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On self-concordant convex–concave functions 关于自调和凸凹函数
IF 2.2 3区 数学
Optimization Methods & Software Pub Date : 1999-01-01 DOI: 10.1080/10556789908805755
A. Nemirovski
{"title":"On self-concordant convex–concave functions","authors":"A. Nemirovski","doi":"10.1080/10556789908805755","DOIUrl":"https://doi.org/10.1080/10556789908805755","url":null,"abstract":"In this paper, we introduce the notion of a self-concordant convex-concave function, establish basic properties of these functions and develop a path-following interior point method for approximating saddle points of “sufficiently well-behaved” convex-concave functions—those which admit natural self-concordant convex-concave regularizations. The approach is illustrated by its applications to developing an exterior penalty polynomial time method for Semidefinite Programming and to the problem of inscribing the largest volume ellipsoid into a given polytope.","PeriodicalId":54673,"journal":{"name":"Optimization Methods & Software","volume":"22 1","pages":"303-384"},"PeriodicalIF":2.2,"publicationDate":"1999-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76998946","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}
引用次数: 17
LOQO user's manual — version 3.10 LOQO用户手册-版本3.10
IF 2.2 3区 数学
Optimization Methods & Software Pub Date : 1999-01-01 DOI: 10.1080/10556789908805760
R. Vanderbei
{"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":"18 1","pages":"485-514"},"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}
引用次数: 130
Perturbed path following predictor-corrector interior point algorithms 扰动路径跟踪预测校正内点算法
IF 2.2 3区 数学
Optimization Methods & Software Pub Date : 1999-01-01 DOI: 10.1080/10556789908805751
J. Bonnans, C. Pola, Raja Rébaï
{"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":"38 1","pages":"183-210"},"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}
引用次数: 5
LOQO:an interior point code for quadratic programming LOQO:二次规划的内部点代码
IF 2.2 3区 数学
Optimization Methods & Software Pub Date : 1999-01-01 DOI: 10.1080/10556789908805759
R. Vanderbei
{"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":"12 1","pages":"451-484"},"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}
引用次数: 568
Globally convergent block-coordinate techniques for unconstrained optimization 无约束优化的全局收敛块坐标技术
IF 2.2 3区 数学
Optimization Methods & Software Pub Date : 1999-01-01 DOI: 10.1080/10556789908805730
Luigi Grippof, M. Sciandrone
{"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":"86 1","pages":"587-637"},"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}
引用次数: 142
Lipschitz programming via increasing convex-along-rays functions * 通过增加沿凸函数的Lipschitz规划*
IF 2.2 3区 数学
Optimization Methods & Software Pub Date : 1999-01-01 DOI: 10.1080/10556789908805740
A. Rubinov, M. Andramonov
{"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":"61 1","pages":"763-781"},"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}
引用次数: 35
Some theoretical properties of Feng-Schnabel algorithm for block bordered nonlinear systems 块边界非线性系统的Feng-Schnabel算法的一些理论性质
IF 2.2 3区 数学
Optimization Methods & Software Pub Date : 1999-01-01 DOI: 10.1080/10556789908805741
G. Zanghirati
{"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":"45 1","pages":"783-801"},"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}
引用次数: 1
On long-step predictor-corrector interior-point algorithm for semidefinite programming with Monteiro-Zhang unified search directions 具有Monteiro-Zhang统一搜索方向的半定规划长步预测校正内点算法
IF 2.2 3区 数学
Optimization Methods & Software Pub Date : 1999-01-01 DOI: 10.1080/10556789908805747
M. Shida
{"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":"58 1","pages":"53-66"},"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}
引用次数: 0
Computing multiple turning points by using simple extended systems and computational differentiation 利用简单扩展系统和计算微分计算多个拐点
IF 2.2 3区 数学
Optimization Methods & Software Pub Date : 1999-01-01 DOI: 10.1080/10556789908805731
G. Pönisch, U. Schnabel, H. Schwetlick
{"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":"52 1","pages":"639-668"},"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}
引用次数: 0
The vertical linear complementarity problem associated with P o-matrices 与P - o矩阵相关的垂直线性互补问题
IF 2.2 3区 数学
Optimization Methods & Software Pub Date : 1999-01-01 DOI: 10.1080/10556789908805739
Aniekan Ebiefung
{"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":"4 1","pages":"747-761"},"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}
引用次数: 3
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