solving a perturbed problem when the associated matrix is a vertical block P o-matrix. The main conditions do not depend on nondegeneracy of the problem or on the matrix class to which the P o matrix may also belong. In the special case of the K o-matrices, the least element solution of the perturbed problem converges to the least element solution of the original problem.","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":"81924876","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: 10.1080/10556789908805748
J. Haeberly, M. V. Nayakkankuppam, M. Overton
We discuss extensions of Mehrotra's higher order corrections scheme and Gondzio's multiple centrality corrections scheme to mixed semidefinite-quadratic-linear programming (SQLP). These extensions have been included in a solver for SQLP written in C and based on LAPACK. The code implements a primal-dual path-following algorithm for solving SQLP problems based on the XZ + ZX search direction and Mehrotra's predictor-corrector method. We present benchmarks showing that the use of the higher order schemes yields substantial reductions in both the number of iterations and the running time of the algorithm, and also improves its robustness.
{"title":"Extending Mehrotra and Gondzio higher order methods to mixed semidefinite-quadratic-linear programming","authors":"J. Haeberly, M. V. Nayakkankuppam, M. Overton","doi":"10.1080/10556789908805748","DOIUrl":"https://doi.org/10.1080/10556789908805748","url":null,"abstract":"We discuss extensions of Mehrotra's higher order corrections scheme and Gondzio's multiple centrality corrections scheme to mixed semidefinite-quadratic-linear programming (SQLP). These extensions have been included in a solver for SQLP written in C and based on LAPACK. The code implements a primal-dual path-following algorithm for solving SQLP problems based on the XZ + ZX search direction and Mehrotra's predictor-corrector method. We present benchmarks showing that the use of the higher order schemes yields substantial reductions in both the number of iterations and the running time of the algorithm, and also improves its robustness.","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":"84222341","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: 10.1080/10556789908805756
Yin Zhang
LIPSOL stands for Linear programming Interior-Point SOLvers. It is a free, Matlab-based software package for solving linear programs by interior-Point methods. It requires Matlab version 4.0 or later to run. The current release of LIPSOL is for 32-bit UNIX platforms. LIPSOL is designed to solve relatively large problems. It utilizes Matlab’s sparse-matrix data-structure and Application Program Interface facility, and at the same time takes advantages of existing, efficient Fortran codes for solving large, sparse, symmetric positive definite linear systems. Specifically, LIPSOL constructs MEX-files from two Fortran packages: a sparse Cholesky factorization package developed by Esmond Ng and Barry Peyton at ORNL and a multiple minimum-degree ordering package by Joseph Liu at University of Waterloo . Built in the high-level programming environment of Matlab, LIPSOL enjoys a much greater degree of simplicity and versatility than codes in Fortran or C language. On the other hand, utilizing efficient Fortran codes for computationally intensive tasks, LIPSOL also has adequate speed for solving moderately large-scale problems even in the presence of overhead
{"title":"User'S guide To Lipsol linear-programming interior point solvers V0.4","authors":"Yin Zhang","doi":"10.1080/10556789908805756","DOIUrl":"https://doi.org/10.1080/10556789908805756","url":null,"abstract":"LIPSOL stands for Linear programming Interior-Point SOLvers. It is a free, Matlab-based software package for solving linear programs by interior-Point methods. It requires Matlab version 4.0 or later to run. The current release of LIPSOL is for 32-bit UNIX platforms. LIPSOL is designed to solve relatively large problems. It utilizes Matlab’s sparse-matrix data-structure and Application Program Interface facility, and at the same time takes advantages of existing, efficient Fortran codes for solving large, sparse, symmetric positive definite linear systems. Specifically, LIPSOL constructs MEX-files from two Fortran packages: a sparse Cholesky factorization package developed by Esmond Ng and Barry Peyton at ORNL and a multiple minimum-degree ordering package by Joseph Liu at University of Waterloo . Built in the high-level programming environment of Matlab, LIPSOL enjoys a much greater degree of simplicity and versatility than codes in Fortran or C language. On the other hand, utilizing efficient Fortran codes for computationally intensive tasks, LIPSOL also has adequate speed for solving moderately large-scale problems even in the presence of overhead","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":"79054252","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: 10.1080/10556789908805742
