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":"21 1","pages":"243-273"},"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}
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":"24 1","pages":"729-743"},"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: 10.1080/10556789908805764
B. Borchers
The CSDP software package consists of a subroutine library for solving semidefinite programming problems, a stand alone solver for solving problems in the SDPA sparse format, some examples showing how to use CSDP, and utility programs for converting between SDPA sparse problem format and the SDPpack problem format. This user's guide describes how to install and use the software.
{"title":"CSDP 2.3 user's guide","authors":"B. Borchers","doi":"10.1080/10556789908805764","DOIUrl":"https://doi.org/10.1080/10556789908805764","url":null,"abstract":"The CSDP software package consists of a subroutine library for solving semidefinite programming problems, a stand alone solver for solving problems in the SDPA sparse format, some examples showing how to use CSDP, and utility programs for converting between SDPA sparse problem format and the SDPpack problem format. This user's guide describes how to install and use the software.","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":"64 1","pages":"597-611"},"PeriodicalIF":2.2,"publicationDate":"1999-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85138622","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 : 1998-04-01DOI: 10.1080/10556789808805720
K. Murota, Mark Scharbrodt
This paper presents an improved algorithm for computing the Combinatorial Canonical Form (CCF) of a layered mixed matrix which consists of a numerical matrix Q and a generic matrix T. The CCF is the (combinatorially unique) finest block-triangular form obtained by the row operations on the Q-part, followed by permutations of rows and columns of the whole matrix. The main ingredient of the improvements is the introduction of two precalculation phases. Computational results are also reported.
{"title":"Computing the Combinatorial Canonical Form of a Layered Mixed Matrix","authors":"K. Murota, Mark Scharbrodt","doi":"10.1080/10556789808805720","DOIUrl":"https://doi.org/10.1080/10556789808805720","url":null,"abstract":"This paper presents an improved algorithm for computing the Combinatorial Canonical Form (CCF) of a layered mixed matrix which consists of a numerical matrix Q and a generic matrix T. The CCF is the (combinatorially unique) finest block-triangular form obtained by the row operations on the Q-part, followed by permutations of rows and columns of the whole matrix. The main ingredient of the improvements is the introduction of two precalculation phases. Computational results are also reported.","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":"13 1","pages":"373-391"},"PeriodicalIF":2.2,"publicationDate":"1998-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79026064","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 : 1998-01-01DOI: 10.1080/10556789808805715
M. Kojima, L. Tunçel
We present primal–dual interior-point algorithms with polynomial iteration bounds to find approximate solutions of semidefinite programming problems. Our algorithms achieve the current best iteration bounds and, in every iteration of our algorithms, primal and dual objective values are strictly improved.
{"title":"Monotonicity of primal–dual interior-point algorithms for semidefinite programming problems","authors":"M. Kojima, L. Tunçel","doi":"10.1080/10556789808805715","DOIUrl":"https://doi.org/10.1080/10556789808805715","url":null,"abstract":"We present primal–dual interior-point algorithms with polynomial iteration bounds to find approximate solutions of semidefinite programming problems. Our algorithms achieve the current best iteration bounds and, in every iteration of our algorithms, primal and dual objective values are strictly improved.","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":"9 1","pages":"275-296"},"PeriodicalIF":2.2,"publicationDate":"1998-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76359632","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 : 1998-01-01DOI: 10.1080/10556789808805707
Ernesto G. Birgina, Yuri G. Evtusenko
Automatic differentiation and nonmonotone spectral projected gradient techniques are used for solving optimal control problems. The original problem is reduced to a nonlinear programming one using general Runge–Kutta integration formulas. Canonical formulas which use a fast automatic differentiation strategy are given to compute derivatives of the objective function. On the basis of this approach, codes for solving optimal control problems are developed and some numerical results are presented.
