interiormethods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interior-point techniques were popular duri...
详细信息
interiormethods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interior-point techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their use for linearprogramming was not even contemplated because of the total dominance of the simplex method. Vague but continuing anxiety about barrier methods eventually led to their abandonment in favor of newly emerging, apparently more efficient alternatives such as augmented Lagrangian and sequential quadratic programmingmethods. By the early 1980s, barrier methods were almost without exception regarded as a closed chapter in the history of optimization. This picture changed dramatically with Karmarkar's widely publicized announcement in 1984 of a fast polynomial-time interior method for linearprogramming;in 1985, a formal connection was established between his method and classical barrier methods. Since then, interiormethods have advanced so far, so fast, that their influence has transformed both the theory and practice of constrained optimization. This article provides a condensed, selective look at classical material and recent research about interiormethods for nonlinearly constrained optimization.
The commutative class of search directions for semidefinite programming was first proposed by Monteiro and Zhang (Ref. 1). In this paper, we investigate the corresponding class of search directions for linear programm...
详细信息
The commutative class of search directions for semidefinite programming was first proposed by Monteiro and Zhang (Ref. 1). In this paper, we investigate the corresponding class of search directions for linearprogramming over symmetric cones, which is a class of convex optimization problems including linearprogramming, second-order cone programming, and semidefinite programming as special cases. Complexity results are established for short-step, semilong-step, and long-step algorithms. Then, we propose a subclass of the commutative class for which we can prove polynomial complexities of the interior-point method using semilong steps and long steps. This subclass still contains the Nesterov-Todd direction and the Helmberg-Rendl-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro direction. An explicit formula to calculate any member of the class is also given.
This paper presents a differential-algebraic approach for solving linearprogramming problems. The paper shows that the differential-algebraic approach is guaranteed to generate optimal solutions to linearprogramming...
详细信息
This paper presents a differential-algebraic approach for solving linearprogramming problems. The paper shows that the differential-algebraic approach is guaranteed to generate optimal solutions to linearprogramming problems with a superexponential convergence rate. The paper also shows that the path-following interior-pointmethods for solving linearprogramming problems can be viewed as a special case of the differential-algebraic approach. The results in this paper demonstrate that the proposed approach provides a promising alternative for solving linearprogramming problems.
Self-scaled barrier functions are fundamental objects in the theory of interior-pointmethods for linear optimization over symmetric cones, of which linear and semidefinite programming are special cases. In this artic...
详细信息
Self-scaled barrier functions are fundamental objects in the theory of interior-pointmethods for linear optimization over symmetric cones, of which linear and semidefinite programming are special cases. In this article we classify the special class of self-scaled barriers which are defined on irreducible symmetric cones. Together with a decomposition theorem for general self-scaled barriers this concludes the algebraic classification theory of these functions.
Due to its many applications in control theory, robust optimization, combinatorial optimization and eigenvalue optimization, semidefinite programming had been in widespread use even before the development of efficient...
详细信息
Due to its many applications in control theory, robust optimization, combinatorial optimization and eigenvalue optimization, semidefinite programming had been in widespread use even before the development of efficient algorithms brought it into the realm of tractability, Today it is one of the basic modeling and optimization tools along with linear and quadratic programming. Our survey is an introduction to semidefinite programming, its duality and complexity theory, its applications and algorithms. (C) 2002 Elsevier Science B.V. All rights reserved.
linear optimization (LO) is the fundamental problem of mathematical optimization. It admits an enormous number of applications in economics, engineering, science and many other fields. The three most significant class...
详细信息
linear optimization (LO) is the fundamental problem of mathematical optimization. It admits an enormous number of applications in economics, engineering, science and many other fields. The three most significant classes of algorithms for solving LO problems are: pivot, ellipsoid and interiorpointmethods. Because ellipsoid methods are not efficient in practice we will concentrate on the computationally successful simplex and primal-dual interiorpointmethods only. and summarize the pros and cons of these algorithm classes. (C) 2002 Elsevier Science B.V. All rights reserved.
A linear program has a unique least 2-norm solution, provided that the linear program has a solution. To locate this solution, most of the existing methods were devised to solve certain equivalent perturbed quadratic ...
详细信息
A linear program has a unique least 2-norm solution, provided that the linear program has a solution. To locate this solution, most of the existing methods were devised to solve certain equivalent perturbed quadratic programs or unconstrained minimization problems. We provide in this paper a new theory which is different from these traditional methods and is an effective numerical method for seeking the least 2-norm solution of a linear program. The essence of this method is a (interior-point-like) path-following algorithm that traces a newly introduced regularized central path that is fairly different from the central path used in interior-pointmethods. One distinguishing feature of our method is that it imposes no assumption on the problem. The iterates generated by this algorithm converge to the least 2-norm solution whenever the linear program is solvable;otherwise, the iterates converge to a point which gives a minimal KKT residual when the linear program is unsolvable.
linear optimization (LO) is the fundamental problem of mathematical optimization. It admits an enormous number of applications in economics, engineering, science and many other fields. The three most significant class...
详细信息
linear optimization (LO) is the fundamental problem of mathematical optimization. It admits an enormous number of applications in economics, engineering, science and many other fields. The three most significant classes of algorithms for solving LO problems are: pivot, ellipsoid and interiorpointmethods. Because ellipsoid methods are not efficient in practice we will concentrate on the computationally successful simplex and primal-dual interiorpointmethods only. and summarize the pros and cons of these algorithm classes. (C) 2002 Elsevier Science B.V. All rights reserved.
Semidefinite relaxations of quadratic 0-1 programming or graph partitioning problems are well known to be of high quality. However, solving them by primal-dual interiorpointmethods can take much time even for proble...
详细信息
Semidefinite relaxations of quadratic 0-1 programming or graph partitioning problems are well known to be of high quality. However, solving them by primal-dual interiorpointmethods can take much time even for problems of moderate size. The recent spectral bundle method of Helmberg and Rendl can solve quite efficiently large structured equality-constrained semidefinite programs if the trace of the primal matrix variable is fixed, as happens in many applications. We extend the method so that it can handle inequality constraints without seriously increasing computation time. In addition, we introduce inexact null steps. This abolishes the need of computing exact eigenvectors for subgradients, which brings along significant advantages in theory and in practice. Encouraging preliminary computational results are reported.
A finite element formulation of the limit analysis of perfectly plastic slabs is given. An element with linear moment fields for which equilibrium is satisfied exactly is used in connection with an optimization algori...
详细信息
A finite element formulation of the limit analysis of perfectly plastic slabs is given. An element with linear moment fields for which equilibrium is satisfied exactly is used in connection with an optimization algorithm taking into account the full nonlinearity of the yield criteria. Both load and material optimization problems are formulated and by means of the duality theory of linearprogramming the displacements are extracted from the dual variables. Numerical examples demonstrating the capabilities of the method and the effects of using a more refined representation of the yield criteria are given. (C) 2002 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved.
暂无评论