M. Kiehl
The solution y(t,t 0,y 0) of an initial-value problem (IVP)[ydot](t)=f(t,y,p) with initial value y(t 0)=y 0 at a point t is a differentiable function of the initial value y 0 and the parameter vector p, provided f y and f p are continuous. The computation of y(t,t 0,y 0) and the sensitivity matrix play an important role in the efficient numerical solution of boundary-value problems, optimal-control problems and for parameter identification in dynamical systems. There is a wide variety of algorithms for the solution of IVPs but till now, there are just a few efficient implementations for the computation of , which is the even more important part. Here a number of implementations are introduced, treating non-stiff, stiff and differential algebraic equations. The basic new idea is to regard the numerical approximation of as the solution of the variational differential equation of a linearised IVP with approximation of the linear right-hand side by the difference quotient of the original non-linear f. For fur...
{"title":"Sensitivity analysis of ODEs and DAEs — theory and implementation guide","authors":"M. Kiehl","doi":"10.1080/10556789908805742","DOIUrl":"https://doi.org/10.1080/10556789908805742","url":null,"abstract":"The solution y(t,t 0,y 0) of an initial-value problem (IVP)[ydot](t)=f(t,y,p) with initial value y(t 0)=y 0 at a point t is a differentiable function of the initial value y 0 and the parameter vector p, provided f y and f p are continuous. The computation of y(t,t 0,y 0) and the sensitivity matrix play an important role in the efficient numerical solution of boundary-value problems, optimal-control problems and for parameter identification in dynamical systems. There is a wide variety of algorithms for the solution of IVPs but till now, there are just a few efficient implementations for the computation of , which is the even more important part. Here a number of implementations are introduced, treating non-stiff, stiff and differential algebraic equations. The basic new idea is to regard the numerical approximation of as the solution of the variational differential equation of a linearised IVP with approximation of the linear right-hand side by the difference quotient of the original non-linear f. For fur...","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":"89618027","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: 10.1080/10556789908805758
C. Mészáros
The paper describes the convex quadratic solver BPMPD Version 2.21. The solver is based on the infeasible–primal–dual algorithm extended by the predictor–corrector and target–following techniques. The discussion includes topics related to the implemented algorithm and numerical algebra employed. We outline the presolve, scaling and starting point stategies used in BPMPD, and special attention is given for sparsity and stability issues. Computational results are given on a demonstrative set of convex quadratic problems.
本文描述了凸二次解算器BPMPD Version 2.21。该求解器基于基于预测校正和目标跟踪技术扩展的不可行原对偶算法。讨论包括与实现算法和所采用的数值代数相关的主题。我们概述了BPMPD中使用的解析、缩放和起点策略，并特别关注了稀疏性和稳定性问题。给出了凸二次问题的一个证明集的计算结果。
{"title":"The BPMPD interior point solver for convex quadratic problems","authors":"C. Mészáros","doi":"10.1080/10556789908805758","DOIUrl":"https://doi.org/10.1080/10556789908805758","url":null,"abstract":"The paper describes the convex quadratic solver BPMPD Version 2.21. The solver is based on the infeasible–primal–dual algorithm extended by the predictor–corrector and target–following techniques. The discussion includes topics related to the implemented algorithm and numerical algebra employed. We outline the presolve, scaling and starting point stategies used in BPMPD, and special attention is given for sparsity and stability issues. Computational results are given on a demonstrative set of convex quadratic 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":"72681176","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: 10.1080/10556789908805763
Nathan W. Brixius, F. Potra, Rongqin Sheng
Mehrotra type primal-dual predictor-corrector interior-point algorithms for semidefinite programming are implemented, using the homogeneous formulation proposed and analyzed by Potra and Sheng. Several search directions, including the AHO, HKM, NT, Toh, and Gu directions, are used. A rank-2 update technique is employed in our MATLAB code so that the computation of homogeneous directions is only slightly more expensive than in the non-homogeneous case. However, the homogeneous algorithms generally take fewer iterations to compute an approximate solution within a desired accuracy. Numerical results show that the homogeneous algorithms outperform their non-homogeneous counterparts, with improvement of more than 20% in many cases, in terms of total CPU time.