{"title":"Automatic differentiation and spectral projected gradient methods for optimal control problems","authors":"Ernesto G. Birgina, Yuri G. Evtusenko","doi":"10.1080/10556789808805707","DOIUrl":"https://doi.org/10.1080/10556789808805707","url":null,"abstract":"Automatic differentiation and nonmonotone spectral projected gradient techniques are used for solving optimal control problems. The original problem is reduced to a nonlinear programming one using general Runge–Kutta integration formulas. Canonical formulas which use a fast automatic differentiation strategy are given to compute derivatives of the objective function. On the basis of this approach, codes for solving optimal control problems are developed and some numerical results are presented.","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":"21 1","pages":"125-146"},"PeriodicalIF":2.2,"publicationDate":"1998-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81549696","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 : 1998-01-01DOI: 10.1080/10556789808805685
A. El-Bakry
Primal-dual interior-point methods for linear programming are often motivated by a certaijn nonlinear transformation of the Karush-Kuhn-Tucker conditions of the logarithmic Barrier formulation. Recently, Nassar [5] studied the reciprocal Barrier function formulation of the problem. Using a similar nonlinear transformation, he proved local convergence fir Newton interior-point method on the resulting perturbed Karush-Kuhn-Tucker systerp. This result poses the question whether this method can exhibit fast convergence ral[e for linear programming. In this paper we prove that, for linear programming, Newton's method on the reciprocal Barrier formulation exhibits at best Q-linear convergence rattf. Moreover, an exact Q1 factor is established which precludes fast linear convergence
{"title":"Convergence rate of primal dual reciprocal Barrier Newton interior-point methods","authors":"A. El-Bakry","doi":"10.1080/10556789808805685","DOIUrl":"https://doi.org/10.1080/10556789808805685","url":null,"abstract":"Primal-dual interior-point methods for linear programming are often motivated by a certaijn nonlinear transformation of the Karush-Kuhn-Tucker conditions of the logarithmic Barrier formulation. Recently, Nassar [5] studied the reciprocal Barrier function formulation of the problem. Using a similar nonlinear transformation, he proved local convergence fir Newton interior-point method on the resulting perturbed Karush-Kuhn-Tucker systerp. This result poses the question whether this method can exhibit fast convergence ral[e for linear programming. In this paper we prove that, for linear programming, Newton's method on the reciprocal Barrier formulation exhibits at best Q-linear convergence rattf. Moreover, an exact Q1 factor is established which precludes fast linear convergence","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":"11 1","pages":"37-44"},"PeriodicalIF":2.2,"publicationDate":"1998-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84265661","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 : 1998-01-01DOI: 10.1080/10556789808805678
H. Goldberg, F. Tröltzscht
An optimal control problem governed by the heat equation with nonlinear boundary conditions is considered. The objective functional consists of a quadratic terminal part aifid a quadratic regularization term. On transforming the associated optimality system to! a generalized equation, an SQP method for solving the optimal control problem is related to the Newton method for the generalized equation. In this way, the convergence of tfie SQP method is shown by proving the strong regularity of the optimality system. Aftjer explaining the numerical implementation of the theoretical results some high precision test examples are presented
{"title":"On a lagrange — Newton method for a nonlinear parabolic boundary control problem ∗","authors":"H. Goldberg, F. Tröltzscht","doi":"10.1080/10556789808805678","DOIUrl":"https://doi.org/10.1080/10556789808805678","url":null,"abstract":"An optimal control problem governed by the heat equation with nonlinear boundary conditions is considered. The objective functional consists of a quadratic terminal part aifid a quadratic regularization term. On transforming the associated optimality system to! a generalized equation, an SQP method for solving the optimal control problem is related to the Newton method for the generalized equation. In this way, the convergence of tfie SQP method is shown by proving the strong regularity of the optimality system. Aftjer explaining the numerical implementation of the theoretical results some high precision test examples are presented","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":"198 1","pages":"225-247"},"PeriodicalIF":2.2,"publicationDate":"1998-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83463935","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 : 1998-01-01DOI: 10.1080/10556789808805687
I. Konnov
A general approach to constructing iterative methods that solve variational inequaliti under mild assumptions is proposed. It is based on combining and modifying ide contained in various relaxation methods. The conditions under which the proposed metho attain linear convergence or terminate with a solution are also given
{"title":"On the convergence of combined relaxation methods for variational inequalties","authors":"I. Konnov","doi":"10.1080/10556789808805687","DOIUrl":"https://doi.org/10.1080/10556789808805687","url":null,"abstract":"A general approach to constructing iterative methods that solve variational inequaliti under mild assumptions is proposed. It is based on combining and modifying ide contained in various relaxation methods. The conditions under which the proposed metho attain linear convergence or terminate with a solution are also given","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":"43 1","pages":"77-92"},"PeriodicalIF":2.2,"publicationDate":"1998-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76836604","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 : 1998-01-01DOI: 10.1080/10556789808805712
Z. Gáspár, N. Radics, A. Recski
Bolker and Crapo gave a graph theoretical model of square grid frameworks with diagonal rods of certain squares. Baglivo and Graver solved the problem of tensegrity frameworks where diagonal cables may be used in the square grid to make it rigid. The problem of one-story buildings in both cases can be reduced to the planar problems. These results are generalized if some longer rods, respectively some longer cables are also permitted.
{"title":"Square grids with long “diagonals”","authors":"Z. Gáspár, N. Radics, A. Recski","doi":"10.1080/10556789808805712","DOIUrl":"https://doi.org/10.1080/10556789808805712","url":null,"abstract":"Bolker and Crapo gave a graph theoretical model of square grid frameworks with diagonal rods of certain squares. Baglivo and Graver solved the problem of tensegrity frameworks where diagonal cables may be used in the square grid to make it rigid. The problem of one-story buildings in both cases can be reduced to the planar problems. These results are generalized if some longer rods, respectively some longer cables are also permitted.","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":"7 1","pages":"217-231"},"PeriodicalIF":2.2,"publicationDate":"1998-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81368699","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：