{"title":"Sdpha: a Matlab implementation of homogeneous interior-point algorithms for semidefinite programming","authors":"Nathan W. Brixius, F. Potra, Rongqin Sheng","doi":"10.1080/10556789908805763","DOIUrl":"https://doi.org/10.1080/10556789908805763","url":null,"abstract":"Mehrotra type primal-dual predictor-corrector interior-point algorithms for semidefinite programming are implemented, using the homogeneous formulation proposed and analyzed by Potra and Sheng. Several search directions, including the AHO, HKM, NT, Toh, and Gu directions, are used. A rank-2 update technique is employed in our MATLAB code so that the computation of homogeneous directions is only slightly more expensive than in the non-homogeneous case. However, the homogeneous algorithms generally take fewer iterations to compute an approximate solution within a desired accuracy. Numerical results show that the homogeneous algorithms outperform their non-homogeneous counterparts, with improvement of more than 20% in many cases, in terms of total CPU time.","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":"77469119","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: 10.1080/10556789908805765
B. Borchers
This paper describes CSDP, a library of routines that implements a predictor corrector variant of the semidefinite programming algorithm of Helmberg, Rendl, Vanderbei, and Wolkowicz. The main advantages of this code are that it can be used as a stand alone solver or as a callable subroutine, that it is written in C for efficiency, that it makes effective use of sparsity in the constraint matrices, and that it includes support for linear inequality constraints in addition to linear equality constraints. We discuss the algorithm used, its computational complexity, and storage requirements. Finally, we present benchmark results for a collection of test problems.
{"title":"CSDP, A C library for semidefinite programming","authors":"B. Borchers","doi":"10.1080/10556789908805765","DOIUrl":"https://doi.org/10.1080/10556789908805765","url":null,"abstract":"This paper describes CSDP, a library of routines that implements a predictor corrector variant of the semidefinite programming algorithm of Helmberg, Rendl, Vanderbei, and Wolkowicz. The main advantages of this code are that it can be used as a stand alone solver or as a callable subroutine, that it is written in C for efficiency, that it makes effective use of sparsity in the constraint matrices, and that it includes support for linear inequality constraints in addition to linear equality constraints. We discuss the algorithm used, its computational complexity, and storage requirements. Finally, we present benchmark results for a collection of test 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":"74440368","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: 10.1080/10556789908805768
I. Maros, C. Mészáros
The introduction of a standard set of linear programming problems, to be found in NETLIB/-LP/DATA, had an important impact on measuring, comparing and reporting the performance of LP solvers. Until recently the efficiency of new algorithmic developments has been measured using this important reference set. Presently, we are witnessing an ever growing interest in the area of quadratic programming. The research community is somewhat troubled by the lack of a standard format for defining a QP problem and also by the lack of a standard reference set of problems for purposes similar to that of LP. In the paper we propose a standard format and announce the availability of a test set of collected 138 QP problems.
{"title":"A repository of convex quadratic programming problems","authors":"I. Maros, C. Mészáros","doi":"10.1080/10556789908805768","DOIUrl":"https://doi.org/10.1080/10556789908805768","url":null,"abstract":"The introduction of a standard set of linear programming problems, to be found in NETLIB/-LP/DATA, had an important impact on measuring, comparing and reporting the performance of LP solvers. Until recently the efficiency of new algorithmic developments has been measured using this important reference set. Presently, we are witnessing an ever growing interest in the area of quadratic programming. The research community is somewhat troubled by the lack of a standard format for defining a QP problem and also by the lack of a standard reference set of problems for purposes similar to that of LP. In the paper we propose a standard format and announce the availability of a test set of collected 138 QP 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":"76222935","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: 10.1080/10556789908805736
B. Christianson
In this paper we discuss Pantoja's construction of the Newton direction for discrete time optimal control problems. We show that automatic differentiation (AD) techniques can be used to calculate the Newton direction accurately, without requiring extensive re-writing of user code, and at a surprisingly low computational cost: for an N-step problem with p control variables and q state variables at each step, the worst case cost is 6(p + q + 1) times the computational cost of a single target function evaluation, independent of N, together with at most p 3/3 + p 2(q + 1) + 2p(q + 1)2 + (q + l)3, i.e. less than (p + q + l)3, floating point multiply-and-add operations per time step. These costs may be considerably reduced if there is significant structural sparsity in the problem dynamics. The systematic use of checkpointing roughly doubles the operation counts, but reduces the total space cost to the order of 4pN floating point stores. A naive approach to finding the Newton step would require the solution of ...
{"title":"Cheap Newton steps for optimal control problems: automatic differentiation and Pantoja's algorithm","authors":"B. Christianson","doi":"10.1080/10556789908805736","DOIUrl":"https://doi.org/10.1080/10556789908805736","url":null,"abstract":"In this paper we discuss Pantoja's construction of the Newton direction for discrete time optimal control problems. We show that automatic differentiation (AD) techniques can be used to calculate the Newton direction accurately, without requiring extensive re-writing of user code, and at a surprisingly low computational cost: for an N-step problem with p control variables and q state variables at each step, the worst case cost is 6(p + q + 1) times the computational cost of a single target function evaluation, independent of N, together with at most p 3/3 + p 2(q + 1) + 2p(q + 1)2 + (q + l)3, i.e. less than (p + q + l)3, floating point multiply-and-add operations per time step. These costs may be considerably reduced if there is significant structural sparsity in the problem dynamics. The systematic use of checkpointing roughly doubles the operation counts, but reduces the total space cost to the order of 4pN floating point stores. A naive approach to finding the Newton step would require the solution of ...","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":"81428988","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: 10.1080/10556789908805753
Y. Nesterov, Olivier Péton, J. Vial
In this paper we consider a homogeneous analytic center cutting plane method in a projective space. We describe a general scheme that uses a homogeneous oracle and computes an approximate analytic center at each iteration. This technique is applied to a convex feasibility problem, to variational inequalities, and to convex constrained minimization. We prove that these problems can be solved with the same order of complexity as in the case of exact analytic centers. For the feasibility and the minimization problems rough approximations suffice, but very high precision is required for the variational inequalities. We give an example of variational inequality where even the first analytic center needs to be computed with a precision matching the precision required for the solution.
{"title":"Homogeneous Analytic Center Cutting Plane Methods with Approximate Centers","authors":"Y. Nesterov, Olivier Péton, J. Vial","doi":"10.1080/10556789908805753","DOIUrl":"https://doi.org/10.1080/10556789908805753","url":null,"abstract":"In this paper we consider a homogeneous analytic center cutting plane method in a projective space. We describe a general scheme that uses a homogeneous oracle and computes an approximate analytic center at each iteration. This technique is applied to a convex feasibility problem, to variational inequalities, and to convex constrained minimization. We prove that these problems can be solved with the same order of complexity as in the case of exact analytic centers. For the feasibility and the minimization problems rough approximations suffice, but very high precision is required for the variational inequalities. We give an example of variational inequality where even the first analytic center needs to be computed with a precision matching the precision required for the solution.","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":"81796223","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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